The Volume of Tubes in Homogeneous Spaces Citation Howard, Ralph Elwood (1982) The Volume of Tubes in Homogeneous Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/r71m-yj10. https://resolver.caltech.edu/CaltechTHESIS:05152018-120303815 Abstract Let M̃ (dim(M̃ ) = m + n) be an oriented Riemannian manifold and M a compact oriented submanifold of M̃ . The tube M(r) of radius r about M is the set of points p that can be joined to M by a geodesic of length r meeting M perpendicularly. We give a formula for the volume of M(r) in the case M̃ is a naturally reductive Riemannian homogeneous space (this includes all Riemannian symmetric spaces) and M is such that for each point p of M there is a totally geodesic submanifold of M̃ of dimension complementary to M through p and perpendicular to M at p. To be more specific, [Equation included in scanned thesis' abstract, p. iii.] Here h j is a function of the point p ε M and the real number r. Also h j (p,r) is a homogeneous polynomial of degree j in the components of the second fundamental form of M in M̃. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Conn, Jack Frederick Thesis Committee: Conn, Jack Frederick (chair) Aschbacher, Michael De Prima, Charles R. Fuller, F. Brock Defense Date: 7 May 1982 Funders: Funding Agency Grant Number Caltech UNSPECIFIED Record Number: CaltechTHESIS:05152018-120303815 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05152018-120303815 DOI: 10.7907/r71m-yj10 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10890 Collection: CaltechTHESIS Deposited By: Mel Ray Deposited On: 25 Jun 2018 21:30 Last Modified: 29 Jul 2025 22:42 Thesis Files Preview PDF - Final Version See Usage Policy. 44MB Repository Staff Only: item control page

The Volume of Tubes in Homogeneous Spaces Thesis by Ralph Elwood Howard In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1982 (Submitted May 7, 1982) ;; ACKNOWLEDGEMENTS I would like to thank my advisor Jack Conn for his encouragement of my research over the last several years. He was also a great help in the preparation of this thesis. I would also like to thank Lillian Chappelle for typing this thesis on short notice and for her patience with my poor job of proof reading. Finally, the California Institute of Technology has been very generous in its support during my stay as a graduate student. greatly appreciated this support. I have iii ABSTRACT Let M(dim(M) =· m + n) be an oriented Riemannian manifold and M a compact oriented submanifold of · M. r about M is the set of points geodesic of length The tube M(r) of radius p that can be joined to M by a r meeting M peroendicularly. We give a formula for the volume of M(r) in the case M is a naturally reductive Riemannian homoge~eous space (this includes all Riemannian symmetric spaces) and M is such that for each point p of M there is a totally geodesic submanifold of through M of di.mension complementary to M p and perpendicular to M at p. To be more specific, n VO l(M ( r) ) Here h. J = .6 J hJ. (. p 'r) QM ( p) . J=O M is a function of the point p e M and the real number r. Also hj(p,r) is a homogeneous polynomial of degree j components of the second fundamental form of M in M. in the iv CONTENTS Acknowledgements ii ....... Abstract . . . . . ii i Introduction . 1 Chapter 2. Connections on the frame bundle of a manifold. . 5 Chapter 3. Connections preserving the metric and geodesic of a Riemannian manifold . 32 Chapter 4. Some geometry of sub.manifolds. 40 Chapter 5. Riemannian homogeneous spaces. 44 Chapter 6. Geometry of syrranetrically embedded submanifolds of naturally reductive Riemannian homogeneous spaces . . . . . . . . . . 55 Chapter 7. Some multilinear algebra . 71 Chapter 8. The tube formula . . • • . 84 Chapter 9~ Parallel hypersurfaces . 87 Chapter 10. An algebraic reformulation for symmetric spaces. 91 Chapter 11. Tubes in product manifolds . 97 Chapter 12. Examples References . . . . . . . . . . . 106 116 1 1. Introduction Let M be a submanifold of the Riemannian manifold ~. Then a fundamental problem in the geometry of submanifolds is to give invariants of the pair (M,M) of M. r, about M in that relates the geometry of M to that One such invariant is the volume of the tube M(r), of radius M. In the case where ~ is a Euclidean space of dimension and M is compact of dimension vol (M ( r) ) = n, n+m Hennann Weyl [13] proved that m-1 +2k 6 c k n mr 0<2k<n 2 ' ' - - f h2k ( p)r-.~bM ( p ) M where h2k(p) is a polynomial of degree 2k in the components of the second fundamental form (or of the Weingarten map) of M in M. It is also possible to express h2k(p) as a polynomial of degree components of the curvature tensor of M. k in the The in'variants · h2k just defined have proven to be useful in geometry. For example, the first proof of the Gauss-Bonnet theorem for manifolds of dimension greater than two was given by Allendoerfer and Weil [l], and used Weyl 's formula. Another example where the invariants h2k are important is the Kinematic formula of Chern [3] and Federer[6]. This shows that it is of some interest to compute the volume of tubes for more general pairs (M,M) and see if invariants similar to the h2k defined by· weyl can be defined. The results of this paper show that in the case M is a "symmetrically embedded" submanifold of a naturally reductive Riemannian homogeneous space M (definitions below) then it is possible to define, for each real t and each integer k wi'th 0 < k ~ dim M a function 2 p + hk(p,t;M,M) M(r) on M such that a formula for the volume of the tube Specifically, if n = dim M, analogous to Weyl ls holds. n vo 1( M( r) ) = 6 then J hk( p , r; M,M) OM ( p) . k=O M The function hk is a polynomial of degree k in the components of the second fundamental form of M in · M. In section 2. those standard results on the geometry of manifolds which will be needed later are given. For the most part, the exposition follows that of Kobayashi and Nomizu [8]. In section 3, we give formulas to compute the curvature and Jacobi fields of a Riemannian manifold M in terms of the curvature and torsion of a connection on M that preserves the metric of M and has the same geodesics as the Riemannian connection of M. It is also shown there is a bijective correspondence between such connections and the smooth 3-forms on M. The results of this section seem to be new, however it is possible they are only of interest when the connection in question is the canonical connection of a naturally reductive Riemannian homogeneous space. In this case they are well known. Sections 4 and 5 are both expository-. Section 4 gives the results on the geometry of submanifolds needed in the sequel. Section 5 gives the results on Riemannian homogeneous spaces that are needed. The calculations of section 3 are used here. Section 6 contains the main results of this paper. First the notion of a symmetrically embedded submanifold of a naturally reductive Riemannian homogeneous space is defined (definition 6.1). Proposition 6.2 then gives a geometric interpretation of what being symmetrically 3 embedded means. The volume of a tube about a compact symmetrically embedded submanifold is then computed. It is the introduction of the fields of linear maps S(t;U), C(t;U), and si(t;U) along geodesics normal to the submanifold which allows the calculation to be done. These linear maps can also be used to compute the Weingarten map of the tube. However, this calculation is not done here. The results of section 7 are algebraic. expand det(A+ B) theorem. The basic problem is to into a sum by something resent> ling the bionomial This was done by Flanders [5]. He uses the universal properties of tensor products in his definition of what is written here as A* B. hard. This makes comparison with formulas in c1ass i ca 1 not ati on The calculations needed to compare the two are done in detail here. In section 8, the algebraic results of section 7 are used to expand the function h(p,t) hk(p,t) = hk(p,t;M,M) of the tube formula of theorem 6.14 into terms homogeneous of degree the Weingarten map of M in M. k in the components of The functions hk(p,t;M,M) are then the natural generalization of the invariants defined by Weyl. It is also shown that, if M is a symmetric space then vanishes hk(p,t;M,M) for k odd. In section 9 the classical results of Steiner [11] on parallel surfaces are generalized to hypersurfaces in a naturally reductive Riemannian homogeneous space. In the case where M is a symmetrically errbedded submanifold of a symmetric space M, it is possible to express the linear maps S(t;U), needed in the tube formula explicitly in terms of and si(t;U) C(t;U), 4 the Lie algebra of a transitive group of isometries of M. This is done in section 10. In section 11 a formula relating the invariants h2k(( Pi, p2), t;M 1 x M2 ,'M1 x M2) to the · invariants of the pairs (M 1 ,'M1) and (M 2 ,'M2) is given. This generalizes the corresponding result for the invariants given by Weyl in the case M1 and 'M2 are Euclidean. This gives more evidence that the invariants introduced here are reasonable generalizations of Weyl ls invariants. In the last section some examples are give. 5 2. Connections on the frame bundle of a manifold. All manifolds will be assumed to be Hausdorff, paracompact and of class ~- If a manifold is not connected, then all connected components are assumed to have the same dimension. The word smooth 11 either manifolds or maps will mean "of class c'° 11 • applied to 11 If M is a manifold, then TM (also written as T(M)) will be the tangent bundle of M and ™p (or T(M)P) wi~~ be the tangent space to M at p. a smooth map between manifolds, then derivative of f at p. If f: M+ N is f*p: TMP + TNf{p) is the The characterization of tensor fields as objects multi·linear over the ring of smooth functions of a manifold will be used (see [8] vol. 1, page 26). For the rest of this chapter, fix some manifold M of dimension n and a real vector space ~ of the same dimension as the bu:ndle of linear frames over M, of M. For ea ch M. We now define or, more briefly' the frame bundle p i n M, 1et L( M) p be the set of a11 l i near isomorphisms of m onto ™p· An element of L(M)P will be called a frame at L(M) with p. p in The frame bundle M. For each p in M, fibre of L(M) over p. A map elements of L(M)p to onto M. Let GL(m) p. TI: is the disjoint union of the L(M)P the set L(M)P is called the L(M) + M is defined by taking all This map is called the projection of L(M) be the group of all linear automorphisms of the vector space m with its usual structure as a Lie group. a natural right action of GL(M) on L(M) Then there is by (u,a) 1+ u 0 a, where u € L(M) and a€ GL(m). We now wish to make L(M) smooth manifold in such a way that the projection TI into a and the action of 6 GL(~) on L(M) U of M let are smooth. By way of notation, for each open subset L(U) = n- 1(U). Definition 2.1. over Let U be an open subset of M. Then a moving frame U is a function e : U -+ L( M) such that: (1) Tioe=identityon U; (2) If ep is the value of e at p, then for all function p-+ ep(v) Remark. Let ~: is a smooth vector field on the U. U -+Jn be a diffeomorphism of the open subset U of M with the open subset ~(U) of Jn. Then, under the standard identification of tangent spaces to Jn with Jn, . ~: U -+ L(M) v in Jn, the function defined by is a moving frame over U. Therefore every point of M is in the domain of some moving frame. Proposition 2.2. There is a unique structure of a differential manifold on L(M) such that: (1) The projection TI: L(M)-+ M is smooth; (2) the right action of GL(rn) (3) every moving frame e: U-+ L(M) on L(M) given above is smooth; over some open subset U of M is a smooth function. Outline of the proof. If the three conditions of the proposition hold, then it is straightforward to check that, for each moving frame 7 e: U + L(M) U x GL (In) over some open subset onto TI - l ( U) U of M, the map ~e from given by = ep o a is a diffeomorphism. This determines the smooth structure of L(M) the open subset TI-l(U) L(M) of L(M). is covered by such sets. in By the remark before the proposition, Thus, the smooth structure on L(M) is unique, provided it exists. Let e.: U. + L(M) J subsets J j = l, 2 be two moving frames over the open u1, u2 of M. Then it is not hard to check that -1 0 ~e ~e l 2 is a di ffeomorph ism. : (U 1 n U2 ) x GL(rn) + (U 1n U2 ) x GL(rn) Therefore, the maps ~ e, where e is a moving frame, can be used to define an atlas for L(M). This finishes the proof. The proof of the following is left to the reader. Proposition 2.3. With notation as above, (l) The dimension of L(M) (2) the projection for all TI is n2 + n; is a submersion (that is, TI*U is surjective u in L(M)); each fibre L(M)P is a closed embedded submanifold of L(M) diffeomorphic to GL(rn) and the action of GL(tn) (3) on the fibre L(M) is simply transitive; (4) the tangent space to a fibre L(M) p .at a frame u is the kernel 8 We now define the class of geometric objects on which most of our calculations will be done. then If G is a closed subgroup of G also has a right action on L(M) GL(~), in an obvious way. Definition 2.4. Let G be a closed subgroup of GL(m). G-strueture on M (also called a reduction of L(M) to G) is an embedded submanifold P of L(M) (1) Then a such that; The restriction of the projection TI to P is a submersion of P onto M; (2) for each p in M the fibre PP, defined to be PP = L(M)P n P, is an embedded submanifold of P such that the action of G on PP is simply transitive. Some elementry facts about G-structures are given in the following. Proposition 2.5. Let P be a G-structure on M; then, (1) The dimension of P is (2) Each fibre of P is diffeomorphic to (3) If TI: P + M is the projection, then, for each in the fibre dim(M) + dim(G). G. p in M and u Pu, T(Pp)u =kernel (TI*U). (4) Each point of M has an open neighborhood e: U + L(M) U and a moving frame defined on U such that ep e P for all p in U. (Such moving frames are called sections of P over U.) Proof. The first three parts are easy. Because TI is a submersion of P onto theorem lets us find a smooth function M the implicit function e: U + P, defined in an open 9 neighborhood of any given point of M, with no e = identity on U. It is not hard to verify that e is a section of P over U. Examples. (1) It is clear that L(M) GL(~) is a structure on M. Recall that a Riemannian metric on M is an assignment of an (2) inner product ( , )p on each tangent space ™p to M, way that if X and Y are smooth vector fields on M, in such a then the function pH- (X(p),Y(p))p is smooth. Put an inner product ~ ( , ) on and let group of all automorphisms of this inner product. isomorphic as a Lie group to the group of all matrices. in M, Thus be the O(~) is n x n real orthogonal Let M have a Riemannian metric ( , _). Then, for each ~ let O(M)P be the set of all isometries of Define O(M) O~) onto to be the union of the O(M)p with p in M. O(~)-structure on M, can be verified that O(M) is an ™p· Then it called the M. bundle of orthogonal frames of O(~)-structure Conversely, given an p P on M we can define a Riemannian metric on M by (X,Y)p = ( u-1 X,u -1 Y) where ~; u is any element of PP and ( , ) is the inner product on this inner product is well-defined because any two frames . in are related by the right action of an element of O(~). Then P will be the bundle of orthogonal frames for this Riemannian metric. giving an O(~)-structure on metric on M. Pp Thus, M is the same as giving a Riemannian 10 (3) Suppose M is a complex analytic manifold of complex dimension m (and thus real dimension n = 2m). has an atlas {(cp ,U): a€A} a a Recall that this means that M such that for each a e A is a cp('J '-" diffeomorphism of the open subset Ua of M onto the open subset cp a (Ua ) of [m so that for each pair a,6 e A the function is holomorphic. ™p' If M is such a manifold, then each tangent space to M has the structure of a complex vector space. Multiplica- tion of a tangent. vector X e TM p by a complex scalar a can be described as follows: Choose a chart (~a'Ua) from the defining atlas or M with _ p € U , then __.et This can eas i lY be checked to be well-defined· by using- that if · -· a,S e A and p e ua n u , then (ep a 0 epS 1 )~(fl is complex 1inear. 8 S(p) Now assume that th is a complex vector space. For each p in M, let C(M)P be the set of all complex linear isomorphisms of ~ ™p' p in M. and let C(M) be the union of all of the C(M)P with onto If GL ([,th) is the group of a 11 complex 1i near automorphisms of th, then is a GL(rL,tn) C(M) structure on th called the bundle of holomorphic frames over M. (4) ( , ) If M is a complex analytic manifold, then a Hermitian metric on M is a choice of a Hermitian inner product ( , >p on each tangent space ™p to M such that for all smooth vector fields on M the complex valued function p >-+ (X(p),Y(p))p X,Y 11 is smooth. Assume that rn inner product is a complex vector space with Hermitian ( , ) and that automorphisms of ( , ). U(rn)-structure U(M) U(rn) is the group of all complex linear Then it is possible to define a on M in a way that should be clear from the last two examples. This unitary frames over M. U(rn)-structure is called the bundie of We now record some facts we will need about GL(rn) subgroups. Let . ge(rn) of rn with Lie bracket given by and its closed be the Lie algebra of all linear endomorphisms [A,B] = AB - BA. Then 91,(tn) is the Lie algebra of GL(rn). For any A€ g,e(rn) define eA by its power series eA = ~ Ak L.J k=O -k, • . Then every continuous homomorphism from the group of additive real nurrbers to GL (rn) is of the form for some A i n g,e (rn ) . If G is a closed subgroup of GL(rn) then, by the "closed subgroup theorem" of E. Cartan ([12] Theorem 3.42, page 11), an embedded submanifold of GL(rn). GL(rn). Then tJ:r, Let 1 be the identity element of the tangent space to G at of G and is a Lie subalgebra of g,e.(Jn). tangent space to G at G is 1, is the Lie algebra By parallel translating the to the origin (zero element) we can and often will view elements of {)y- as linear transformations on rn. It 12 should also be noted that f!f is the set of all that etA is in G for all real A in ge(m) such t. The adjoint representation of G on Cf.{ is given by at+ Ad(a) where Ad(a)A = aAa-l. It is easy to check that Convention 2.6. 11 P11 Unless stated otherwise, for the rest of this chapter wi 11 denote some fixed G-s tructure on M where G is some fixed closed subgroup of Definition 2.7. a on P by (2) For each GL(rn) (1) A in with For each OJ Lie algebra {)J. a in G define right translation by define the fundamental vector field A* on p by d A* (u) ~ dtl Proposition 2.8. (1) t=O ue tA The flow of the vector field A* is retA. (For the definition of the flow, or local 1-parameter group generated by a vector field see [12] 1.49 Definitions, page 39.) (2) For A e {)]' and a e G ra*A * = (Ad(a -1 )A) * . (3) The map A 1-+ A* is a Lie algebra homomorphism of Lie algebra of all smooth vector fields on (4) For each u in P the map A~ A* (u) tf.r into the P. is injective. 13 (5) For each Proof. (1 ) u in P the tangent space to the fibre Pnu at u is It is easy to check that r tA e 0 r sA = r ( t+s) A· e e The result now follows from the deftni'tion of a flow. (2) The tangent vector to the curve t ~ uetA at t = O is A* (u). Therefore * d ra A (u) = dtl * t=O ra(ue tA ) This proves (2). (3) For each u in P define a map cru: G-+ P by cru(a) = ua. tangent vector to _the curve tr-+ etA at t = 0 is A; d cru*lA = dtl t=O = _Q_l dt t=O crue The therefore, tA ue tA =A* (u). The map cru*l * is linear, which shows A1-+ A{_u) It follows that . A1+ A* is linear. is linear for all u. 14 Let s.,A* be the Lie derivative with respect to A* (see [12], pages 69 and 70 for the definition of Lie derivative and for a proof of the equality s.,xY = [X,Y]). A* is -r tA' e Using (2) and the fact that the flow of we have * * * [A , B ] ( u) = (J: A*B ( u) d = dt I * r -tA B ( ue t=O e ·* We have just shown the map CJ+ C* (u) T(P)u. tA ) to be linear from ~ to is any smooth curve in (JT it follows Therefore, if t 1-+ Ct that This yields = [A, B] * ( u) . This completes the proof that A rt- A* is a Lie algebra homomorphism. (4) Let A e ~ and u e P with A* (u) = 0. of A* is r tA Then because the flow it follows that e r tA(u) = uetA e for all real t. =u The action of G on fibres is simply transitive; therefore etA = 1 for all t. This implies A = 0. This along with 15 linearity of the map (5) * A~ A (u), By (3) and (4) we see that the same dimension as G. dimension. proves (4). {A * (u): A e ()j} The vector space T(P TIU ) U is also of this Thus to show ·the two are equal it is enough to show the first is a subspace of the second. TI 0 r a = TI. is a linear space of If a e G then it is clear that Consequently * d TI*uA . (u) = dtl t=O TIUe tA d = dtl t=O TIU = 0. Thus {A * (u): A e (Jj} s. Kernel(Tiau) = T(PTIU )u. This finishes the proof. Definition 2.9. Vectors tangent to some fibre PP of P will be called vertical. Remark. It will be convenient to use the formalism of vector valued differential forms. The following 1ist of definitions is given so as to fix our conventions on what constants are used in the definitions of exterior derivative and wedge product. Then a V-vaZued r-form w on Let V be a real vector space. M is a smooth assignment for each in M of an r-linear alternating function wp on ™p with values in r = 0, w is defined to be a smooth function with values in V. When V. In the case V = IR, w is just ca 11 ed an r-form. derivative p dw of w is the V-va 1ued The exterior (r + 1 )-form given on smooth 16 0 , ... , Xr vector fields x + by . i+j A A .6.(-1) w([Xi,Xj],X 0 , ..• ,Xi, ... ,X., ... ,X ). 0.::_1 <J2_r J r where "' means the term is omitted. For r = 0 and l this becomes dw(X) = Xw, dw (X , Y) = Xw ( Y) - Yw( X) - w ( [ X, Y] ) • If w i,s a V- val ue d r- fo nn , e i s a W- va1ued s - form and ( , ) a bilinear function on V x W with values in the vector space S then the wedge product of w and e is the s-va 1ued ( r + s )-fonn given by (w~e) (x 1 , ... ,xr+s) = r~s ! ~ ( - l)cr (w(Xcr ( 1 )' · · • •\r (r p,e(Xcr ( r+ 1)' · .. ,Xa ( r+s) )} where the sum is over all permutations cr of the set {l , ... ,r+s} (-l)cr is the sign of the permutation cr. and It can be checked that d(w"'e) = (dw~e) + (-l)r(w"'de) . In the case in which both V and W are the real numbers and ( , ) is multiplication of real numbers, we just write Definition 2.10. The value of w on vertical vectors is given by wu(A * (u)) =A for all A in Cf.! and u in P. for (w4'e) . P is a smooth Qr- valued one-form P that satisfies the following two conditions; w on (1) A connection on W/\8 17 (2) w transforms under the action of a e G by * = Ad(a -1 )w. raw Definition 2.11. If w is a connection on P then for each u e P let H = kernel(n U Then *LI ). Hu is called the space of horizontal vectors at u or more briefly the horizontal space at Proposition 2.12. u. Let {H : u € P} u vectors, for the connection w on (1) · {H : u e P} u be the set of all horizontal P. Then, is a smooth distribution on (2) For all a e G and u e P (3) For all u e P M. (direct sum). Conversely, let {Hu: u e P} satisfy (1), (2) and (3) and define w · P given by wu(A * (u)) = A for A and wU(X U) = 0 if XU is in Hu. Then w is a connection to be the O,r-valued one-form on in VJ on P and the horizontal spaces defined by !!r:r w are {Hu: u e P}. Proof. See proposition 1.1 on page 64 of vol. 1 of [8]. Remark. A connection is often defined to be a smooth distribution {H : u e P} u satisfying (1), (2) and (3) of the last proposition. w is defined as above and is called the connection form of the connection. Then 18 (2) Let ~ connection on be a connection on L(M). P if and only if, for each of horizontal vectors determined by Proof. ~· u at in P, the space Hu u is tangent to P. The first part is a special case of proposition 6.1 on page 61 of vol. 1 of [8]. Remark. Then w is the extension of a The second part is straightforward. Some of the definitions below, such as parallel translation along a curve or the curvature and torsion tensors on M, given in terms of either a connect ion w on connection on L(M). can be P or the extended It will be left to the reader to show these definitions ~re independent of which of these two connections is used. Definition 2.14. c: (a,B) + M be Then a piecewise smooth curve 2: (a,S) + P Let w be a connection on any piecewise smooth curve. is Called a horizontal lift Of is horizontal for all Proposition 2.15. if and Only if C unique hori zonta 1 1 ift 1TO C: C and C (t) 1 t. Let w · be a connection on piecewise smooth curve, P and t 0 € (a,6) and P, c: (a,S) + M a u e Pc(to)" Then there is a c of c to P with 2( t 0 ) = u0 . If a e G, then the horizontal lift y: (a,S) + P with y(t 0 ) = u0 a is given by y(t) = c(t)a. Proof. This follows from proposition 3.1, page 69 of [8]. Definition 2.16. Let w be a connection on P, c: (a,B) + M a piecewise smooth curve and t 1 , t 2 e (a,6). Then parallel translation along c from ™c(tl) to ™c(t ) is defined by 2 tl A -1 Tt = c(t 2 )c(t 1 ) A 2 19 /'\ where c is any horizontal lift of c to P. t, is a Clearly Tt 2 1 i near i soma rph ism of ™c ( tl ) onto ™c ( t ) · 2 By the 1as t proposition any other hori zonta 1 lift of c i.s of the form t t+ c(t)a. It follows that parallel translatton is independent of the choice of the horizontal lift of c. that if t 1 , t 2 , t 3 are in It is also easy to check then (a,6) t, t2 tl Tt Tt = Tt . 3 2 3 Definition 2.17. Let Y be a smooth vector field defined on some open subset U of M and X(p) a tangent vector to M at a smooth curve c: (-s,s) 7 U for some s > 0 with p e U. Choose cl(Q) = X(p). Then define where -r6: ™c(t) 7 ™c(O) is the parallel translation along c defined by the connection w. Remarks. (1) For all ts (-s,s) the vector -r~Y(c(t)) finite dimensional vector space TMp. The derivative is in the Jt -r6Y(t) is computed as the tangent vector to a curve in a vector space. (2) The vector vX(p)y with c'(O) = X(p). (3) To compute vX(p)Y it is enough to know the values of Y along c See pages 114 and 115 of vol. 1 of [8]. any curve c that fits Proposition 2.18. is independent of the choice of the curve X(p) The map in the sense of the definition. (X(p),Y) t+vX(_p)y the following five relations: defined above satisfies 20 (1) "Vx (p)+x (p)Y = "Vx (p)Y + vx (p)v. 2 2 1 1 ( 2) 'V cX ( P) Y ( 3) 'Vx(p)(v,+Y2) = 'Vx(p)vl + 'Vx(p)Y2· ( 4) 'V X( P) ( bY). = = c'V X(p) Y for a11 real c. b( p) v X( p) Y + ( X( p) b) Y( p) fa r a11 smooth real val ued b with the same domain as Y. ( 5) If X and Y a re smooth vector fie 1ds on the open subset U of M, then so i s p 1-+ 'V x( p)Y. Proof. See proposition 1 .1, page 114 of vol . 1 of [8]. Definition 2.19. Let ~ be the set of all pairs (X(p),Y) where Y is a smooth vector fl e1d on some open subset of M and X(_p) is a vector tangent to M at some point Then a p in the domain of Y. (X(p) ,Y) >+ 'VX(p)y defined on ~ function and satisfying the five conditions of 2. 18 is ca 11 ed a covariant derivation on M. If 'V is defined from a connection w then v is called the covariant derivation of w. Proposition 2.20. (1) Two connections on P with the same covariant derivation are equal. (2) Every covariant derivation on M ts the covariant derivation of a (unique by (1)) connection on Proof. L(M). See proposition 7.5, page 143 of vol. 1 of [8]. We now describe parallel translation in terms of the covariant derivation of a connection. Definition 2.21. w on P, Let 'V be the covariant derivation of the connection and c: (a,S) -+ M a smooth curve. Then a vector field 21 t t-+ Y(t) for all along c is. called paraUe"l if and only if t (a,S). in Proposition 2.22. on Let w he a connection with covariant derivation v P, c: (a,S) ~ M a smooth curve, and t 0 e (a,S). parallel translation defined along c by w, vector field t t-+ Y(t) If T is the then every parallel along c is of the form to Y(t) = Tt Y0 for some v0 in ™c(t )• Therefore, for every Y0 in ™c(t ) 0 there is a unique parallel field t ~ Y(t) along c with Y(ta) = Yo. The vector Y(t) 0 is called the parallel translate of Ya along c to c(t). Proof. to (a,S) If Y(t) = Tt Y0 then for any t 1 in d 'Vc•(t )Y(t) = dtl l t=t, t Tt Y(t) l t=t1 Tt Tt 1 Yo to d = cttl ta t d = dtl t=t, T t Yo l = 0. Therefore Y(t) is parallel. Let t t+ Y(t) let xl' ... , xn be a basis of ™c(t )• 0 x1(t), ... , xn(t) along c by be parallel along c and Define fields 22 Then we have just shown each Xj(t) ta ,.t from ™c(to) to ™c(t) is parallel along c. The map is a linear isomorphism,therefore x1(t), ... , Xu(t) is a basis of ™c(t) for all t in ™c(t). Whence, n Y(t) = .611.(t) , 'it, x.(t) 1 1= for some smooth functions y 1 , •.. , Yn on we have This shows Yi = 0, so each Yi (a, S). is constant. By proposition 2. 18, Consequently, n y ( t) = .6 lJ i ( t 0) xi ( t ) i=l . n t · 0 = 6 11 . ( t ) ,. t x. 0 . l 1 1 1= t n = ,. t 0 ( .6 1lJ 1. ( t 0 ) x1. ) 1= This finishes the proof (with v0 = Y(t 0 )). The next several definitions are devoted to defining the curvature and torsion forms on P and the corresponding curvature and torsion tensors on M. Definition 2.23. The canonical form e on P is the rn-valued one-form 23 on P given by eU(X) = u-l'Tf*Ux. Remark. e is defined independently of any P and the kernel of eu is the space of vertical The canoni ca 1 fonn connection on vectors at u. Proposition 2. 24. If e is the canoni ca 1 form on P then e transforms under the action of G on P by r;e a... e. = Proof. 1 Strai-ghtforward. Definition 2.25. space V. Let a be a k~form on P with values in some vector Then the covariant differential connection w on P is the V-valued k+l Dex of ex defined by the form given by (Dex) (_Xl, ..• ,Xk+l) = dex{_hX 1 , ... ,hXk+l) where d is exterior derivative and X = hX + vX is the decomposition of X into its horizontal component vX defined by the connection w. Definition 2.26. Let w be a connection on P and D the covariant differential defined by w. (1) hX and its vertical component Then: The torsion form e of w is the ~-valued two-form given by 8 = De. (2) The curvature form O of w is the eq- -valued two-form given by 0 = Ow. 24 The proof of the next proposi.tion is straightforward. The torsi.on form e and the curvature form o of Proposition 2 ~ 27. a connection P transform under the action of G by w on * -.1 ra e = a e, r: o = Adla- 1 ) o for a in G. Definition 2. 28. Let w be a connection on P. Then, for each p e M, A Xe ™p and u e Pp we define the horizontal lift X(u) of X to u A by letting X(u) be the unique hodzontal vector at u with A Remark. A It is easy to check using 2.12 (2) that ra*X(u) = X(ua). Definition 2.29. Define the torsion tensor T and the curvature tensor R of a connection w on P by A A TP(X,Y) = u(eu(X(u),Y(u))) A where A X,Y,Z e ™p' TIU= p and X(u), Y(u) are the horizontal lifts of X and Y to P. Elementary calculations using proposition 2.27 and the remark preceding the definition show that the definitions are independent of the choice of u with Proposition 2.30. TIU = p. The two tensors T and R defined above are related to the covariant derivation v of the connectton w by 25 T(X,Y) ~ - ~xy - vyX - [X,Y], R( X, Y) Z = v XJ y Z - v yV XZ - V [ X, y i where x~- y, Z are Proof. This is theorem 5.1, page 133, vol. 1 of [8]. smooth vector fields define·ct on some s_ub-set- of M. We now define the covariant derivatives of (vxT) and (vxR) in the usual way, .which is by requiring the product rule to hold, i.e., (v ~T )( Y, Z) = V7 X(T( Y, Z)) - T(v XY, Z) - T( Y,v Xz) , (vxR) (Y ,Z)W = v x(R(Y ,Z)W) - R(v xY ,Z)W where Y, Z, W are smooth vector fields on some open subset of M. Proposition 2.31. Let T be the torsion tensor and R the curvature tensor of a connection w on P. Then the following hold: First Bianchi Identity. ~(R(X,Y)Z) = ~(T(T(X,Y)~Z+ bxT)(Y ,Z)) Second Bianchi Identity. ~((vxR)(Y ,Z) + R(T(X,Y) ,Z)) = 0 where Proof. is cyclic sum over X, Y and Z. ~ This is theorem 5.3, page 135 of vol. 1 of [8]. Definition 2.32. g. P with covariant Then a smooth curve g: (a,S) + M is a geodesic of w if and only if t 1-+ g' (t) is a parallel vector field along derivation v. (or of v) Let w be a connection on That is, g is a geodesic of w if and only if Vg'(t)g'(t) = 0 26 for all t in (a,S). Definition 2.33. Let w be a connection on rn. Then the basic vector field B(v) on P and v a vector in P determined by v e rn is defined by letting B(v)u be the unique horizontal vector at ,u with An equivalent definition is . _ /\~ B(v)u = u(v)(u). Proposition 2.34. A curve g =··(a,S) -+ M is a geodesic for the connection w on P if and only if y: (a,S) -+ P B(v). g is of the form noy, where is an integral curve of one of the basic vector fields Consequently, for each tangent vector X(p) to M there is a unique geodesic g defined in a maximal connected neighborhood of zero in the rea 1 numbers Proof. lR with y ( O) = p and y 0) = x( p). 1 ( See proposition 6.3 and theorem 6.4 on page 139 of vol. 1 of [8]. Definition 2.35. Let w be a connection on P then the exponential map determined by w is defined as fo 11 ows. For X e ™p write t ~ expp(tX) for the unique geodesic with expp(oX) = p d dt l t=O exp ( tX) = X. p Then the exponential map from TM p to M is the function Xt-+ exp P(X) = exp P( 1 • X) . This is defined in a neighborhood of zero in ™p· 27 We will need to take derivatives of the exponential map. This task is reduced to computations with ordinary differential equations by the following definition and proposition. Definition 2.36. Let g: (a,b} + M be a geodesic for a connection with covariant derivation v. (1) A vector field Y(t) Then along g is a Jaeobi field along . g if and only if it is a s.o Jution to the Jacobi equation 2 vg'(t)Y(t) +'Vg•(t)(T(Y(t},g'(t))) + R(Y(t},g 1 (t}}g 1 (t) = 0 along (2) g. Here T and R are the torsion and curvature tensor of v. A variation of g through geodesics is a smooth function a: (-s,s) x (a,b} -+ M (for some s > 0) such that a(O,t) = g(t) for all s e (-s,s) Proposition 2.37. the map t ~ a(s,t) and is a geodesic. Let g: (a,h) + M be a geodesic for a connection with covariant derivation 'V· Then: (1) A Jacobi field Y along and g is determined by the values of Y(tO) (vg'(t)Y)(t 0) for any t 0 e (a5b) specified arbitrarily. (2) and these values can be If a: (-s,s) x (a,b) + M is a variation of g through geodesics then t t+ ~~(0,t) is a Jacobi field along Proof. (1) g. The Jacobi equation is a homogeneous linear second order ordinary differential equation; therefore, (1) follows from standard results. (2) See theorem 1.2, page 64 of vol. 2 of [8]. 28 Proposition 2.38. Let v smooth vector fields X and 'V '·. and Y on be covariant derivations on M. M~ For let Then C is a tensor field of type (1,2) (called the difference tensor of v and v '). The covariant derivations v and v have the same geodesics 'if and only if C is alternating. Proof. See propos_iti on-1.5 · on page 271 of vol . 2 of [l 0 l. We now turn to connections on Riemannian manifolds. Proposition Z.39. Let M be a Riemannian manifold with metric ( , ) and let O(M) be the bundle of orthogonal frames over M. connection on L(M) If w is a then the following are equivalent: (1) w is the extension of some connection on (2) Parallel translation along any smooth curve in M is an isometry O(M). between tangent spaces of M. (3) If v is the covariant derivation of w and X, Y, Z are smooth vector fields on M thEn Proof. The equivalence of (1) and (2) is the content of proposition 1.5 on page 117 of vol. 1 of [8]~ Suppose (2) holds and let Y,Z be smooth vector fields on M. Let X be any tangent vector to M and choose a smooth curve c: (-c:,E:) + M such that c'(O) = X. along c. Choose an orthonormal basis 0 ej (t) = ,.tej. Because Let ,. be the parallel translation e1 , ... ,en of ™c(O) . an isome . t ry, ,.t0 is e 1 (t), ... , en(t) and let is an 29 orthonormal basis of ™e(t)' for all y1 , ... , yn and z1 , •.. , zn on t. (-s,s) There are smooth functions with n Y(c(t)) = .6 y i ( t ) eJ. ( t ), 1 =l n Z(c(t)) = 6 zJ. (t) e . ( t) . j=l J Therefore, c'(t)(Y(c(t)),Z(c(t))) d = dt n n 1 =l J=l <.6 yi(t) e;(t), ~ zJ.(t) eJ.(t)} d n 6 y k( t ) z k( t) = dt k=l n n i=l j=l J = ( 6 yk(t ) e1. (t ) , 6 z . ( t) eJ. ( t )) n n + ( 6 y . ( t) e . ( t) , 6 z ~ ( t) e . (.t) ) i=l 1 1 J j=l J + (Y(c(t)), ~c'(t)X(c(t))). Noting that c'(O) = X shows (2) implies (3). Now assume (3) holds. Let c: [a,b] ~ M be a smooth curve and the parallel translation along c. Then Let Y,Z be vectors in ™c(a)· T 30 a a +<rt Y,-vc (t)rt Z) = 0. Therefore (rta Y,rta Z) is constant as a function of t. This shows a a a (rb Y,rb Z) = (Y ,Z), whence rb is an i sometry of ™a with ™b. Thus (3) implies (2). Definition 2.40. A connection on a Riemannian manifold that satisfies the three conditions of the last proposition is called metric preserving. Proposition 2.41 (Fundamental lemma of Riemannian Geometry). Every Riemannian manifold has a unique metric preserving connection with vanishing torsion. Remark. This connection is called the Riemannian connection or the Levi-Civita connection. Proof. See theorem 2.2 on page 158 of vol. l of [8]. Definition 2.42. Let M be a Riemannian manifold. Then the geodesics of M a re the ·geodesics of the Riemann i an connection on M. The curvature tensor of M is the curvature tensor of the Riemannian connection. If R is the curvature tensor of M and dimensional subspace of some tangent space ™p curvature of M at P is a two- then the sectional P is K(P) = (R(X,Y)Y,X) where X,Y is any orthonormal basis of P. An easy calculation shows this is independent of the choice of the basis X,Y. 31 Proposition 2.44. If M is a Riemannian manifold with metric ( , ) and curvature tensor R then for all X, Y, Z, W tangent to M at some point (1) (R(X,Y)Z,W) + (Z,R(X,Y)W) = 0. (2) (R(X,Y)Z,W) = (R(Z,W)X,Y). Proof. See proposition 2.1 on page 201 of vol. l of [8]. Remark. (1) of the last proposition tells us that for each the linear map· R(X,Y) on ™p X, Ye TMP is sk~1-symmetric with respect to the inner product ( , >p· Definition 2.45. Let M be a Riemannian manifold with metric ( , ) and c: [a,b] ~ M a smooth curve. Then the length of c is defined ·to be the number b L ( c) = J II c' ( t) II dt a where II c' ( t) 11 = J <c' ( t) , c' ( t) > . If p and q are points of M then the distance from M is defined to be the infimum of the set of numbers is a curve from Proposition 2.46. p to p to L(c) q in where c q. The geodesics in a Riemannian manifold locally are the curves of minimum length, in the sense that every point of M has an open neighborhood U such that any two points p and q of U can be joined by a unique geodesic contained in U and the length of this geodesic is the distance between Proof. p and q. See proposition 3.6 on page 116 of vol. 1 of [8]. 32 3. Connections preserving the metric and geodesics of a Riemannian manifold. It will be convenient to speak of both a connection on the frame bundle L(M) and of its covariant derivation as a connection. Because of the bijective correspondence between covariant derivations and connections on L(M) to any confusion. given by proposition 2.20, this should not lead For the rest of this section 11 M11 will denote a Riemannian manifo1d with metric ( , ). Definition 3.1. A connection with covariant derivative D will be called a geom'etric connection if and only if D preserves the metric of M and has the same geodesics as the Riemannian connection on M. We will refer to D and not its connection as the geometric connection. Examples of geometric connections will be given below. Proposition 3.2. manifold M. Let D b~ a geometric connection on the Riemannian Let T be the torsion tensor and B the curvature tensor of D. Let R be the curvature tensor of the Riemannian connection on M. Then, for all smooth vector fields (1) The connections D and ~ X, Y, Z on M: are related by 1 ~xY = DxY - 2 T(X,Y). (2) The torsion tensor T of D satisfies (T(X,Y),Z) + (Y,T(X,Z)) = 0. (Thus the map Y ._. T(X,Y) is skew-symmetric.) (3) R(X,Y)Z = B(X,Y)Z -¥ T) ( DX + Y, Z) + }( Dy T )( X, Z) - ~ T (T ( X, Y) , Z) t T(X,T(Y,Z)) - } T(~,T(X,Z)) ~ 33 (4) R(X,Y)Y = B(X,Y)Y + ¥ovT)(X,Y) - ~ T(T(X,Y),Y). (5) The sectional curvatures of M can be computed by (R{X,Y)Y ,X) = (B(X,Y)Y ,X) + ~IT(X,Y)ll 2 . Let C(X,Y) = DxY - vxY be the difference tensor of D Proof. (1) and v. The connections D and v have the same geodesics; therefore, proposition 2.38 yields that C(X,Y) is alternating. Whence, T(X,Y) = DxY - DyX - [X,Y] = vxY + C(X,Y) - vyX + C(Y ,X) - [X,Y] = (vxY - vyX - [X,Y]) + 2C(X,Y) = 2C(X,Y), where we have used that v has vanishing torsion. This shows C(X,Y) = 2l T(X,Y) and proves ( 1). For (2) we use that both v and D are metric preserving. any smooth vector fields For X, Y, Z X(Y,Z) = (DxY,Z) + (Y,DxZ> = <vxY + 21 T(X,Y),Z) + (Y,vxY + 2l T(X,Z)) 1 - <vxY,Z) + (Y,vxZ) + i<<T(X,Y),Z) + (Y,T(X,Z))) = X(Y,Z) + ~(T(X,Y),Z) + (Y,T(X,Z))). Therefore, (T(X,Y),Z) + (Y,T(X,Z)) = 0. (3) Let X(p), Y(p), Z(p) be vectors tangent to M at some point p. Extend these to smooth commuting vector fields X, Y, Z defined on a 34 neighborhood of p. Then R( X, Y) Z = \J X\J yZ - \J y'l XZ, B(X,Y)Z = DxDyZ - DyDxZ, T(X,Y) = DxY - DyX. Now compute R(X,Y)Z = 'lx\/yZ - \Jy\JxZ = Dx(DyZ-} T(Y ,Z)} - } T(X ,DyZ -1 T(Y ,Z)) - Dy (DX Z- ~ T( X, Z) ) + ~ T( Y, DX Z- 1T( X, Z) } = DxDyZ - ~DxT)(Y,Z) - ~ T(DxY,Z)_:~ l T(X,T(Y,Z)) - DYDXZ + ~DyT)(X,Z) + 1T(DyX,Z) +~ - ~) + + ~)_ ~ ~ -r(Y ,T(X,Z)) 1 = (DxDyZ - DyDxZ) - ¥Dx T)(Y ,Z) + - l2 T(DxY- DyX,Z) + t T(X,T(Y,Z)) - } T(Y,T(X,Z)) = B(X,Y)Z - ~DXT)(Y,Z) + ~DyT)(X,Z) - } T(T(X,Y),Z) + t T(X,T(Y,Z)) - } T(Y,T(X,Z)). Evaluation at p finishes the proof of (3). (4) Set Z = Y in (3) to get R(X,Y)Y = B(X,Y)Y - ~DXT)(Y,Y) + }(DyT)(X,Y) ~} T(T(X,Y),Y) + ! T(X,T(Y,Y)) -}T(Y,T(X,Y)). 35 T(Y,Y) = 0 and But = 0. Consequently, R( X, Y) Y = B( X, Y)Y + ¥DyT )( X, Y) - i T(T( X, Y) , Y) 1 + .4 T(T(X,Y),Y) = B(X,Y)Y + ~DyT)(X,Y) - l T(T(X,Y),Y). Thi s proves ( 4) . To prove (5), use (4) to get - 1 (R(X,Y)Y,X) = (B(X,Y)Y,X) + "2 ((DyT)(X,Y),X) -t (T(T(X,Y),Y},X). By (2) (T(X,Y),X) = 0, whence ((DyT)(X,Y),X) = Y(T(X,Y),X) - (T(DyX,Y),X) - (T(X,DyY),X) - (T(X,Y),DyX) = 0 + (T(Y,DyX),X) - 0 + (DyX,T(Y,X)} = 0' where (2) has been used in this calculation. Also by (2) (T(T(X,Y),Y),X) = -(T(Y,T(X,Y)),X} = (T(X,Y),T(Y,X)) = -(T(X,Y),T(X,Y)) = -llT(X,Y)!!2· 36 The above expression for (R{X,Y)Y,X) thus reduces to (B{X,Y)Y,X) + ~jT(X,Y)ll 2 . This finishes the proof. Proposition 3.3. Let g: [a,b] ~ M be a geodesic and let U(t) = g'(t) be the tangent vector field along g. Then the Jacobi field t H X(t) along g defined by can be defined in tenns of the geometric connection D by (2) (Du) 2 x + Du(T(X,U)) + B(X,U)U = 0 X(a) = x,' - -- - 1 (DuX){a) = x, + 2 T(U,Xa) where R is the curvature tensor of "V and T is the tors ion and B the curvature tensor of D. Proof. By (1) of the last proposition 2 1 1 1 ("Vu) x = Du(DuX-2 T(U,X)) - 2 T(U,DuX- 2 T(U,X)) = 2 1 1 l (Du) x - 2 Du(T(U,X)) - 2 T(U,DuX) + 4 T(U,T(U,X)). Using (4) of the last proposition and that DuU = 0 we find 2 2 l 1 ("Vu) x + R(X,U)U = (Du) x - 2 Du(T(U,X)) - 2 T(U,DuX) 1 1 1 + "4 T(U,T(U,X)) + B(X,U)U + ~DUT)(X,U) - -zr T(T(X,U),U) = (Du) 2x + 2l Du(T(X,U)) + 2l T(X,DuU) + 21 T(DuX,U) + ~DUT)(X,U) + B(X,U)U = (Du) 2x + 2l Du(T(X,U)) + 2l Du(T(X,U)) + B(X,U)U = (Du) 2x + Du(T(X,U)) + B(X,U)U. 37 Also note x(a) = x0 , (v uX )(a) = (DuX)(a) - 1T(U,X(a)) = x, if and only if 1 X(a) = Xa' (DUX) (a) = Xl + 2" T(U, x0 ) • This finishes the proof. The rest of this section is devoted to proving there is a bijective correspondence between the geometric connections on M and the smooth three-forms on M. Lemma 3.4. Let T be a smooth tensor field of type (1,2) on M such that, for al 1 X, Y, Z tangent to M at some point, the following hold (1) T(X,Y) + T(Y,X) = 0 (2) (T(X,Y),Z} + (Y,T(X~Z)) = 0. Then the connection where v D defined by is the Riemannian connection is a geometric connection with T as its torsion tensor. Thus there is a bijective correspondence between the geometric connections on M and the tensor fields of type (1,2) satisfying (1) and (2). Proof. that Because T is alternating it follows from proposition 2.38 D has the same geodesics as that D is metric preserving. \7· The following computation shows 38 X(Y,Z) = ('VxLZ) + (Y,'VxZ) = ('V XY, Z) + ( Y;v XZ) + "21 ( T( X, Y) Z) + "21 ( Y, T( X, Z)) l = ('V XY+ 2 T ( X, Y) , Z) 1 + ( Y,'V XZ+ '[ T ( X, Z)) = (DxY,Z) + (Y,DxZ). Therefore D is geometric. That T is the torsion tensor of D now follows from proposition 3.2 part (1). Lemma 3.5. This finishes the proof. For every smooth three-form a on M there is a unique smooth tensor field Ta of type (1,2) satisfying (1) and (2) of the 1as t 1emma wfth a(X,Y,Z) = (Ta (X,Y),Z). Moreover, every smooth tensor field T of type (1,2) satisfying (1) and (2) of the last proposition is Proof. Ta for some smooth three-form a. It is easy to see there is a unique tensor field Ta of type (1 , 2) with a(X,Y,Z) = (T a (X,Y),Z). Then a(X,Y,Z) + a(Y,X,Z) = 0 implies (1) and a(X,Y,Z) + a(X,Z,T) = 0 implies (2) of 3.4. If T is a tensor field of type (1,2) satisfying (1) and (2) of 3. 4 then define a(X,Y,Z) = (T(X,Y),Z). Then a is alternating in X and Y by 3.4 (1), and alternating in Y and Z by 3.4 (2). clear that T = Ta . Therefore a is a three-form on M and it is 39 Proposition 3.6. Let v. be the Riemannian connection on M. Then, using the notation of the last lemma, there is a bijective correspondence between the geometric connections D on M and the smooth 3-forms on M given by DxY ~ vxY + ~ Ta(X,Y). Proof. This follows immediately from the last two lemmas. 40 4. Sorre geometry of submanifolds. In this section we record some of the facts we need about submanifolds of Riemannian manifolds. M be a Riemannian manifold Let of dimension m + n with metric ( , ), and M be an embedded submanifold of M of dimension induced metric from · M. 11 ( , )". n. It will be assumed M has the The metric on M wi 11 a1so be denoted by The following notation will be used: = Riemannian connection on M; "V = Riemannian connection on M; ,...., ,...., R = curv'ature tensor on M; R = curvature tensor on M; t.LM = normal bundle of M in ~. roJ 'J Let p e M and g(p) e T.LM p Definition 4.1. A(g(p)): TMP ... TMP then the Weingarten map is given by . A(g(p))X = orthogonal projection of ~xs where s is any 1oca1 extension of g ( p) Remarks. (1) Let section of T.LM. onto ™p' to a smooth section of T.L M. X be a smooth vector field on X and g a smooth Then an elementary calculation shows that the map (X,g) ... (orthogonal projection of ';xs onto TM) is bilinear over the smooth functions on M, independent of the extension of g(p) (2) to whence A(g(p)) is s· The above definition differs by a sign from the usual definition. This choice of sign purges latter formulas of enough factors of -1 to justify it. Proposition 4.2. With notation as above, for any smooth vector fields 41 X, Y on M and smooth section s of TLM (1) ~XY =orthogonal (2) (A(g)X,Y) = -(vxY,g) = (A(g)Y,X). Thus A(g(p)) the following hold: projection of 'ixY onto TM. ™p· is a self-adjoint map on Let e 1 , ... , em be on orthonormal basis of :TLMP. X, Y, Z, W in TMP (3) Then, for (~(X,Y)Z,W) = (R(X,V)z;W) m + Proof. 6 ((A(e.)X,Z)(A(e.)Y,W) - (A(eJ.)X,W)(A(e.)Y,Z)). j=l J J .J See [10] where(l) follows from the last formula on page 46, and (2) and ( 3) fo 11 ow from formulas on page 51 . It will be convenient to restate (3). If V is any finite dimensional real vector space with inner -produce ( , ) then A2(V) is also an inner product space with the inner product on A2 (V), also denoted by ( , ), given by _ (X A y ,Z AW) - (Y,Z)~ -(X,Z) detl(x ,W) (Y ,w) . = (X,Z)(Y,W) - (X,W)(Y,Z). Any linear endomorphism A of V determines a linear endomorphism A2(A) of A2(V} given on decomposable elements by A2(A)(X AY) = (AX) A (AY). Let R be the curvature tensor at some point p of M. as R(X,Y) is an alternating function of X and Y, linear endomorphism A(R) of A2TMP by (A(R)(XAY),Z/\W) = (R(X,Y)Z,W). Then, R induces a 42 Usually R and A(R) are both written as simply When A(R) to be referred to, we will say "view Ras a linear map on /\ 2TM 11 • 11 R11 • is Proposition 4.3 (Equation of Gauss). View R as a linear map on 2 2 A TM and R as a linear map on A 1M. Let Pp be the orthogonal 2 projection of A 1'Mp onto its subspace A2TMP. orthonorma 1 basis e 1 , ••• , e ,...., m Then, for any of T.L M P n 2 pp RP - RP = ;121 /\ (A ( e j ) ) . Proof. This is a restatement of (3) of the last proposition. Definition 4 .4. The excess tensor Hp of M in linear endomorphism of A 2TMP given by 'fVf at p e M is the ,...., Hp = PpRp .. Rp • RP 2,...., is viewed as a linear map on /\ TMP, R is viewed as a 2 1i near map on A TMP and P is. the orthogona 1 projection of A 2TMP where 2 onto /\ ™p. We will be interested in product submanifolds of product manifolds. We recall the definitions. Let M1 , M2 be Riemannian manifolds. ( , }i If pi: M1 x M2 _. M; ( , ) on M x M2 by 1 be the metric on M;. define the product metric Let is projection then The proof of the following is straightforward and is left to the reader. be the covariant deri~tion of the Riemannian Proposition 4.5. Let V'i connection on M.' 1 and connection on Then the covariant derivation V' Mi. R. 1 be the curvature of the Riemannian and the curvature 43 R of the Riemannian connection on M1 x M2 are defined by ('V p xP1 *y 'P1 *Z) l* 1 + ('V p xP2* y 'P2*Z) ' 2* 2 (R{X,Y)Z,W) = (R 1 (pl*X'pl*Y)pl*Z'pl*W) l In the first equation X, Y, Z are smooth vector fields on M1 x M2 so that pi*X' P;:*Y, pi*Z are vector fields on Mi, for i = 1, 2; in the second equation, X, Y, Z, W can be any vectors tangent to M1 x M2 at some point. Let M1 be a submanifold of ~i and let Ai be the Weingarten map of Mi in · 'Ml. for i = 1, 2. Then the Weingarten map Proposition 4.6. where X is tangent to M1 x M2 , U is norma 1 to M1 x M2 , and Y is tangent to M1 x M2 at some point of M1 x M2. Proof. A straightforward calculation using the last proposition. 44 5. Riemannian homogeneous spaces. Let M be a connected Riemannian manifold, and J(M) group of isometries of M. That is, J(M) be the is the group of all diffeomorphisms of M whose derivatives preserve the length of tangent vectors. We give J(M) the compact-open topology. For each p in M let J(M)p be the subgroup of J(M) consisting of those isometries which fix is called the isotropy p. The subgroup J(M)p subgroup of J(M) at p. The following is well known. Proposition 5.1. If J(M) i.s the isometry group of the connected Riemannian manifold M then: (1) J(M) is a Lie transformation group on M. (That is J(M) has the structure of a Lie group and the map (a, p)-+ ap from J (M) x M to M is smooth). (2) Each isotropy subgroup J(M)p (3) If M is compact then so is J(M). (4) If g e J(M)P then g is the identity. Proof. is compact. For the first three see [8],vol. 1, page 239, theorem 3.4. The last part follows easily from the formula This formula is clear as g(expp(X)) = expp(g*pX). exp is defined in terms of the Riemannian metric and g preserves the metric. The manifold M will be called a Riemannian homogeneous space if ~ and only if J(M) is transitive on M. Since it is not always easy to work with the full group of isometries we make the following: Convention 5.2. For the rest of this section, we assume that M is a Riemannian homogeneous space and that G is a closed subgroup of the group of isometries of M such that 45 (1) G is transitive on M; (2) Each isotropy subgroup and Gp = [ g e G: g(.p) = P} is a compact subgroup of G. The following will also be useful. · Notation 5.3. For the rest of this section we fix some point o in M and call it the origin of M. Also let· H = [ g e G: g(o) = O} be this isotropy subgroup of G at the origin. ~ = TM 0 = tangent space to M at the origin. Then the frame bundle L(M) fibre of M can be assumed to have as its L(M)P over p the set of linear isomorphisms of ~ onto ™p· With this convention it follows that: Proposition 5.4. The map g .+'g*o ·i s a diffeomorphism of G onto a closed embedded submanifold of L(M). map G(M). Then G(M) definition 2.4. Call the image of G under this is an H-structure over M in the sense of The fibre G(M)p over p = g(o) is the image of the coset gH. Proof. See chapter X of volume 2 of [8]. Convention 5.5. We will, when convenient, identify G with G(M) via the diffeomorphism of the last proposition a~d use this identification to move the algebraic structure of G over to G(M). The identity element of G goes over to the identity map on ~· The tangent space to G(M) at the identity will be written as Cf.!, have its usual structure as a Lie algebra. space to H at the identity. and be assumed to Let h be the tangent Then h is a Lie subalgebra of Dy- and, 46 by proposition 2.8, part (5), vectors at 1. h is also the space of vertical To make the notation look like that of section 2 the exponential map from Definition 5.6. OJ" to G will be written as AH eA. For g e G let L be left translation on 9 G = G(M). That is, Proposition 5.7 . . .For A eh by A on G{M) the fundamental vector field determined is A* (g{o)) =Lg* A. Proof. This is an easy computation * · . g(.o)e tA d A (g(o)) = dtl t=o d I = dt t=O Lge tA = Lg* A. Proposition 5.8. (1 ) ~ = 1h (2) 111 Proof. 0 0 (±) There is a subspace 111 0 of Of such that h ( di rec t sum) is invariant under the adjoint action of H on tJ:r. See page 199 of volume 2 of [8]. Convention 5.9. We now fix some 111 0 as in~roposition 5.8. If n: G(M) ~ M is the projection then is easily seen to be a linear isomorphism. identified with 1h by this isomorphism. From now on 111 0 will be 47 Definition 5.10. and l\n the ~ component of A relative to the splitting of ~ as fJT = ~ ~ h. (1) For any vector A in DJ" let Ah be the h-component Then Define a ~-valued one-form e on G(M) by eg ( X) = (L _l X)lh • g* (2) Define an h-valued one-form w on G(M) Proposition 5.11. by The form e is the canonical form on · G(M) is a metric-preserving connection on G(.M). and - w This connection will be called the canonical connection on M. Proof. is By definition the value of the canonical form at X i TG(M) -1 9 g* n*g X. therefore, But . e9 (X) = g~-1 n~g X = n*l Lg - 1* X The last line holds because the convection 5.9 makes projection of ~ onto n*l into the ~. It follows directly from proposition 5.7 that w (A * (g)) =A for 9 * every fundamental vector field A on G(M). Let a e H, g e G(M) and X e TG( M) g. 48 Then * (X) = w a(ra*X) (raw) 9 9 = (L -1 -1 ra*X)h a g * = (La-l*ra*(L -1 (L -1 X))h g * g * = (Ad(a- 1 )(L -l X))h g * = Ad(a- 1((L _ *X)h) 1 g = Ad(a- 1) w (X), 9 where we have used the following facts: L _1 ra = raL _1 , g g 1 Ad(a- ) = L 1 r *' a- * a The last of these holds because ~ is Ad(H) invariant. This completes the proof that w· is a connection. Because G is a group of i sometri es of M the H-structure G(M) is a submanifold of O(M), the bundle of orthogonal frames on M. Therefore w can be extended to a conn~tion on O(M). 2.39 now yields that w is metric preserving. Proposition 5.12. the canonical form. Proposition This finishes the proof. Let w be the canonical connection on Let ~ and Y~ be as in 5.10. Then G(M) and e 49 ~ = TG(M) 1 by (1) The torsion form of w is given at (2) The curvature form of w is given at (3) The torsion tensor of w is given on TM0 =~ (4) The curvature tensor of w is given on TM 0 = ~ by Proof. ~ = TG(M) 1 by If X is a left invariant vector field on by G(M) (that is . .Lg*X = X for all g e G) then it follows directly from the definitions that e(X) and w(X) vector field on X(l)~ G(M) are constants on M. then let and likewise for Xh. If X is a left invariant X be the left invariant extension th Then for left invariant vector fields X,Y e(X,Y) = de(\n,Y~) = \ e(Y~) - Y~ e( xm) - e( [ \ , Y~ J) = 0 - 0 - e([\n,Y~]) = As the point 1 e G(M) -e ([x~, v~ J) . this reduces to (1). A similar calculation proves (2). The convention 5.9 shows that a vector in TM0 = ~ horizontal lift to 1 in G(M). is its own Putting this into the definition of the torsion tensor and using (1) yields 50 = -[X,Y~. Part (4) follows from (2) the same way (3) followed from (1). This completes the proof. Proposition 5.13. through For the canonical connection on M the geodesics o are the curves t ,_. netX where translation along the geodesic tX by ( e )*l. Proof. on X is in tn • Para 11 e1 t H netX from o to ne tX is given It is easy to check that the left invarient vector fields G(M) · with X(l) (see definition 2.33). in tn are the basic vector fields on G(M) Therefore the integral curves of the basic vectors that pass through 1 are the .curves · rn. X t t-+ etX where X is in The first statement of the proposition now follows from proposition The curve t ~ etX is horizonal so the second part follows from 2.34. the definition of parallel translation. Proposition 5.14. Let connection. D, T (the torsion tensor) and B (the curvature Then D be the covariant derivation of the canonical tensor) are all invariant under G. invariant under G then OS = 0. Proof. If S is any tensor field on M Thus OT = O and DB = 0. It is clear that w is invariant under G. Each of D, T and B is defined in terms of w and therefore they ire also invariant. Let X e rn. Define a vector field · l ,..., d X(p) = df I t=O on M by .etX (p). The flow of this vector field is clearly at(p) = etX(p). Therefore S 51 is invariant under the flow of X and thus the Lie derivative of S X is zero. But (etX)* is parallel translation with respect to along the geodesic t H netx. Thus ( DXS ) 0 = ( Dx'5 ) 0 = (i1s )o = 0. This shows OS vanishes at the origin of M. invariant and G is transitive, so DS But OS is G vanishes everywhere. This completes th~ proof. Definition 5.15. The natural connection on M is naturally reductive if and only if it has the same geodesics as the Riemannian connection on M. Because the natural connection on M is metric preserving we see that D is naturally reductive if and only if it is geometric in the sense of section 3. Proposition 5.16. The canonical connection on M is naturally reductive if and only if, for all Proof. X, Y and Z in rn, If the canonical connection is naturally reductive then the above equation follows from proposition 3.2 (2) and proposition 5.12 (3). To prove the converse, note that by 5.12 (3) the above equation can be rewritten as where T0 is the torsion tensor of the canonical connection at 0. D be the covariant derivation of the canonical connection. Then Let 52 define a new covariant derivation 0 on smooth vector fields y X and by Then a straightforward calculation shows that 0 is metric preserving and torsion free. o = v, the covariant derivation of the Thus Riemannian connection. But then the difference tensor of D and v is alternating, so D and v have the same geodesics by proposition This finishes the proof. 2.38. Proposition 5.17. Assume the canonical connection on M is naturally reductive and that D is its covariant derivation. Let T be the torsion tensor and B the curvature tensor of D. Let R be the curvature tensor of v. (1) Then For smooth vector fields X and Y on M vxY = DxY - }T(X,Y) (2) (T(X,Y),Z) + (Y,T(X,Z)) = 0 ( 3) R(X,Y)Z = B(X,Y)Z - } T(T(X,Y)Z) + t T(X,T(Y,Z)) - 1T(Y,T(X,Z)) (4) R(X,Y)Y = B(X,Y)Y - -} T(T(X,Y),Y) (5) (R(X,Y)Y,Y) = (B(X,Y)Y,X) + ~!T(X,Y)lj 2 . Proof. The connection D on M is geometric, therefore this proposition is just proposition 3.2 plus the extra information that OT = 0. Proposition 5.18. Assume the canonical connection on M is naturally reductive and let T, Band R be as in the last proposition. For any 53 vector U e ™p define linear endomorphisms Tu, Bu and Ru by Tu(X) = T(X,U) , Bu ( X) = B( X, U) U, Ru(X) = R(X,U)U. Then 1 2 Ru + 8u - 2 Tu, both Proof. Ru and Bu are symmetric and Tu is skewsymmetric. That Ru = Bu - ~ TG is (4) of the last proposition. The skew-symmetry follows from (2) of the last proposition. The Ricci identity (proposition 2.44 (2)) shows Ru is symmetric. The square of a skewsymmetric map is syrrunetric therefore Bu = Ru + 2l Tu2 is the sum of symmetric maps and thus symmetric. Proposition 5.19. With notation as in the last proposition, if g: (a,s) ~ M is a geodesic and U(t) = gl(t) is the tangent along g then the initial value problems (1) (2) define the same Jacobi field along Proof. g. This is proposition 3.3, where we also use that (OUT) = 0 and DuU = 0 so that Du(T(X,U)) = T(DuX,U). Definition 5.20. A submanifold N of a Riemannian manifold totally geodesic if and only if every geodesic of metric is a geodesic of M. M is N in the induced 54 Proposition 5.21. · Let M be a naturally reductive Riemannian homogeneous space and p e M. Let S be a vector subspace of ™p· Then there is a totally geodesic submanifold N of M passing through p with TNP = S if and only if for all and B(X,Y)Z are in S. X, Y, Z in S both T(X,Y) (Here Tis the torsion tensor and B the curvature tensor of the canonical connection on M). Proof. See [2], .theorem 3. 2, page 57. The following defines a class of Riemannian manifolds that has been very much studied. Definition 5.22. A Riemanntan manifold is a symmetric space if and only if it is a naturally reductive Riemannian homogeneous space in which the Riemannian connection equals the canonical connection. Proposition 5.23. If M is a symmetric space, then, with the notation of proposition 5.17, T = 0, Proof. Clear from proposition 5.17. B = R. 55 6. Geometry of symmetrically embedded submanifolds of naturally reductive Riemannian homogeneous spaces. In this section the following notation will be maintained. First M wil1 be an oriented naturally reductive Riemannian homogeneous space of dimension m + n. Then M will be an oriented submanifold of M of dimension n. Because most of what follows is local, M will be assumed compact with smooth (possibly empty) boundry. D = covariant derivation of the canonical connection on M, T = torsion tensor of D, B= curvature tensor of D, 'i = Ri emanni an connection on · ~, · v = Riemannian connectfon on M, R = curvature tensor of v, R = curvature tensor of v. Define, for each U e ™p' linear maps from ™p to itself by T'u(X) = 'T(x,u), Bu(x) = B'(x,u)u, Ru(X) = 'R(x,u)u. Definition 6.1. The submanifold M is symmetrically erribedded in if and only if for a11 M p e M and U e T.L Mp the vector space T.L Mp is stable under both Bu and Tu. Examples of symmetrically embedded submanifolds of homogeneous spaces will be given after the following proposition . 56 Proposition 6.2. -M: The following are equivalent for a submanifold M of (1) M is symmetrically embedded in· (2) For all p e M under both Tu and (3) For all p e M M. and U e T.LMP the vector space ™p is stable Bu. there is a totally geodesic submanifold N of M passing th rough .p with TNP = l.L Mp. Proof. A symmetric or skew-symmetric linear map on an inner product space stabilizes a subspace if and only if it stabilizes its orthogonal complement. The map Bu is symmetric and the map by proposition 5.18. is skew-symnetric This proves the equivalence of (l) and (2). By proposition 5 .21, i'f ( 3) holds then for a11 . Tu p e M and U, X e T.t M , p lu(X) = ~(X,U)U e ™p' Tu(x) = T'(x,u) e ™p· The re fore ( 3) imp l i es ( l) . To finish it is enough to show (l) implies (3). By proposition 5.21 it is enough to show that if M is symmetrically embedded in and X, Y, Z e ·T.tMP then B(X,Y)Z e ™p· M Therefore suppose M is symmetrically errbedded in M and that X, Y, Z e T.tMP. Then B(X,Y)Z + B(X,Z)Y = B(X,Y+Z)(Y+Z) - B(X,Y)Y = B(Y ,Z)Z € T.LMP. Combining the fact that OT = 0 with the first Bianchi identity (proposition 2.31) yields B(X,Y)Z + ~(Y,Z)X + B(Z,X)Y = T(T(X,Y)Z) + 1(1(Y,Z),X) + T(T(Z,X),Y) 57 Adding these gives (B(X,Y)Z + B(Y,Z)X + B(Z,X)Y) + (B(X,Y)Z + B(X,Z)Y) = 2B(X,Y)Z + B(Y ,Z)X + (B(Z,X)Y + B(Y ,Z)X) = 2B ( x'y' )z + B(.Y 'z) x € T.L Mp • X._. Y, Y.._. X, Z ,_. Z in B(X,Y)Z + B(X,Z)Y Doing the permutation shows B(Y,X)Z + B(Y,Z)X = -B(X,Y)Z + B(Y,Z)X e l.LM • .. p Therefore 3B(X,Y)Z = (2B(X,Y)Z + B(Y,Z)X) - (-B(X,Y)Z + B(Y,Z)X) € T.LM p· This finishes the proof. Examples. (1) M is called a hypersurface of M if the codimension .. of M in M is one. If p is a point of M then there is a geodesic of ~ p and perpendicular to M. passing through By (3) of the last proposition this shows all hypersurfaces of any naturally reductive homogeneous spaces are symmetrically embedded. (2) T=0 Let M be a space of constant curvature and K. Then by definition B = R is given by B(X,Y)Z = K((Z,Y)X- (Z,X)Y). Thus From this it is easy to check that every submanifold of M is symmetrically embedded. (3) Let M be a complex analytic manifold of constant holomorphic curvature. Then calculations that will be done later show that every complex submanifold of M is symmetrically embedded. 58 Other examples will ge given later. Convention 6.3. From now on M will be assumed to be a symmetrically embedded submanifold of M. -D and v have the same geodesics and therefore the Recall that same exponential map. The common exponential map for these two connections wi 11 be denoted by exp. used to study the. image of Definition 6.4. (2) (1) For U e S.LM The fo 11 owing notation wi 11 be under the exponent i a1 map. :s.t M Let 1T: S.tM.-. M be the. bundle projection. let g(t;U) = expp(tU). (3) Set U(t) = g'(t;U). (4) J(t;U) = D-parallel translate of ™nu along g(•;U) (5) h(t;U) = Orthogonal complement of the span of U(t) to g(t;U). and J(t;U) in TMg(t;u)· Proposition 6.5. Let IRU(t) be the span of the vector TM g( t ; U) . Then each of J ( t ; U) , h ( t ; U) g(·;U) and TMg(t;U) Also ;r(t;U), h(t;U) and IRU(t) i s par a11 e1 a 1ong are all stable under all three of the and Ru(t)• The field of spaces J(t;U) definition. IR U( t ) in is the orthogonal direct sum of these spaces. linear maps TU(t)' Bu(t) Proof. and U(t) The spaces IR(t) is D-parallel along are D-paral lel along g(·;U) g( • ;U), g(·;U) is a geodesic and U(t) is its tangent vector. h(t;U) is also D-parallel along g(·;U), because Therefore as it is the orthogonal complement of D-parallel spaces and D is metric preserving. by 59 Because ;r(t;U), h(t;U) g(·;U) to show ™g(t;U) and IRU(t) are D-parallel along is the orthogonal sum of the three it is enough to show for a particular value of t. At t = 0 this is easily checked. Since we are assuming M is symmetrically embedded in M it follows from proposition 6.2 that both ;r(O;U) = TM p and Ti.M1Tu= (h(O;U)<±)lRU(O)) are stable under both Bu and Tu. But B'u(u) ='B(u,u)u = o, Tu(u) ~ r(u,u) = o. Therefore is also stable under both Bu and IRU(O) 1u. But ~U is r.J symmetric and Tu is skew-symmetric. Therefore the orthogonal complement of U which is h(O;U), both Bu and Tu. This shows ;r(t;U). h(t;U) under BU(t) and 1U(t) when t = 0. D-parallel along g(•;U) are stable under Bu(t) TMg(t;U) ";< "?< 1 ....., completes the proof. Definition 6.6. Let and IRU(t) it follows that ;r(t;U), h(t;U) and TU(t)• 2 are stable But as all of these are and IRU(t) That the three subspaces of in question are stable under Ru(t) KU(t) = oU(t) - 2(TU(t)) is also stable under follows from the equation which is given in proposition 5.18. This 60 Ru(t) = Ru(t)I ~t(t> = 1u(t)I Bt(t) = ~U(t)I R-U(t) = Ru(t)I ;r(t;U) h(t;U) h(t;U) h(t;U) , , ' . Then define linear maps S(t;U): ;r(t;U)-+ ;r(t;U), c( t; u) : J ( t; u) ... J ( t; u), si(t;U): h(t;U)-+ h(t;U), as the unique solutions to the initial value problems: (1) 2 (;U(t» S(t;U) + Ru(t) S(t;U} = 0 S(O;U) = o, (~U(t)S)(O;U) = (id)ITM, p 2 (9'u(t)) C(t;U) + Ru(t) C(t;U) = 0 C(O,U) = (id)TM ,(;U(t)C)(O;U) = 0, p 2 (9u(t)) Sl.(t;U) + Ru(t) 'S 1 (t;U) = 0 Sl.(O;U) = 0, (~U(t)S 1 )(0;U) = (id) lh(O;U)" By proposition 5.19 these can also be defined by (1 ') 2 (DU(t)) S(t;U) + Tu(t)(DU(t)S)(t;U) + Bu(t) S(t;U) = 0 s(o;u) = o, (Du(t)s)(o;u) = (id)TM , (DU(t/ C(t;U) + Tu(t){DU(tf)(tlU) + Bu(t) C(t;U) = 0 C(O;U) = (id)TM ,(DuC)(t;U) = -} Tu(O), p 61 (Du(t)) 2 1 s1 (t;U) + Tt(t)(Du(t)'S.1.)(t;u) + st(t) S.1.(t;U) = o S.1.(0;U) = O; Remarks. g(·;U) 1 (Du(t)S )(0;U) = (id)h(O;u)· (1) if t ,_. X(t) is any D-parallel vector field along with X(O) in J(O;U) then both t 1-+ C(t;U)X(t) are Jacobi fields along S(t;U)X(t) C(t;U)X(t) and t...,. S(t;U)X(t) g(•;U). In this case are both in J(t;U) similar statement · is true for tr+ S1(t;U)X(t) D-parallel vector field along g( •;U) with and for all when X(O) A t. X(t) is a in h(O;U). These facts fol low directly from the definitions. (2) C and 'S1 If the differential equations defining S, are written with · respect to 0-parallel fields, then the differential equations in (1 ') have constant coefficients. Definitibn 6.7. For each number r, define a map fr: 'sJ. M _. M by The image M(r) by fr is the tube of radius We now compute the derivative of then the fibre 5.L M. fr· r If about M. U e SJ.M and p = nU, SJ.MP is an embedded submanifold of the total space Th us, the tangent T(SJ.Mp) U to the fib re can be viewed as a subspace of the tangent space T(SJ.M)u to SJ. M. But the sphere 5J. Mp is al so embedded in the vector space 'TJ. MP as the set of all vectors of unit length. Therefore, the tangent space T(SJ.Mp)U to the fibre can be identified with the set of all vectors in ™p orthogonal to can be identified U. But this is h(O;U). Thus h(O;U) that are 62 with a subspace T(SiM)u. check that h(O;U) Under this identification it is easy to is the kernel of Lemma 6.8. Consider h(O;U) as a subspace of . T(SiM)u as above. Xe h(O;U) and X(t) g(•;U) to g(t;U) Proof. Without loss of generality we may assume If is the D-parallel translation of X along then X is a unit vector. Then define a curve by c(s) = cos(s) U + sin(s) X. Because U and X are orthogonal vectors, this is a curve from the reals to siM. Clearly C(O) = U and (fr*) U X = Js C'(O) = X. Therefore, Is=O fr (c(s)) d j = d"S:t . exp ( re ( s ) ) , s=O -· p where p = TIU. Define a(s,t) by a(s,t) = expp(tc(s)). Then f r(c(s)) = a(s,r) so that (fr)*U X = ~0,r). Clearly (see definition 3.26) a(s,t) a(O,t) = expp(tU) = g(t;U) through geodesics. 2.37, the vector field ~~(O,t) si(t;U)X(t) is a variation of along g(·;U) is also a Jacobi field along lemma it is enough to prove . ~~(O,t) initial conditions at t = 0. Thus, by proposition is a Jacobi field. g(•;U). But Thus, to prove the and ·s i(t;U)X(t) have the same (See proposition 2.37). We now compute 63 ~as(0,0) = _Q_I <JS s=O u exp (0) p =0 ~ The covariant derivation oa has no torsion and the vector fields (Ja os , ot commute thus, This yields (~u(t ) ~ ~ )( 0 '0 ) = = (~oa ~~)(O ,0) oS - ox ot /"-1 - 'Vaa oS s=O expp(tc(s)) t=O c(s) = 'Voa 05 s=O = c'(O) =x = ~U(t) t=O si(t;U)X(t). This finishes the proof. Definition 6.9. Let A be the Wiengarten map of M in ~ definition 4.1). Then, for each U e S.LM, let A(t;U) D-parallel translate of A(U) A(t;U) along g{·;U) is a linear transformation on J(t;U). to g(t;ur (see be the Therefore 64 A Lemma 6.10. Let A Xe T(S.LM)u, X = 1T*UX, D-parallel translate of X along and g(·;U) . to X(t) be the g(t;U). Then A (fr)*U X = (C(r;U)+S(r;U)A(r;U)) X(r) + (an element of h(r;U)). s : (-E:, s) _. 5.L M Proof. Choose a smooth curve from some neighborhood (-E:,E:) of 0 such that g(O) = U and g'(O) = X. A c =TIO s· Set p = nU and Then 'c: (-E:,E:) _. M is a smooth curve with c(O) = p and Define a(s,t) = expc(s)(tg(s)). Then the last equation can be written as But, as in the last lemma, a(s,t) through geodesics and thus ~~(0,t) is a variation of a(O,t) = g(t;U) is a Jacobi field along g(t;U). We now find its initial conditions. ~0,0) = ~sls=Oexpc(s)(O) = _Q_I c(s) 05 s=O = c• ( O) = x. The curve T.LM along c. s can be viewed as a section of s.t M and, thus, of 65 Therefore, ';aa g(s) = ;'c'(O) g(s) 0sls=O =,;'xg(s) = A(g (0) )X +(element of h(O;U)) = A(U)X + (element of h(O;U)). This yields, (vu( t)~) (o ,o) = Voa : t / - exp c ( s) (ts ( s)) -1 t-0 as s=O = ~a ~'s=O g(s) = A(U)X + (element of h(O;U)). rJ Where, as in the last lemma, we have used the facts that v is without torsion and that ~~ and ~~ commute . Let J(t) be the vector field along g(·;U) defined by J(t) = (C(t;U) + S(t;U)A(t;U))X(t). Then J is a Jacobi field and from the definitions of C and J(O) = X(O) = X, (~UJ)(O) = A(O;U)X(O) = A(U)X. Thus, if Y(t) = ~0,t) - J(t), g(· ;U) with Y(O) = 0 and g then Y is a Jacobi field along (~UY)(O) € h(O;U). Hence, Y(t) is in 66 h(t;U) for all t. in terms of ~~' "" This, together with the expression for (f r)*UX completes the proof. its volume form 1 as a unit sphere in T.LMP. o ·s.tM p If OM is the volume form on M and n : s.L M-+ M is the bundle projection, then a vo 1 ume form o SLM is defined on s.L M by p = nU. where We choose the orientations so that Fubini 's theorem holds with the following choice of signs where f is any compactly supported continuous function. Proposition 6.11. Let r e IR and U € ;S.L M. Assume det(C(r;U)+ S(r;U)A(r;U))det(S1 (r;U)) 1 O. (*) Then (fr)*U . is injective, and thus U in S.LM K(r) at into a hypersurface f r(U) K(r) of f r(U) = J(r;U) EB h(r;U). is the volume element on K(r), then f* OK( ) = det(C(r;U)+ S(r;U)A(r;U))det(S1 (f;U))n (U). r r S.LM Proof. Let x1 , ... , Xm-l be an oriented orthonormal basis of T(S.LM )U = h(O;U) (with p = nU) p orthonormal basis of T(S.LM)u. and Y1 , ... , Yn be an oriented "" "" of Choose elements Y1 , ... , Yn ' K of M. The tangent space to is T(K(r)) If OK(r) fr maps some neighborhook 67 A T(S.LM)u with n*Uyi =Y; i =l, ... , n. Bythelasttwolemrnas (fr)*U X; = s.t(r;U)X;(r) (fr)* Yj = (C(r;U)+S(r;U)A(r;U))Y/r) + Zj where Xi(t) and Yj(t) and Zj is the parallel field along g(·;U) with is the parallel field along g(·;U) is an element of h(r;U). that ·s.t(r;U)Xi(r) Xi(O) = x0 ; with Yj(O) = Yj The condition (*) easily implies for l < /\ < m - 1 is a basis of h(r;U), and that (C(r;U) + S(r;U)A(r;U))Yj(r) is a basis of J(r;U). Therefore (fr)*Uxi' (fr)*Uyj is a basis of J(r;U) e;> h(r;U). 1 < i < m - 1, This proves (f r)*U is injective. The statements that U has a neighborhood hypersurface K(r) 1 '$.. j < n K mapped into a of M and that the tangent space to this hypersurface is as claimed now follow from the implicit function theorem. It is now easy to check that /\ A (fr)*Uxl /\ ••• A(fr)*Uxm-1 A (fr)*Uyl A ···A (fr)*UYN = de t (C( r ; U) + S ( r ; U)A( r ; U) ) de t ( A si (r ; U) ) X1 ( r) /\ • • •/\ Xm- 1( r) /\ Y1( r )/\ • • •/\ Yn( r) . A x1 /\ • • • /\ X 1 A v1 /\ • • • /\ Y is dual to o (that is, mn S.LM * o .LM(x 1 , ... ,Xm-l'Yl, . . . ,Yn)= l) the given formula for fr oK(r) holds. 5 This completes the proof. But as A Corollary 6.12. A If the condition the volume of the tube M(r) (*) holds for all of radius U in ·s.L M, then r about M is 68 vol(M(r)) = J det(C(r;U)+ S(r;U)A(r;U))det(S.L(r;U))o S.LM Proof. (U). S.LM Clear from the last proposition. Convention 6.13. From here on,the volume of the tube M(r) will be defined by the formula of the last corollary, even when the condition (*)of proposition 6.11 does not hold. The followin§ result restates what we said above without having to compute any parallel translations. Theorem 6.14. of Let M· be a compact symmetrically embedded submanifold M with smooth boundary. For each U in 5.L M, set Define linear maps s(t; u) ' c(t; u) : ™p _. ™p' and by the initial value problems S"(t;U) + TuS'(t;U) + BuS(t;U) = 0 S(O;U) = 0, S'(O;U) = (id)TM , p p = nU, and, 69 C (t;U) + TuC'(t;U) + BuC(t;U) = 0 11 C(O;U) = (id)TM , C'(O;U) = p cs.L) 11 ( -l TU' t ; u) + tt cs.L) ' ( t; u) + st (s.L )( t ; u) = o S.t(O;U) = 0, (~.t)(O;U) = (id) T.LM . p h : Mx IR -. IR be the function defined by Let h(p,t) = t J's.tM . det(C(t;U)+ S(t;U)A(U))det(S.t(t;U))O .tMp (U), S p A is the Weingarten map of · M in · M. where tube M(r) of radius Then the volume of the r about M is VO 1( M( r) ) = J h( p 'r )OM ( p) . M Remarks. (1) The derivatives, denoted as primes, are to be taken in the usual sense of a function from the real numbers to a finite dimensional real vector space. It should be noted that Tt ~ Tt(o) (2) T.LMP, as rt has as its domain while ·Tt(o) has h(O;U) for its domain. Proof of the theorem. Let r(t;U) be 0-parallel translation along g(·;U) from p to g(t;U). Then, because TU(t) and Bu(t) D-parallel along g(t;U), we have S(t;U) = r(t;U)-l I (t·U)~(t;U)r(t;U) ITM , J ' p C(t;U) = r(t;u-l I (t·U)C(t;U)r(t;U) I™ , J ' p 1 A(U) = r(t;ur IJ(p;U)A(t;U)r(t;U) l™p' Therefore, det(C(t;U) + S(t;U)A(U)) = det(C(t;U) + S(t;U)A(t;U)), are 70 and likewise, However, we have T.l Mp = h(O ;U) ~ lRU. compute S.L(t;U) X"(t) = O; on Therefore it only remains to mu. Let X(t) = tu. Then X'(t) = U and also Tu(X'(t)) = O as Tu(U) = o, and ~Bt(x(t)) = O as ·st(u) = O. Thus, X(t) is a solution to x t) + rt (x t) ) + Bu ( x(t) ) = 0 II ( f ( X(O) = 0, X'(O) = U. But s.L(t;U)U is also a solution to this initial value problem. Therefore S.l(t;U)U = tU. Using this with what we know about ~.L(t;U) lh(O(U) yields det(SJ.(t;U)) = det(SJ.(t;U) lh(O;U)) det('SJ.(t;U) !mu) = de t ts.t (t ; u) )t , whence det(C(t;U)+ S(t;U)A(t;U)) det(S.l(t;U)) l = det(C(t;U)+ S(t;U)A(U)) det(S.t(t;U). The result now follows from corollary 6.12 or convention 6.13 and Fubini 1 s theorem. 71 7. Some multilinear algebra. The results of this section are inessential variants of the algebraic results in [5]. written in Flanders as What is here written as "AB". "A* 8 11 is His definition of A* B differs from that given here; instead, he uses proposition 7.2 as its definition. If W is a real vector space, then of all linear endomorphisms of W. end(W) Throughout this section be an n-dimensional real vector space, /\k(V) exterior power of V and S£ (1 , ... ,£]. lf a will be the algebra V will will be the k-th is the group of all permutations of is a permutation, then (-1) 0 will denote the sign of cr. Definition7.l. If Aeend(Aa(V)) and Beend(Ab(V)) then let A*B be the endomorphism of Aa+b (V) defined on decomposable elements by ( A*B Hx1"· • ·Ax a+b ) If a is a real valued alternating b-form on v, then a can be viewed as a linear functional on Ak(V) by a(x1A•••/\xk) = a(x,, ... ,Xk). Conversely it is clear that every linear functional on /\ k(V) is of Let e 1 , ... , en be a basis of V. Then e. /\ • •• /\ e. , where i 1 , ... , ik range over all k-tubles of positive this form, for some a. 11 1k integers with 1 < i 1 < ••• < i k <- n is a basis of /\k(V). Therefore, our remarks about linear functionals tell us that every element of end(Ak(V)) can be written as 72 where each ai ···; l means is a real valued alternating k-form on V. This k A is given on decomposable elements by 6 a. ,. (x1,···'xk)e.A •••A . < • • • <"l k 11 ... k 11 11 A(x1A···Axk) = Let e1 , ... , en be a basis of V and Proposition 7.2. 6 A= a. i l < • • •< i a 1 . e. 1 1 1· · · a A ···A e. 1 1 e end(Aa(V)) a Then Proof. x1, ... , Xa+b e V. Let 1 (A*B )( Xl'" • • ""Xa+b) = a ! b ! ~s (- 1 )cr A(xcr ( l )"···A\,-( a) )AB (xcr (a+ l )" • .."Xcr ( a+b )) L cr j 6 1 < • • • <j Then, SJ· J. (x (a+l)' ... 'X (a+b))eJ. A···AeJ. ) l · · · b cr cr l b b 1 =. 6 . (a! b! 6 (-l)cro:i ... i (Xcr(l)' ... 'Xcr(a))Sj ... jb(Xcr(a+l)'"""'Xa(a+b))) 1 1 < • • • <1 cre 5L -. 1 a . .1 .a J < •• •<J l b e.1 1 "···A e.1 · /\ e. A •••A 6 = 1· <···<1· .1 .a J < •• •<J 1 J1 a b (a.1 . "s . 1··· 1 a . )(x1· , ... ,X a+b) e.1 1 " • • ·" e.1 a " eJ. 1 A • ··A eJ. b . J, ... Jb 73 This finishes the proof. Proposition 7.3. Let A€ end(/\a(V)), B € end(/\b(V)) C € end(/\c(V)). Then the map and (A,B) HA* B is bilinear, and A*B=B*A (A*B)*C = A*(B*C). Proof. That A* B is a bilinear function of (A,B) is clear. prove the other two statements, we use the last proposition. To Let e1 , ... ,en be a basis of V and A = 6 a. . e. /\ ••• /\ e. . . 11···1 11 la . 11<· ··<ia a ... " Then (A*B) = = ( -1 ) ab ( 1 ) ab - 6 i <···<i ~. .a J < •• •<J .1 1 J, ... Jb. " a.1 1 ··· 1. aeJ1. " • • • /\ eJb. /\ eJ1. " • • •/\ e.1 a . b = (B*A). The associativity of * follows from proposition 7.2 and the associativity of /\ by a similar calculation. Recall that if A e end(V), endomorphism of /\k(V) then "k(A) This completes the proof. is the 1inear given on decomposable elements by 74 Definition 7.4. If A € end(Aa(V)) then define A*k = A* A * • • • * A Proposition 7.5. A*k € end(Aak(V)) ( k factors). A, B, A , ... , Ake end(V), If by then 1 (A1*···*Ak)(X1A···Axk) = 11~\{-l)crAlXa{l) A ••• A AkXcr(k) (1) = cr~\Acr ( 1 )X1 A • • • A Acr ( k )X k ; (2) A*k = k! Ak(A); ( 3) Ak( A+ B) = ~ Aj (A) * Ak- j ( B) , (where A0 (A) = 1) j=O n . . det(A+B) = 6AJ(A)*An-J(B); j=O k (BA l ) *(BA 2 ) * ••• * (BA k ) = /\ (B) o (A *·· ·*Ak), 1 · (4) (A 1B)*(A 2B) * ••• * (AkB) = (Ai*···*Ak)oAk(B). Proof. To show (1) we use induction. of pennutations on [a , ... ,ak}· 1 Let perm(a , ... ,ak) 1 be the group Assume (l) holds for (k-1). Then (A 1*· • ·*Ak Hx 1A • • ·Axk) = ( (A 1*· • ·*Ak-l) * AkHx 1A • • ·Axk) = (k~lj ft~ (-l}cr(A1*···*Ak-l )(Xa(l)"···Axcr(k-l))A AkXcr(k) crE\ = 1 ( )cr ( ) e( A· .."A x ( ))A A x ( ) ( k-1) ! ~S -1 6 -1 AlXecr ( l) k-1 ecr k-1 k cr k cr k eeperm(cr(l),~ •. ,cr(k-1)) (k-1)! = (k-1}! p P~\ (-1} A]Xp(l} A ••• A AkXp(k) 75 The second line of (1) follows from the first by a change of Now (2) follows from (1) by letting A1 = A2 = ••• = Ak =A. For (3) we remark that * is commutive and associative, so that variable. (A+ B)*k can be expanded by the binomial theorem. = l k k! A*j * s*k-j rr},? j!(k-j)! 0 = {; (~ A*j) * ( = {; Aj(A) * Ak-j(B). j=O J. 1 (k-j)! B*(k-j)) j=O The second line of (3) follows from the first and that det(A) = An(A). To prove (4) we use (1). (BA 1) *. • • • * ( BA k ) (x l A • • ·AX k ) = "k (B) (er~\ (-1 )crAl'Xcr (1 )" •• ·AAk\r ( k)) = Ak( B) o (Al * .. •*A k)( Xl A• •• AX k) . The second line of (4) follows by a similar calculation. This completes the proof. Remark. It follows from (3) that Ak(A) * An-k(I) is crk(A), k-th elementary symmetric function in the eigenvalues of A. this, note that (3) implies the To see 76 det(x I+ A) = 6 crk (A) x n-k k =6 /\ k(A) * /\ n-k(I) Xn--k. k Definition 7.6. Let e 1 , ... ,en be a basis of V and A e end(Ak(V)). Then the components j 1 ••• j k A., , ••• 1.k of A in the basis e1 , ... ,en are defined by j, ... j~ . . e . /\ . • • • /\ e . . . J' , ,- .... .•<JJk -· . • . l l : •• l k· J 1 Jk 1 :0 Ae . /\ • • • /\ e 1. · - -k1 l1 j 1 •••.j k where A. . 11 ... l k of .. k . A• . is an alternating function of i 1 , ••• , ik and also j 1 ' ... ' j k. If we restrict ourselves to increasing sequences 1 < i 1 < i 2 < • • • < i k ~ n and 1 ~ j 1 < • • • < j k ~ n, components of A in the basis e1 ' ... ' en of then the v are components of the matrix of A in the basis [ e. /\ • • •/\e. } of /\ k (V). , , l k It follows that tr(A) = 6 i <•••<i 1 k i 1 •.• i k A. . 1 1··· 1 k il ... ik 1 = -k.1 . 6 . A.,, ... lk . 11 ... 1 k i, ... ik We will write 0 . . J 1 ..• J k k for the component of /\(I), map on /\k(V). The components of /\k(I) of basis of V. It is easy to check that 0 . the identity are the same for any choice i 1 ••. i k . J 1 ••. J k vanishes unless i 1 , ... , ik are all distinct and the sets [i 1 , ... ,ik} are the same. In this case, its value is the sign of the permutation taking each i 1 for 1 ~ 1 ~ k. 77 If A e end(Aa(V)), Proposition 7.7. then A* An-a(!) = tr(A). jl · · · j a Let A. Proof. V. of . 11 .•• 1 a be the components of A in the basis e , ... , en 1 Then, A*An-a(I)(el'\···Aen) = a!(n~a)! a~S (-l)crA(ecr(l)l\ ...l\ea(a» n /\ ecr(a+l) /\ ···A ecr(n) 1 ;, ... ia (-l)o". 6. A cr(l)·. ·cr(O) e. A .. ·A e. A e ( +l) cr es n 11 ... , a ,1 , a cr a = a!a!(n-a)! , 6 A • • • /\ ecr(n) = 1 "\'"'\ (- 1) 0 a!a!(n-a)! cr~es n "\'"'\ ~ [ i 1 , ... 'i al ;, ..• ia A e A • • • ''" e i A e (a+l) cr (1) ... cr (a) i 1 a cr = [ cr ( l) , ... ,cr (a)} A • • • /\ ecr(n) = a! cr cr(l)···cr(a) A···Ae Ae i( )' 6 (-1) A e (l) cr(a) cr(a+l) a ( 1 ) ... cr ( a) cr a . a . n- a . cr es 1 n A • • • /\ ecr(n) cr(l)···cr(a) 1 6 A e 1 /\ = a ! ( n-a ) ! cr es cr ( 1 ) · · · cr ( a) ••• A e n 1 - aT ;, ... ia . e 1 A ••• /\ e n 1 ; ... ia 1··· a 1 6 A.1 = tr(A) e 1 A • •• /\ en. n 78 We now relate our formulas to those in the literature. Proposition 7.8. (1) Let He end(A 2(V)). H*k(x,A···Ax2k) = Then -t-2 creSk 6 (-l)crH(x (l)"X (2)) cr er A ••• A H(xcr(2k- l) /\ Xcr(2k)) (2) If Hij k,e are the components of H in the basis e1 , ... ,en of V then H*k *A n-2k( I) = tr(H*k) = -1 k 4 i l ... i 2k o. 6 . j 1j2 j 3j 4 H. . H. . 1 1 1 1 ~i···~2k J1···J2k 1 2 3 4 J 1 ... J 2k Proof. We show (1) by induction. 1 = (2k-2)!2! creS E = 2k (-l)cr H*(k-l)(xcr\'1)" 0 • "/\Xcr (2k-2))/\ H(xcr (2k-l)AXcr (2k)) 1 ' ' E ( - l f _l_ E ( - 1) eH(Xecr ( 1 ) " Xecr ( 2 ) ) (2k- 2 )· 2 · cr€S2k 2k-leeperm(cr(l), ... ,cr(2k-2) This proves (1). H*k. To prove (2), we use (1) to find the components of 79 *k e.1 /\ ••• /\ e.1 1 2k H 6 (-l)crH(ecr(i1)" ecr(i2))" ··•" H(ecr(izk~l )" ecr(i2k» = ;k a€perm( i 1 , ... 1" 2k) 1 = -k 2 jl ... j2k 6·1 . H(e. /\ e. ) /\ ••• /\ H(e. "e. ) j,, ... ,j2k 1··· 1 2k J1 J2 J2k-l J2k 6 Therefore so by the last proposition 1 = (2k) l 1. 6 *k i, ... i2k ) i 1 ... i 2k . (H 1 ' ... 'l 2k This completes the proof. 80 Proposition 7.9. Consider mm with its standard inner product and 1et A be a linear map from IRm into end(V). Then for any orthonormal basis e 1 , .•• ,em of .IRm define Then H is independent of the choice of orthonormal basis, and for k odd and vol (Sm-l) H*k. Jsm-1 /\ 2k( A( u)) 0 s m- 1( u) = k ! m( m+ 2 ) · · · ( m+ 2k- 2 ) Proof. The independence of H from the choice of orthonormal basis k = l. follows from the second integral formula with the 1eft side is independent of the basis. 1 This is because he fffs t i ntegra 1 formula is clear, as Ak(A(u)) is an odd function of u and the integral of an odd function over the sphere Sm-1 is zero. To prove the second integral formula we need; Lemma. If (a 1 , ..• ,am) a= is a multi-index (that is each aj is a nonnegative integer) then J u2a 0 m-l(u) = sm-1 Here s 2a = u2a 1 u2a 2 u 1 2 (2a)! vol(Sm-1). m(m+2) · · · (m+2lal-2)2 la~ 2a u m m (where u = ( u1 , u2, ... , um) ) and a! = a1 .I a2.I ..• am! lal = a 1 + a2 + ' ... + am. 81 2a Proof. S X lRm If e-x~ o ( ) is i.ntegrated in polar coordinated IRm X (and re ca 11 i ng that J ta e co -t 2 1 dt = 2 r +l cy) ) we f i nd 0 J X2a e-x -X o m(x) IRm IR ={ J 0 5 = (ru) 2a e - r2 0 m- l ( u) rm- l dr s m-1 2 Joo r21aI+m-1 e- r 2 dr J 5m-1 0 u a o m- 1 ( u) s l r ( I I + ~) Jm-l 2 os 1 ( u) = .}_ m(m+2) · · ·(m+2lal-2) _r(m) J = a u a m- 5 2 2lal 2 rn-1 u2a 0 s (u). sm-1 But this integral can also be computed using Fubini 's theorem: 2a -x ·x J x e m co 2a. 1=1 -co dx = IRm r( i =l m = TI ( . 1 i= 1 2 e-t dt 2a. -1 m J x IRm = .TI J t = TI 2a1 ~ ) ( 2a.) ! 1 ) Cl. 4 1 . Cl.! 1 1 r(-)) 2 1 2am · · · xm 2 -x 1 e 2 -xm ••• e dx 1 • • • dx m 82 By equating these two expressions for J m IR x 2a e-x )( o lRm (x), when a= (0, ... ,0) we see that Now, for any a, equate the two expressions for the integral and use the formula for vol(Sm-l) to finish the proof of the lemma. We now finish the proof of proposition 7.9. Using the multi-index notation of the lemma, the multinomial theorem can be written as ! l (x1+ • • ·+xm) = al a2 6 g_ x x lal=la! am • · · xm · 1 2 Now let e 1, •.. , em be an orthonormal basis of mm. We then write 2 1. elements of sm-l as where u2 + ···+um= 1 Let u = (u 1 , ... ,um); then the multinomial theorem and 7.5 (2) imply J A2k(A(v))o m-l(v) sm-1 s 83 and any 13. J is odd, then f m-1 u13 osm- l (u) = 0 s by summetry. f 5 Using this fact and the lemma yields A2k(A(v))o m-l(v) s m- l = = (2a)! vol(Sm-l) *2a1 *2a A(e ) * • • • * A(e ) m 1 Ia I= k( 2a) ! m( m+ 2) · · · ( m+ 2 Ia 1-2 ) 2 Ia Ia ! m -- k' *2 *2a Vo 1(· sm~1 ) k 6 - j A( e1) al * · · · * A( e ) m m m( m+ 2 ) · · · ( m+ 2k- 2 ) 2 k! Ia I=k a · _ - 6 1 vol(Sm-l) k (A(e 1 ) m(m+2)·· ·(m+2k-2)2 k! vol(Sm-l) m(m+2)···(m+2k-2)2 k! k 2 = ---~----=---=--k- 2 (A _ vol(Sm-l) H*k - m(m+2)···(m+2k-2)k! This finishes the proof. *2 +••• +A(em) *2*k ) 2 *k (A( e1 )) + ···+A (A( em))) 84 8. The tube formula. In this section the algebraic results of the last section are used to restate theorem 6.14. Theorem 8.1. Let M, ~' C(t;U), S(t;U), si(t;U) For each with 0 < k < n, k hk{p,t) = be as in theorem 6.14. define hk: M x IR_. 1R by t JSLM /\k(S(t;U)A(U))*/\n-k(C(t;U))det(si(t;U))os.LM(U). p p Then, the volume of the tube M(r) n vol(M(r)) = 6 of radius r about M is J hk(p,r)oM(p). k=O M Proof. By theorem 6.14 vol(M(r)) = J h(p,r)oM{p) M where h(p,t) = i J det(C(t;U)+S(t;U)A(U))det(S.L(t;U))o, Mp (U) .L S.LM S p = In this computation we have used proposition 7.5 (3) to expand det(C(t;U) + S(t;U)A(U)). Remarks. (1) We can use the formula /\ k(S 0 A) =/\ k(S)o/\ k(A) to rewrite the formula for hk(p,t) as 85 =if· .t (Ak(S(t;U)) Ak(A(U))*An-k(C(t;U))det(:S"t(t;U))O .t (U). · SM hk(p,t) 0 S Mp p This shows that hk(p,t) is a linear function of the map U14 Ak(A(U)). Both S(t,.U) vanish to order one at t = 0. (2) and is.t(t;U) for any U, Ak(S(t;U)A(U)) vanishes to order at least vanishes to order m at t = 0. and det(S.L(t;U)) easy to see that · hk{p,t) k at Thus, t = 0, Therefore, it is vanishes to order at least m + k - 1, for al 1 p in M. · The above formula becomes simpler if M is a symmetric space. Theorem 8.2. If 1V1 is an oriented symmetric space, and M is a compact symmetrically embedded submanifold of 1V1 with smooth boundary, then, for each U e s.t M let ,...,, where R is the curvature tensor of the Riemannian connection of M. Define C(t;UL S(t;U): TMP (p =TIU), -+ TMP 5.t ( t ; U) : t.tM p -> ™p, by S11 (t;U) + RuS(t;U) =: -0 s(0; u) = 0' s 0 ; u) = (; d) TM ' C (t;U) + RuC(t;U) = 0 C(O,U) = (id)TM , C (0;U) = 0, II ( p 11 11 p (S.t)"(t;U) + RuS(t;U) = 0 s.t(O;U) = 0, (S.t)"(O;U) =(id) T.tM . p 86 Then the volume of the tube M(r) r about M is is the greatest integer in ~· where Proof. of radius T = 0 and B = R by proposition 5.21. In a symmetric space, Therefore, by 8.1, it is enough to show that if hk(p,t) vanishes for M is symmetric, then k odd. Note that R (X) = R(X,-U)(-U) -U This shows Ru = R( -U) and R(-u) = H(-u). It then fo 11 ows from the defining equations of C(t;U), S(t;U) and :s i(t;U) are even functions of U. But A(U) is a linear function of U and thus an odd function of U. Thus, is an odd function of U for over the sphere s~MP is zero. k and finishes the proof. k odd. that all three The integral of an odd function This shows hk(p,t) vanishes for odd 87 9. Parallel hypersurfaces. In this section we will use the notation of section 6 with the ~ extra condition that m = 1. Then M is a hypersurface of M. assume that M is compact and oriented with smooth boundary. We If this is the case, it is possible to choose a smooth unit normal field U a1ong M. For each p in M the vector space T.LM p is one-di mens i ona 1 and therefore contains exactly two vectors of unit length. Therefore, S.tMP = ( U( p), - U(p)J. Define ~he parallel hypersurface at a distance r from M by PM(r) = (expp(rU(p)): p e M}. It is then clear that the tube M(r) the union of PM(r) Proposition 9.1. of radius r about M is and PM(-r). With notation as in 8.1, n + vol(.PM(r)) = E J hk(p,r)oM(p), k=O M where h;(p,r) = /\k(S(r;U(p))A(U(p))) * /\n-k(C(r;U(p))). Proof. If M is an oriented hypersurface then S.LM is the disjoint union of + S M = [ U( p) : p e M} and S-M = (-U(p): p e MJ. If S.LM + is replaced by SM, then all the results of section 6 go through as before, except that we will be computing the volume of PM(r) 88 rather than M(r). TU(p) (U(p)) The same holds true of theorem 8.1. and st(p) (U(p)) Because both vanish_, the initial value problem defining ·sJ.{t;U{p)) becomes (SJ. ) II ( t ; u( p) ) : 0 s.t(O;U(p)) = 0, (S.L)'(O;U(p}) =(id). Therefore SJ.(t;U(p))U(p) = tU{p}. This yields det(SJ.( t;U{_p))) = t. Because S+MP has only the one point U{p}, is just evaluation at this point. integration over S+Mp Theorem 8.1 now yields n vol(PM(.r)) = k~O J f k.(p,r)oM(p}, M where . fk(p,t) = tJ t\k(S(t;U)A(U))* t\n-k(C(t;U))det(S.1.(t;U))n .L (U) S+Mp S Mp = t\k(S(r;U(p))A(U(p})) * t\n-k(C(r;U(p))) This completes the proof. Remark. - In the case M is the Euclidean space of dimension n + 1, then both and S(t;U) B and T vanish. Using this in the definitions of c(t;U} shows C(t;U(p}} = (id)TM p S(t;U(p}) = t(id)TM p Whence h~(p,t} = Skt\k(A(U(p))) * t\n-k(_I) = sktr(t\k(A(U(P))) = skcrk(A(U(p)} 89 where crk(A(U(p))) is the k-th elementary symmetric function in the eigenvalues of A(U(p)). of 7.5. This follows from the remark after the proof This yields n vol(PM(r)) = 6 rk J crk(A(U(p)))oM(p), k=O M a formula due to Steiner, [11]. If M is a hypersurface of the symmetric space M Proposition 9.2. then the volume of M(r), the tube of radius r about M, is where Here H is the excess tensor of M in M defined in definition 4.4. This shows each h2k is a linear function of H*k and that vol(M(r)) only depends on the excess tensor of M in M, but is otherwise independent of the embedding of M in Proof. M. By theorem 8.2, where h2k(p,t) = ~ J "2k(S(t;U)A(U))*An- 2k(C(t;U))det('s.t(t;U))n .l (U) s.tM S Mp p = /\ 2k(S(t;U(p))A(U{p)}) * /\n- 2k(C(t;U(p)) + /\ 2k(S(t;-U(p))A(-U(p))) * /\n- 2k(C(t;-U(p)) 2 2 2 = 2 (/\ k ( S( t ; U( p) ) ) o /\ k (A ( U( p) ) ) ) * /\ n - k ( C( t ; U( p) ) ) . 90 We have used the facts th.at det(S.L(t;U)) = t, that integration over S.LMp is the sum of the evaluations at U(p) and -U(p), S(t;U), C(t;U) and A2k(A(U)) are even functions of U. and that In the case at hand, the excess tensor is given by Set A= A(U(p)). Then, by proposition 7.5 (2) we have A 2k l *2k (A) = (2k) ! A = ~A*2)*k t2kJ!' = ~2A2(A))*k 2k *k = (2k) ! H • Putting this into the above formula for h2k yields the result. Remark. l"Y If M is not a symmetric space then it is easily seen from the differential equations defining C(t,U) not even functions of U. and S(t;U) that they are Therefore there is no reason to expect the last proposition to hold in any space other than a symmetric space. 91 10. An algebraic reformulation for symmetric spaces. Let M be an oriented symmetric space. M satisfying the two conditions of convention group of isometries of elements of G that fix o. o. M, Let o be the origin of 5.2. Let G be a transitive and H be the subgroup of all Let ffi be the tangent space to M at Then, as in proposition 5.8 and convention 5.9, we identify fh (the Lie algebra of G) so that ffi is invariant ~ with a subspace of under the adjoint action of H, and where h is the Lie algebra of G. then denote by •n..l Definition 10.1. II If 1h is a vector subspace of the Orthogona 1 Comp 1ement Of 1h m, in ffi • A second order germ of a manifold (or briefly a second order germ) is a pair (lh,A) where 1h is a vector subspace of 1h and A is 1i near map from lh.L to the symmetric 1 i near maps on 1h. The dirrension· of (m,A) is defined to be the dimension of ffi. linear map A is called the Weingarten map of (lh,A). germs (lh ,A ) 1 1 and The Two second order (lh 2 ,A 2 ) will be considered equivalent if and only if there is an element a in H so that for a 11 X in lhl ' .L • and Y in m 1 Definition 10.2. If M is a submanifold of second order germ (m,A) of M at Choose any element g in G with M and p e M, then the p will now be defined. g(p) = 0. A is the Weingarten map for the manifold Then 1h = T(gM) 0 , gM at o. It is clear and 92 g{p) = o give equivalent second that different choices of g with order genns in the sense of the last definition. Definition 10.3. Let (tn,A) be a second order germ, 1{ the curvature tensor of M at o viewed as a linear map on /\ 2T'M 2 P the orthogonal projection from /\ TM curvature tensor · R of (m,A) 0 0 onto /\ 2(rn). n L: /\2 (A ( e . )) i=l J where e1, ..• ,em is any orthonormal basis of H of (rn,A). is defined to be m 2 H = 6A (A(e.)) = PR - ~L. The excess tensor ~ j=l J Let M be a submanifold of tangent space at Then the is defined to be R = PR - ·Remark. and R M passing through o whose o is tn and whose Weingarten map at o is A. Then proposition 4.3 and definition 4.4 imply that the curvature of M at o, viewed as a linear map on /\ 2TM , is the same as the 0 curvature of the second order germ (tn,A). Definition 10.4. The second order germ (rn,A) X and U in It is easy to check that a submanifold M of M is symmetrically err/bedded if and only if, for all vector R(X,U)U is also in Remark. is said to be tn.l, the tn.l. symmetrically embedded if and only if its second order germ at each of its points is symmetrically embedded. Definition 10.5. germ. Let (rn,A) be a symmetrically embedded second order Define for a11 U € tn.l 1i near maps '1n, U: th -+ th, ~, U: tn.l -+ Ih.l 93 by Now def i ne sin ( t ; u) , Sn (t ; u) : rn ... rn and :s~ ( t ; u) : rn ... rn.L by the initial value problems: S~(t;U) + f1n·,uSrn(t;U) = 0 Sln(O;U) = 0, '1n, ucrn (t; u) = o ~) t ; 1.n + '\n ,us~ (t ; u) = o cm (o; u) = (id )rn, c~ (o; u) = o, s~ (o; u) = o, )·(o; u) = ( i d)m . ~ ( t; u) + 11 ( ( Proposition 10.6. M and (tn,A) cs~ = (id)rn' Let M be a symmetrically embedded submanifold of the second order germ of M at function h2k(p,t) h2k ( p , t) = tJ ·s.LM S~(O,U) p e M. Then the of theorem 8.2 can be computed by 2 11 k ( slh ( t ; u) A( u ) ) p * 11 n- 2k ( 1n {t ; u) ) de t ( s~ ( t ; u) )n (u) . . s-Tu 's-Tu is the unit sphere of tn.L. Here Proof. By definition there is a g e G with . g(p) = o and such that rn = T(gM) 0 and A is the Weingarten map of gM at o. Let A1 be the Weingarten map of M at p. ,...., M, we see for all U e ~iMP, Then, because g is an isometry of that g*PU e T.L(gM) 0 , and 94 Set P1 = g I *P TM and p = g I . It then follows from the initial 2 *P T.tM p p value problems defining the linear maps involved that, S~(t;P 2 U) = P1S(t;U)Pll C~(t;P 2 U) = P1C(T;U)Pll . S~(t;P 2 U) 2 detCS~(t;P U)) This shows 1 = P2s.t(t;U)P2 . = det(S.L(t;U)). We now use proposition 7.5 (4) to compute "2 k ( s~ (t .; P2u) A( P2u) ) * "n- 2k( c(t ; P2u) ) = /\ 2k(P2S( t ;U)Al (U) P2 l) * /\ n2-k( P2C(t;U)P2 l) 2 = /\ n ( P2) "2 k( S( t; U) Al ( U)) * "n-2 k( C( t; U) ) /\ n( P l ) = /\ 2k(S(t;U)A (U)) *An- 2k(C(t;U)). 1 The function h2k(p,t) = i SS.LM /\ h2k(p,t) is then given by =} f /\ 2k(S(t;U)A,(U))*An- 2k(C(t;U))det(S.l(t;U))o ~.LM (U) s.tMP p 2 2 k ( sh\ ( t ; P u) )A( P u) ) * /\ n- k( 2 2 Sn (t ; P2u) ) de t (~ ( t ; P2u) )n p The map .t ( u) . S Mp U,,... .P2u is an isometry of · S.LMP with 's-ln. The result thus follows by a change of variables in the integral. We now compute Lie algebra tr· C~(t;U), S~(t;U) and 'S~(t;U) in terms of the for X €Of define a linear map ad(X): ~-+ Oj by ad(X)Y = [X,Y]. The map X1-+ ad(X) is called the adjoint representation of a Lie algebra homomorphism of OJ. OJ- into the Lie algebra of all It is 95 derivations of ()T·. Proposition 10.7. Let (rn,A) germ, and U € rn.L. Then be a symmetrically embedded second order l\n, U: rn -+ ffi and ~, U: rn.L -+ rn.L a re given by 2 R = -ad(U) 1 ·rn,u rn 2 R.t. = -ad(U) 1rn .L. ·rn,u Also if cosh(t ad(U)) and ~d(U)-l sinh(t ad(U)) are defined by their power series, that is o:> t 2k cos.h(t ad(U)) = k12o ( 2k)! (ad(U)) ad(U)- 1 ~ 2k , t 2k+l sinh(t ad(U)) = 6 (2k+l)! (ad.(U)) k=O 2k , then Sm(t;U): ad(U)-l sinh(t ad(U))lm Crn(t;U) = cosh(t ad(U)) Im' Sm ( t ; u) = ad ( u) - Proof. 1 s i nh ( t ad ( u) ) lrn .L ~ By proposition 5.23, the torsion tensor T of M is zero and the curvature tensor is the same as that of the canonical connection. Therefore, by proposition 5.12 (3), for X,Y in ~' 0 = T(X,Y) = -[X,Yin. Thus [X,Y] eh. Using this in 5.12 (4) yields, for R(X,Y)Z = -[[X,Y]h,Z] = -[[X,Y],Z]. X, Y, Z e fh, 96 So, if x, u € fn' then 'R'u(X) = 'R(x,u)u = -[[X,U],U] = -[U,[U,X]] = -ad(U) 2X. This proves the statements about fk,u = -ad(U) 2 1rn' it is easy to check that From the formula cosh(t ad(U)} lrn Crn(t;U). \,u and ~,u· is a solution to the initial value problem defining The other formulas are proved in the same way. Corollary 10.8. Let · germ and U e rn...L. (rn,A) be a symmetrically embedded second order Then, for any rea 1 number a, Sn(t;aU) = Crn(at;U), 1 Srn(t;aU) =a ).(at;U) s~ (t ; au ) = ~ srn (at ; u) . Proof. By the formulas of the last proposition S(t;aU) = ad(aU)-l sinh(t ad(AU)) = ~ ad(U)-l sinh((at)ad(U)) = ~ S(at;U). The other two equations are proved in the same way. 97 11. Tubes in product manifolds. For a = 1, 2 let Ma be a compact oriented symmetrically errbedded submanifold of dimension na Ma: of dimension na + ma. Let in the oriented symmetric space (ha) 2k ·(p,t) be the function given by theorem 8.2, so that In this section we prove Theorem 11.1. The submanifold M = M1 x M2 is a symmetrically errbedded submanifold of = ~l x 'M2 • Let n = n1 + n2 and 1Vf VO 1(M ( r) ) = 6 J h2k( ( p 'q) 'r )OM ( ( p 'q)) 0<2k<n M where h2k((p,q),t) is as in 8.2. Then with the convention that VO l ( M( r)) = r (ha) 2£ = 0 if 2L > na. sTI/2 1(Ml ( r cos e) ) 1( M2 ( r sin e) ) de. VO VO 0 Proof. Assume the formula for h2k( (p,q), t). Then vol(M(r)) = Therefore, ~ 0~2k<n f h2k((p,q),r)oM((p,q)) 0 98 rr/2 J ( 6 ( h1) 2k( p, r cos e )( ~ ( h2 )( q , r s i n e)) oM x M ( p, q ) de 0<2J <n 2 ·1 2 a Mlx M2 0<2k<n - - 1 - - =r J rr/2 = r J 0 (J 6 (h 1 )2k(p,rcos e)oM (p)J(J ~ (h 2) 2j(q,rsin e)°m (q)}de Ml 0<2k.2_n 1 1 . M0<2J2n 2 2 2 rr/2 vol(M1 ( r cos e) )vol(M 2( r sin e) )de. = r J 0 The proof that the formula for series of lemmas. h2k(p,t) holds will be done in a It will be more convenient to work with the second order germs of submanifolds than with the submanifolds thenselves. p e M1 , q e M2 , and Let (tn ,A ) 1 1 be the second order germ of M1 at p, and (tn 2 ,A 2) the second order germ of M2 at q. Let ffi'a be the space to ~Ct at 0, its origin. Then, as in the last section, there is a decomposition Let ffi' be the tangent space to ~ at (0,0). Then we can assume tn 1, tn 2, tn1 and rn2 are subspaces of fh in the natural way. lh = lh 1 e tn 2 . Then the orthogona 1 comp 1ement to rn in fii' is that Let rn.1. = lhl e rn2. Convention 11 .2. The letter U always denotes elements of letter V will always denote elements of tn1 and the rn2· Define a linear map A from rn.t. to the symmetric linear_maps on rn by A( u) Im, = A1( u), A( u) Ith· = 0, A( v) Im = o, 1112 A(V) Im 1 2 . = A (v). 2 99 Lemma 11. 3. The second order germ of M= M1 x M2 at ( p ,q) is (rn ,A). ,..., is the origin of M1 and that A1 is the Weingarten map for M1 at 0. Similar assumptions are made for M2 and q. If pCl : M-+ MCl is projection, then our identification of fii'Cl Proof. It can be assumed that p with a subspace of fii' identifies the derivative pa*(O,O) gonal projection of fii' onto fTi'Cl • with ortho- It is clear that the tangent space to M1 x M2 at · (0,0) is rn = rn 1 ern 2. Let B be the Weingarten map of M at (0,0). Then because pa* is orthogonal projection, proposition 4.6 becomes where x1 , Y1 e rn 1 and x2 , v2 e rn 2 . This shows B = A, and finishes the proof. Lemma 11.4. ,..., Let R, 'R1 and M2 respectively. and rn2 M, and .'R2 be the curvature tensors of 'M1 _t Then for. all u, v' a 11 four of rn, ' lh2' 1n 1 are stable under R(U+V)' I and This shows that M = M1 x M2 is symmetrically embedded in Proof. Let x1 , Y1 e m 1 , x2 , Y2 e tn 2 . lemma proposition 4.5 yields ,..., M. Using the notation of the last 100 <I\ U+ v) ( x1+X2 ) 'y 1 + y2) = (R(X +x 1 2 ,U+V)(U+V), Y + Y ) 1 2 = <R,(x, ,U)U,Y1) + (R2(X2,V)V,Y2> = ((Rl)UXl ,Y1> + ((R2)vX2,Y2>· The result now follows easily. Let M 2 C(t;U+V), c (t;U) 1 and c2(t;V) respectively as in theorem 8.2. s, s , s 2 and 1 be defined for M, M1 , and Make analogous definitions for S , 's.L1 , s.L2 . ' .L Lemma 11. 5. C(t;U+ V) = c (t;U)@ c (t;V), S(t;U+ V) = s (t;U)@ c (t;V), 1 1 2 2 si (t , u+ v) = :st (t ; u) (f) s2 ( t ; v) , where the notation means c(t; u+ v) lrn = c1 ( t ; u) , 1 c(t ; u+ v) lrn = c ( t ; v) , 2 2 etc. Proof. Using 11 .4 it is easy to check that satisfies the differential equation defining c1 (t;U) $ c2(t;V) C(t;U+ V). cases are similar. Lemma 11.6. Proof. det(s.t(t;U+V)) = det(Sl(t;U))det(S2(t;V)). Clear from 11.5. Lemma 11.7. S(t;U+V)A(U+V) The other 101 Proof. This also follows from 11.5 (and the definition of A). It is possible to view c1(.t;U) as a linear map on rn by extending c1 (t;U) from rn 1 to rn 1 @rn 2 by having c1 (t;U)!,.,_ =O. . c2(t;V) Using a similar convention for 1u2 lets us write C(t;U+ V) .= c1(t;U) + c2(t;U). This convect ion wi 11 be used i.n the fo 11 owing few lemmas. Lemma 11.8. A 2k(S( t ;U+V)A(U+V)) * /\ n- 2k(.C(t ;U+V)) = Proof. Let S = S(t ; U+ V) , A = A( U+ V) , C = C( t ; U+ V) , S1 = S1( t ; U) , etc . Then the last few lemmas and 7.5 (3) yield /\ 2k(SA) * /\ n-2k( C) = /\ 2k(S1Al + S2A2) = * /\ n-2k(Cl + C2) :0 (A i ( S A ) */\ j ( C ) ) * (/\ 2k- i ( S A ) */\ n- 2k- j ( C ) ) • O<i< 2k l l l 2 2 2 O<j_::.n-2k The linear maps s1A1 and c1 take values in a vector space of dimension n1 . Therefore, if i + j > n1 , it follows that /\ i ( s A ) * /\ j ( c ) = 0. 1 1 1 Likewise, if (2k-i) + (n-2k-j) = n1 + n2 - (i+j) > n2 , then /\2k-i(S2A) * /\n-2k-j(C2) = 0. Consequently, the only nonvanishing terms have j by n - i 1 and summing on i i + j = n1 . yields the lemma. Replacing 102 Lemma 11 . 9 . For 0 < i ~ 2k ~ n 1et 1 . 2k-i n,-i =ts /\ i (s,(t;U)A,(U))*A (S2(t;V)A2(V))*A (c,(t;U)) :Sln *" n -2k+i 2 ~ ( c2 ( t; V) )detCsl( t ;U)) det(S2( t; V) )n -ln (u+ V). 5 Then Proof. This ·is 1errnna 11.8 substituted into the definition of h2 k(t) . Lemma 11.10. If f is a continuous real valued function on s-Tu, then J f(U+ V)o. 's1n (U+ V) s-Tu rr /2 ·=S S J 0 S-lnl Sln2 Proof. on a = 1, 2 and . si. = S-ln. Put the product metric = cos eu + sin ev. We now compute the pullback of the volume from o. Let ·si st x s2. ( e,u,v ) e [ 0, TI /2 J x s.L1 x .s.L2 . orthonormal basis of ~l with orthonormal basis of ~~ with ·v 2 , ... , vm 2 e de. Sln2 st x s2 and define cp: [O,rr/2] x st x s2 ~ si by ~(e,u,v) [O,rr/2] x m -1 ( ) (V)cos 1 (e)sin 2 (U)n Slnl Let S.L()'. = S-ln ()'. [O,n/2] x m -1 f(cos(e)U+sin(e)V)o. Let u1 = u and v, = v. Then to u1 , ... ,um let :e' be an 1 v1 , ... , vm 2 u2 , be an . .. ' um ' is an orthonormal basis of the tangent space to 1 103 [O,rr/2] x st x 52 at (9,u,v), and c:p * 0e - -sin e ul + cos e v, ' 0 - cp* u.1 = cos 9 u.1 2 ~ i ~ m, cp* v. = sin e v. J J Therefore, cp* :a /\ cp *u2 /\ •.•• /\ kum/' cp *v2 /\ ... /\ cp* vm2 qi m -1 = cos 1 m -l (e)sin 2 (e)(-sin(e)u 1+cos(e)v 2 ) /\ u2 /\ • • • /\ um /\ v2 /\ ••• /\ vm . 2 1 m +m -1 The length of this vector is an element of /\ 1 2 (rn.L) m-1 m-1 cos 1 . ( e) s i n 2 ( e ) . Thi s s.hows cp *o = o 5.L st m -1 /\ o s2 The function c:p is surjective. of measure zero. Therefore It is also injective outside of a set completes the proof. Lemma 11.12. s be real nunbers. Let c and 2 /\ i ( Sl ( t ; c U) Al ( c U) ) * /\ k - i ( S ( t ; s V) A ( s V) ) 2 2 n -i n -2k+i *" 1 (c1 (t;cU))*A 2 m -1 " cos 1 ( e) sin 2 ( e) de. Using the form of c:p *o 5.L is (c (t;sV)) det(5l(t;cU))detCS2(t;sV)) 2 Then 104 = A1 (s, (ct;U)A, (U)) *A 2 k-i(S2(st;V)A2(V)) cmlsml · 1 n -1 n -2k+i * A 1 ( c (ct; u) ) *A 2 ( c (st; v) ) 2 1 det(Sl(ct,U) )detCS2(st;V)). This is a consequence of corollary 10.7. Proof. We can now finish the proof of the theorem. If in 11.9, then we use the last two lemmas to rewrite H2k,i(t) is as H k,i(t) as 2 H2 k' i ( t) l Tr/2 =1J J . s A. (s,(tcose;U)A,(U))*A 2k -l(S2(tsin9;V)A2(V)) 1 o ·s-Tu 1s"ln 2 n -i * A1 ( c, (t cos e; u)) *A n2-2k+i ( C2 (ts in 9; v) det(Sl( t cos e ;U) )det(S2( t sine; V) )o . 0 ·s1n2 Tr /2 =tJ s-Tu, (U) de (V) cos(e)sine . n -i 1 1 A (S 1 (tcose;U)A 1 (U))*A (c 1 (tcose;U)) (tc~sef o s-ln 1 de t (st ( t cos e; u) )o . ( t s1; n ( u) ) s-Tu, n -2k+i . 2 2 J " k- i ( s2 (ts in e; V)) *A 8 S°ln2 2( de t ( s t si n e; v) )o s1n2 ( v) ) de . ( c2( t cos e ; v)) 105 If i is odd then the integrand for the integral over odd function of U and thus reduces to zero. s1n 1 is an For H2k, 2i(t), the definition of (h a. ) 2l. to see that H2k, 2i(t) = tf rr/2 (h 1 ) 2;(tcos e)(h 2) 2 (k-i)(tsin e)de. 0 The theorem now follows from lemma 11.9. we use 106 12. Examples. ,...., We first consider the case where M is the complete simply connected manifold of constant curvature K of dimension n + m. M is symmetrically embedded. Let M be a Then every submanifold of ,..., compact oriented submanifold of M of dimension n. Using the notation of theorem 8.2, and the form of the curvature tensor for M given in example (2) following the proof of proposition 6.2, we see that if U e siM p , V e TiMp and X e TM p , then Ru(X) = KX Rt ( X) = KX - ( U, V) U. Define two real valued functions m by c, s on c 11 (t) + Kc(t) = 0 c(O) = 1, c'(O) = 0, s 11 ( t) + Kc( t ) = 0 s(O) = 0, s'(O) = 1. Using the initial value problems defining C(t,U) and S(t;U) we see C(t;U) = c(t)(id)TM , p S(t;U) = s(t)(id)TM . p We now compute det(si(t;U)). Note that Ru(U) = 0, and If Ve TiMP and V is perpendicular to U, then Rt(V) = KV. si(t;U)V = s(t)V, and it follows that det(si(t;U)) = ts(t)m-l. Thus, 107 Let A be the Weingarten map of M in M. excess tensor of M in t J /\ 's.tM and let H be the The integral formula of proposition 7.9 can now be used to compute the function h2k(p,t) = M, h2k(p,t) of theorem 8.2. 2k(S(t;U)A(U))*/\n- 2k(C(t;U))det(S.L(t;U))o .L (U) 3 Mp p = s(t)m+2k-lc(t)n-2kf /\2k(A(U))*An-2k(I)o s.tM = s ( t) m+ 2k- l c ( t ) n-2 k s.tM p (J S.tM (U) I\ 2k(A ( U) )0 p ( U) ) * /\ n- 2k( I) s.tM p p = s( t)m+2k- l d t) n-2kvol ( Sm-1) H*k *A n-2k( I). k!m(m+2J·. ·-(m+2k- l) n . M at If the curvature tensor of I\ 2 "' ™p' Let p ·is viewed as a linear map on then it has the form I be the identity map on ™p and view the curvature tensor of M at p as a linear map on I\ 2 ™p· Then, by proposition 4.5 the excess tensor H of M at p is given by H = R + KA 2 (I) Here we have used calculation. r*j = j! Aj(I). This is also used in the following 108 1 H*k*/\n-2k(I) = 1 k (n-2k)! (R+Kr*2)*k* r*(n-2k) 2 k . *. k' * . ~ (-2K) -J .R J * I n-J = (n- 2k)'. j=O L.J •t(k.-J•)i. J. = k! ~ (n-j)! ( K)k-j R*j * /\n-j(I) (n-2k)! j=O j!(k-j)! 2 = k! k (n-j) ! (-2K)(k-j) tr(R*j). (n-2k)! jf2o j!(k-j)! The last line of the above follows from proposition 7.8. The following integral invariants of a Riemannian manifold were introduced by Hennann Weyl [13]. Definition 12.1. If M is a compact oriented Riemannian mantfold with smooth boundary and R is the curvature tensor of M viewed as a . 2 linear map on /\ TM then for each k with 0 -5- 2k -5- dim(M) set Then the following (also due to Weyl) holds. Proposition 12.2. M is the complete simply connected Riemannian If manifold of dimension n + m and M is a compact oriented submanifold of M of dimension r about M, n then the volume of M(r), the tube of radius is given by VO l(M( r)) m-1 = s ( r) m-1 v01 ( 5 ) 6 0<2k<n s(r) 2kc(t)n- 2k k (n-j)! K (k-j) ( n- 2k ) 1 m( m+ 2 ) · · • (m+ 2k - 2 ) .6 j 1 ( k - j ) I ( 2) w2j ( M) · J =O · · 109 where c 11 ( t ) + Kc ( t ) = O c(O) = 1, c'(O) = 0 s 11 ( t) + Ks (t) = O s(O) = 0, s'(O) = 1. Proof. This follows from theorem 8.2 by using the above expression for . the formula given . H*k */\ n-2k (I) in for h2k(p,t). We now turn to complex manifolds of constant holomorphic curvature. Let M be a complex manifold of complex dimension n + m. Recall from example (3) following proposition 2.5 that each tangent space TMP to "' M is a compl~x vector space. Let be the linear map on TMP induced by multiplication by J-::f. be assumed that M has a Riemannian metric ( , ) It will such that (JX,JY) = (X,Y) for all X,Y tangent to M at the same point. A Hermitian metric ( , ) is then given on each tangent space by (X,Y) = (X,Y) + (X,JY) J-::f. The manifold "'M is said to have constant holomorphic curvature if its curvature tensor is given by ~(X,Y)Z = K((X,Y)Z - (Y,X)Z - (Y,X)Z + (Z,Y)X); in this case, Ru(x) = R(x,u)u = - K((U,U)X + (X,U)U - Z(U,X)U). Let M be the complete simply connected space of constant K 110 holomorphic curvature K. Then M is known to be a Riemannian symmetric space (See [8] volume II, example 10.5, page 273 and example 10.7, page 282). M ,..., If K is positive th~n · M is complex projective space. be a compact complex submanifold of M with smooth boundary. for each p in M, If U e S.LMP, both ™p Let Then, ,..., and 'T.LMP a re comp 1ex subspaces of ™p. then Ru(X) = K((u,u)x + (x,u)u - 2(u,x)u) = KX. M. This shows that M is symmetrically embedded in It also shows that Ru = K(i_dTM ) . p So, if we again define functions c(t), s ( t) by the di fferenti a 1 equations c 11 ( t) + Kc ( t) = 0 c(O) = l, c' (0) = 0, s"(t) + K(s(t) = 0 s(O) = 0, s'(O) = 1, then C( t ; U) = c ( t )(_ i dTM ), p s(t; u) = s (t )(id™ ) . p If Ye T.LMP and (Y,U) = 0 then Rt (Y) = KY. Thus s.t(t;U)Y = s(t)Y. Assume that M has complex dimension n. with (Y,U) = 0 has real dimension 2(.m-1). si(t;U) = tU. Finally, note that Then the set of Y e T.LMP As before, 'Rt(u) = 0 so 111 .Rt(JU) = K((U,U)JU + (JU,U)U - 2(U,JU)U) = 4KJU; therefore, si(t;U)JU = } s(2t)JU. Combining these, we obtain det(si(t;U)) = ~ s(t) 2 (m-l)s(2t). We can now use the integral formula of proposition 7.9 to compute roJ Let A ·be the Weingarten map for M in M and H the excess tensor of M in M. Then we have h k( p, t) = ~ I\ 2 f ~M 2k ( S( t; U) A( U)) */\ 2( n-k) (C( t; U)) det (Si ( t; U) )O i SM p = s(~ t) 5 ( t) 2 ( m+ k- 1 ) c( t ) 2 ( n - k ) f siM A 2k (A ( p ( U) p * I\ 2( n- k) (I ) u) )o siMP _ s(2ts(t) 2(m+k-l)c(t) 2 (n-k) 2m-l *k* 2(n-k) ) H I\ (I) - 2(k!)(2m)(2m+2)···(2m+2k-2) vol(S ~ s(2t)s(t)2(m+k-l)c(t)2(n-k) vol(S2m-l) tr(H*k) . 2k+l(k!) m(m+l)· · ·(m+k-1) This yields the following proposition due to R. Wolf ([14]) and F. J. Flaherty ([4]). Proposition 12.3. With notation as above the volume of the tube M(r) about M is vol(M(r)) = s(2t)s(~) 2(m 1) n (( )2(k-l) ( )2(n-k) - vol(S2m-1)6 s kt - ct ftr(H*k)~. k=O 2 m ( m+ l) · · · ( m+ k- 1 ) M 112 Therefore vol(M) only depends on the excess tensor of M in M. As a last example we do a hypersurface in a space of constant holomorphic curvature. To this end let · M be the space of constant holomorphic curvature and complex dimension hypersurface of M. n. Suppose that M is a Let p e M and U e 's.LMP. Then the vector J(U) is perpendicular to U and thus tangent to M at p. Define p = Orthogonal projection of ™p onto 1 . orthogona 1 complement of JU in ™p· p2 -- Orthogonal projection of ™p onto span of JU. If X e TMP and X is perpendicular to JU, then (X,U) = 0. -Thus and so C(t;U)X = c(t)X, S(t;U)X = s(t)X. As above, RU ( JU ) = 4KJ U; therefore C(t;U)JU = c(2t)JU, S(t;U)JU = ~ s(2t)JU. These facts together yield C(t;U) = c(t)P 1 + c(2t)P 2 , S(t;U) = s(t)P 1 + ~l s(2t)P 2. Because P2 has rank one it follows that Aj(P 2 ) = 0 for j ~ 2. 113 Whence /\2n-l-k(C(t;U)) = c(t)2n-l-k/\2n-l-k(Pl) + c(t)2n...:k-2c( 2t)P *A2n-k-2(P 1). 2 We choose a smooth unit normal along M and let A be the corresponding Weingarten map. Then P2A also has rank one; thus Ak(S(t;U)) = s(t)kAk(P A) 1 +} s(2t)s(t)k-l(P 2A)*Ak-l(P 1A). But P2 and. P2A both have the same one-dimensional range, and thus P2 * (P 2A) = 0. Therefore, using the notation of proposition 9.1, we have h~(p,t) = /\k(S(t;U)A) * A2n-l-k(C(t;U}) = s(t)kc(t)2n-l-kAk(P A)*/\2n-l-k(P 1 ) 1 +} s(2t)s(t)k-lc(t)2n-l-k(P2A)*Ak-l(P1l + s ( t) kc ( 2t) c2 n- 2- k ( t )A k( Pl A) * p2 * A2n-2-k ( p1 ) • Choose A so that JU is one of its eigenvectors with eigenvalue a1 and let a2 , ... , a 2n-l be the other eigenvalues of A. Then let crk(a 2 , ... ,a 2n_ 1 ) be the k-th element symnetric function in a 2 , ... , a2n_ 1 . Then Ak(P,A) * /\2k-l-k(Pl) = crk(a2,··· ,a2n-l). However, it is not hard to show that, if Kr 0, then s(t)kc(t) 2n-l-k ( )2n-1-k and is linearly independent of s ( 2t ) st ( ) k-1 ct s(t)kc(2t)c(t) 2n- 2-k. Therefore, we can compute crk(a 2, ... ,a 2n_ 1) from h~(p,t). But this is independent of a1 so h~(p,t) is not a 114 function of the k-th element symmetric function of A. can be proved is that h~(p,t) The best that is a linear function of Ak(A). 115 References [l] C. B. Allendoerfer and A. Weil, The Gauss-Connet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc., VO 1 . 53( 19 4 3), 101 -129 . [2] I. Chavel, Riemannian Symmetric Spaces of Rank One, M. Dekker, New York, 1972. · [3] S. S. Chern, On the kinematic formula in integral geometry, J. Math. and Mech., vol. 16(1966), 101-118. [4] F. J. Flaherty, The volume of a tuhe in corrrplex projective space, Ill. Jour. Math., vol. 16(1972), 627-638. [5] H. Flanders, Development of an extended differential calculus, Trans. Amer. Math. Soc., vol. 75(1951), 311-326. [6] H. Federer, Curvature measures, Trans. Amer. Math. Soc., vol. 93 (1959)' 418-491. [7] S. Helgason, Differentiai Geometry, Lie Groups, and Syrrmetric Spaces, Academic Press, New York, 1978. [8] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience, New York, vol. !(1963), vol. 11(1969). [9] S. Lang, Differential Manifolds, Addison-Wesley, Reading, Mass., 1972. [10] · M. Spivak, A Corrrprehensive Introduction to Differential Geometry, vol. IV, Publish or Perish, Inc., Boston, Mass., 1975. [11] J. Steiner, Uber paralLel Flachen, Mber. Preuss. Akad. Wiss., 1840, 114-118; see also "Collected Works", vol. 2, 173-176, Reimer, Berlin, 1882. [12] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresmann, Glenview, Ill., 1971 . [13] H. Weyl, On the colurne of tuhes, Amer. J. Math., vol. 61(1939), 461-472. [14] R. A. Wolf, The volume of tuhes in complex projective space, Trans. Amer. Math. Soc., vol. 157(1971), 247-371.