Some Applications of Functional Analysis to Parallel Displacement in Riemannian Geometry Citation Johnson, Paul Bennett (1947) Some Applications of Functional Analysis to Parallel Displacement in Riemannian Geometry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7rf1-b173. https://resolver.caltech.edu/CaltechTHESIS:04252025-204100956 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1947 Record Number: CaltechTHESIS:04252025-204100956 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04252025-204100956 DOI: 10.7907/7rf1-b173 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17175 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 30 Apr 2025 16:14 Last Modified: 30 Apr 2025 16:35 Thesis Files PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

SOME APPLICATIONS OF FUNCTIONAL ANALYSIS TO PARALLEL DISPLACEMENT IN RIEM.ANNIAN GEOMETRY Thesis by Paul Bennett Johnson In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1947 Summary This thesis is a study of finite parallel displ ac ement of a contravaria,nt vector according to Levi-Ci vi ta' s di f ferenti al definition. The interesting featu~es of such displacement are shown to depend upon the matriza,nt function, ...D...~ [-f1~ ~ie.J . J. i The precise dependence of the matrizant, and hence the displaced vectors, upon the coefficients of connection of the space and upon the directrix of displacement is studied by means of the Frechet differentials of the matrizant. This is done both by warping the space, and by varying the directrix. Since parallel displacement is a geometric phenomenon it is not surprising that the ma.triza.nt turns out to be a two point tensor. However, the Fr~chet differentia.ls of the matrizant are two point tensors only under rather special conditions. A second surprising result is that only for flat spaces do the Fr~chet differentials of any particular order produced by varying the directrix vanish for arbitrary directrices and variations. The interesting case of a closed directrix is discussed in some detail. In particula,r the 11 fixed 11 vectors are exam- ined, where "fixed" vectors are those which retu rn to their original direction a.fter displacement about the directrix. The theory is shown to generalize immediately to tensors of any order and of any type. Acknowledgements The a.uthor wishes to register here his deep appreciation for the leadership and guidance given him by Professor Aristotle D. Michal. Without his suggestions and kind criticisms this thesis would never ha.ve been writteno The author I s grati tu.de also goes to his wife a.nd her pa,rents for their understanding and f orbea.ra.nce during the months he was completing his graduate studies. CONTENTS Chapter : Page : I. Introduction 1 II . Notations, Conventions and Elementary Theorems 4 III. Integration of the Equations of Levi-Civita Parallel Displacement 8 IV. The Matrizant as a Function on a Banach Ring 22 v. The Change in a Set of Parallel Vectors Produced by Warping the Space VI. VII. 27 The Change in Parallel Vectors Produced by Changing the Directrix 33 The Tensor Character of SL (,~and o-~Jl~. - 57 VIII. The Classification of Riemannian Spaces by Their Degree IX. Applications of Matrix Theorems to Displacement About a Closed Directrix x. XI. 65 86 Extension of the Theory to Tensors of Types Other Than Contravariant Vectors 92 Bibliography 96 1 I. Introduction The problem of parallelism has fascinated geometers through- out the history of mathematics. Euclid faced it in his famous fifth postulate, and Euclidean par-allelism remained the standard until the nineteenth century. With the development of Riemannian geometry as a generalization of Euclidean geometry it soon became evident that no generalization of the notion of parallelism could l· be devised which had all the properties of Euclidean parallelism. This thesis will be concerned with that generalization proposed by T. Levi-Civita. 2 Levi-Civita suggested a differential definition of a generalized parallelism with the property that if a vector is displaced parallel to itself a.long a geodesic of the space, it makes a constant angle with the geodesic. constant components. However, such vectors do not have The change in the components depends in general upon the coefficients of connection (Christoffel symbols) of the space, and on the directrix curve followed in displacing the vector. We shall examine this dependence of the displaced vector on the connection of the space and on the directrix in some detail by means of the theory of Fre'chet differentials in Banach spaces. A substantial amount of study has already been made of LeviCivita parallelism. 1. 2. J. / ' 3 found the change produced by displacing Peres For a discussion of several types of generalized parallelism, see Whittaker (1) Levi-Civita (1) or see any text on Riemannian geometry Peres, J. (1) 2 a vector about an infinitesimal closed curve. Dienes 4 has inte- grated the differential equations essentially by the LiouvilleNeumann 5 method of successive substitutions to obtain the change produced in a vector by displacement over a finite distance. He determines the change produced in the displaced vector by changing the directrix an infinitesimal amount. Applying his result to a 6 / \ closed curve, he obtains the Peres formula. Appel quotes Dienes / ' formula on taking the first terms of the and determines the Peres Liouville-Neumann series. Eisenhart 7 and Thomas 8 find the change produced by displacement about an infinitesimal parallelogram by considering infinib~simal vectors for the sides of the parallelogram. Duscheck-Mayer 9 also integrate the differential equations by the Liouville-Neumann series. The contributions of these authors can be summarized in two statements. First they have indicated the solution to the equat~ons of parallel displacement. Second they have displayed the local dependence of the displaced vector on the space by integrating the equations about an infinitesimal closed directrix. They have dis- played the local dependence upon the directrix by making infinitesimal changes in the directrix. The difficulty of treating a function ex- pressed as a system of series of iterated integrals has probably inhibited more thorough treatment of the vector displaced over a finite distance. 4. 5. 6. 7. 8. 9. Dienes, Paul (1) Whittaker & Watson (1) Appel, Paul (1) Eisenhart, L.P. (1) pp. 65-67 Thomas, T.Y. (1) pp. 38-42 Duschek-Mayer (1) 3 Much of this difficulty can be simplified by regarding the system of differential equations of parallel displacement as .a single matric equation. The solution can then be expressed by means of the matrizant function chapter III. 10 . This technique is used in Some of the earlier results are developed and a tensor expression for the displaced vector is given. A discussion of the Frechet differential properties of the matrizant as a function over a Banach ring is given in chapter IV. By means of these properties of the matrizant, the change in the displaced vector as the underlying space is warped is discussed with examples in chapter V. A similar change as the directrix curve is varied, the space remaining the same, is discussed in chapter VI. The existence of the Fre'chet differentials of the matrizant as a function of the directrix with the change in the directrix as increment is proven, and their generating for:-mula is derived. The tensor nature of the matr:-izant and its differentials is discussed in chapter VII. In chapter VIII spaces are classified according to their degree, where degree is defined in a manner comparable to the degree of a plane curve. The surprising result is obtained that spaces ar:-e of zero or infinite degree as they ar:-e flat or not. Vectors which return to their original position after parallel displacement about a closed directrix are discussed in chapter IX. The theory is extended to mixed tensors of any order in chapter X. 10 . Fr-azer, Duncan and Collar (1) 4 II. Notations, Conventio!ls, and Elementa.£.r_Theorems. In this chapter are listed the general conventions which are followed in this thesis, and the elementary theorems on Frechet differentials which will be used. II.A.Coordinate sistems. are generic coordinates in then dimensional Riemannian space. Where no ambiguity results, the index i will frequently be omitted, as in "the coordinates x" or "A, a point function, is a function of x". II.B.Summation Convention. When an index appears twice in the same term, once as a superscript and once as a subscript, that term represents the sum of n terms obtained by giving the index successively the values 1, 2, ... , n. Parameter values, on the other hand, are not summed. It is usually clear from the context when a symbol is an index and when it is a parameter value. A note is made wherever there is danger of confusion. II.C.Tensor-Matric Notation. The matric notation is used to reduce the number of explicitely expressed indeces. The quantity with n2. may be represented by the matric symbol B . components, B ~ In the matrix the component B~ appears in the rth row and in the cth column. The product of two matric terms A,B is the matric product, ie ( AB ) n.-~ :::. I A'\, ')t1, r.i..,., '-'t • by the column matrix than ni The contravariant vector A . \I. A is represented Similarly if a mixed quantity has more components it is represented by a matric term with indeces, 5 foe- example, the coefficients of connection re,: are Wt'i tten rte. Where ambiguity may result from the matric notation, the indeces are inserted. The term is then said to be expressed in "componentTT notation. I II.D. Fundamental Theorems on Frechet Differential~ . .An ordinary differential exists for numerically valued functions of a numerical variable. The Frichet differential is a generalization of the ordinary differential to functions with values and arguments in normed linear spaces. / The Frechet differ- ential is defined as follows. Definition 1. Let P (AJ be a function on the normed linear space normed linear space Bz. • to a £3, Then if for A=Ao a function 1 cJ F'[Ao•JDJ a:n.d. l-f exists which is linear in D~for Din some nei ghborhood of the origin of B 1 (2.1) , and H p [Ao-tD]- F'[A J- 5 1 P [Ao'>DJ = /1D1/ • E.[A.) DJ 0 and if // c. lAo > DJll ➔O as II 011 ➔ 0 then J'F[A.>o]is the first Fre'c het differential of P LA] at Ao with increment D . Higher order Fre'chet differentials are successive first Frechet differentials . Theorem 2.1. If the Fre'chet differential of F existh_i! is_given bJ:: the Gateaux_differential ( 2. 2) X F'fA.~DJ = L,·'-\ ). ➔ 0 t F l Ao +- >- D} - F [ A.,] 1.:.._::__-:-\--- J /\ Formula (2. 2) is frequently used to compute J' F' [A 0 ; DJ . The existence of the Fre'chet differential is often proved by showing that the Gateaux differential exists linear in D and satisfies (2.1) for Din some neighborhood of the origin of B1 • This proof, 6 / of course, also furnishes the formula for the Frechet differential cf'F= I F . The fo l lowing elementary t heor ems on the Fre'chet differ11 entials are listed without proof I Theorem If the Frechet diffe rential of a function F[AJ exists, 2 . 2 it is unique. A finite linear combination of functi~ Fre'chet Theorem 2.,2, differ entiable at A is itself Frlchet differ entiable at Ao O • Chain Rule. Ir B1 , B2 , B3 are three normed linear Theorem 2.4 s12aces, if s 1 , s are 02en......§_et§. in B 2 1 , B / , !! Pl AJ is a Frechet 2 differentiable function on s1 to s2 with differential J ' F' [ A 'i DJ and if ,tl [T] is a Frechet differentiable function_on / with Frechet differential &'jj [T; s] , then ~ [AJ .Q!1 s2. - 51 to to {3 3 13 3 defined by ! LA; =- ,t [FCAJ] is Frechet differentiable with differential (2.3) Definition 2. Multilinear Function. A function P [A, 1 ... > A p] of p variables A,> " ' J AP on B1 , ... , BP to B where B1 , ... , BP' Bare normed linear spaces is called a multilinear function if Fis additive in each variable and if a real number M exists such that //FLA,> ,.. ; Ar11I {:; M-JIA,11 : llA,.ll· .. 11/.\rll _!heorem 2. 5 A mul tilinear function differential with increments D,) ... ) Dr:i \ I 0 11. F [ A, ) • ' ' J A I' )• 1) • • • 1)f' J I) Michal, A.D. (1) ) = P [ A, 1 • • 1 Ap] ,12_0s sesses a Fre"chet given by .. ,+ F'[ D' > A..,... ) A (' ]-t- F[A IJ D'- J ... ) A,,]+ ,- 7 Notation for the Frechet Differential. The first Frechet differential of a function F of a vat'iable A with increment D, which depends pat"ametrically on the parameters 5 , 11. is wt"i tten J' Fe"'; DI s,1] , The i th Frechet differ- ential with i inct'ements D1 , D2 , ... Di is Wt"itten rF [Aio,; D2- i"•jDi, /s,~] . When all the increments are equal to D, the ith differential may be written t'esults, the parameters o L f[.AjD.i s >tJ When no ambiguity :S ,l'\. and sometimes even A and D will be omitted, writing the ith differ en t ial simply as J iF . 8 III . Integration of the Equations of Levi-C_!.vita Parallel Displacem~nt. III.A.The E,91!?,tions of Parallel Displacement of a Contravariant Vector as a System of Ordinary Dif ferential_~guations. Let us consider a general Ri emannian space with generic Let a new system of generic coordinates x be de- coordinates x. fined by the equations J-, = ,£ (,:,) where the functions x(x) possess first order de rivatives and the inverse functions x = x(x) also possess first order derivatives. A set of n point functions ),/ o:) > i.-=-1/",n are said to de- fine a contrava.r-iant vector at the point x if the functions in the two coordinate systems 3,re related by \~Cx) = A\x.> d x-l,_<x) ~ t,..l The functions may be defined at a point only, along a curve '>-" (x.) or over any other suitable domain of definition. A set of such vectors defined one at each point of a directrix curve -,:./::. ,/c5) are said to be Levi-Ci vi ta parallel if their components, )..'\ x) , . satisfy the system of differential equations J, >,i __ r.1., (3.1) .;.-- J, 5 , I, - J~ ,.\j &..Je. Cl s whe"'e L A [ ,c.C~)J are the components of the vector at the point x ( 5 ) ,i (5) is the point with coordinates x 1 ( 5 ) ' ' y/ == x.~(~) are the parametric equations of the directrix with real numerical parameter, 5 . ~ are the derivatives of the par·a metric equations, .r :: I, ··) it. tls 9 are the Christoffel symbols of the space. They are also called the "coefficients of connection" of the space, and are symmetric in j,Je.. In a gener-al, non-Riemannian, space, the coefficients of connection, ri i.l'l need not be symmetric in il, j. , In such cases two different definitions of Levi-Civita parallelism are possible ·as one writes ~~ ,\J J.,,,I?. or Ts rJl/2. A1e, ~ dg · in (3 .1) . The theories of the two definitions are completely similar, and when is symmetric they are identical. rd.lie We shall consider only the one definition, (3.1), in what follows, and shall assume symmetry. Many of the theorems generalize immediately to the more general non-symmetric case. However, since we do not pursue the non-symmetric theory, there will be no distinction made between those theorems which are valid only for the symmetric case and those which hold more generally. and A~ are functions of the parameter 5 by virtue of the directrix equations. ~ hJ. ll::: ls a,}; ; For example Ti It is sometimes convenient to think of a set of parallel vectot"s as being generated by an initial vector which moves parallel to itself along the directrix, rather than of the set of vectors as a whole. parallel to itself. Such a vector is said to be displaced The set as a whole is said to be generated by parallel displacement of the initial vector. This viewpoint will be generally followed in this discussion. We seek now to integrate equations (3.1) so that, given an initial vector, ·~ [_ )(. (Cl..lJ , we can determine the vectors, ).. L>< tt)J , which are 10 parallel to it at different points ~(t) along the directrix. The solution of equations (3.1) has been given by many authors 12 . In order that this discussion may be self contained we also give the Liouville-Neumann series solution to (3.1). Theorem~ A , be disElaced Let a contravariant vector, par-allel to it§.elf alpng th.§ direc.:t;rix ' ,x,'-::: J:. 1.' ( 5) . ~ ' ,\tc11.) ~ a set of predetermined constants.i_ the ini tia.1 va.lues of the.-.£QmThen th~~lues of >..' (t) , the ~.Q!!!- ponents of t\ at ;ti(a-) • ponent§ of ,,\ at :r, (t) are given i!L.!:h§_region of convergrn by (3 :2) - ... where f ~-1c, [x ('s)] are the coefficients of connection of the space eval:lli!~d along the directrix. Proof: (3.2) is the Liouville-Neumann series solution for the defining equations (3.1) III.B. The Eguations of Parallel Displa.c_§ment of a Contravariant Vector as a §ingle Matric Equation. We may consider (3.1) as a single matric valued equation vl>- tL"5 where }. === - r re b-1e .Ls is the column matrix of the components of the vector being displaced. 12. Ji See introduction. 11 is the square n x n matrix of the components f~~ f .-n, • The suppressed lower index indicates the column position and the suppressed upper index indicates the row position of the term fe~ in the matrix fn.. • Note that it takes n matrices, k-=l, ... , n to list all the coefficients of connection. r is the sum of n matrices, each of which is the square ~/?. re J..~ matric product of a square matrix, &,_n. ti 1 r,e ~ On the other hand, • ·a square matrix, f ,e_ f-1e 'by a scala.r point function, is the column matric product of , multi plied on the right by a column matrix, A . We note that it makes no difference whether is first multiplied by f~. (3. /\ or ~le. d:g That is (rri i\/?_)A = ( rfl A 4 ) ) * I t The solution to (3.3) leads to Toeorem 3 .2. Let_.a contravariant vector 1-Q_!tself along the direptr1,~ -;:,;, : : ,/cs). A be dis£laced paralle+ -Let }. ("--) -be-a --column - matrix of predetermined constants, the initial values QLlhe £.Q!11~,Q!!fil!ts of Qf. A , ( 3 . ~ at ~(~) . A(t) , ·at ,:: lt) are ,given in the ~onvergence bI A(t)-= _n_~ [ _:_ rtl 5 ) Then the column ma~rix of_sQmponents 1/· ,sJ A (IL) where the matri·z ant functi.Q!l ..IL! of an n x n matrix ? is given by -a function of the pare.m~t~ ( 3 . 6 ) _n_!, [Als1= 1 +Jo., Acs)t1-5 -+-J/}.,l'tA ('5) l s Ja..rs A( ~ ) where t I A (5) itself ' . : 1 , + • · · is the unit matrix with one's on the m,2-in diagonal arul zeros elsewhere. Proof: (.3.5) is the rewriting of (3.2) in matric form, and hence is the solution of (3.3). The theorem follows f r om theorem 3.1. 12 III.C. 1llustra~.!.9n. Parallel Displacement on a Sphere in Ordinary Three Dimensional Euclidean Space. III.C .l. ~cement Along a Parallel of Longitude. Theo.rem 2•2.• 1 Let coordinates J:,- } x.2. dimensional Eu~J:Jdean space with the longitude. -r.,' be chosen on a sphere in three the colatitude ~ ;t.z... Let a vector A be displaced by Levi-Civ,! ta _par- allel dj.§p~:?£.§ment alsmg 'J-2. == constant from ~'=-a.. Then ). (o..) and ),., (t) the values of ). . fil M ,:,'-=- -t . z = t resp_e_gti~l!.J_~re 1 ct) related by (3.7) 0 Proof. If r is the radius of the sphere, the metric on the sphere is JA,i. = n,'- h' (3 .8) ""l.. 1.. ' + it., ~ I 'Z. f- ,I.,~ 2... IJVr By direct computation, the two matrices of Christoffel symbols 13 are We may take (3 .10) 6~ I, -xi cl:y,~ -- to be the parameter 1 along the directrix. ~l'c..:: J.,r,t [~ I,, Then where J.5 is Kronecker's delta =I> k=-1)==-o k:¾-l Computing the matrizant from (J.6) 13. For formulas for computing the Christoffel symbols when the coordinate lines are orthogonal, see Michal,(2) page 98 or Eisenhart,(l) page 44. 13 Substituting (3.11) in (3.5) gives the theorem. In component notc.tion, (.3. 7) is written (3.12) X Ct)= X(a.) >.2 (t) =- ~ Cc ,,\ z. (c,_) ~"t The computation can be checked by means of the theorems 14 that parallel displacement along a .g eodesic changes neither the length of the vector nor the angle the vector makes with the geodesic. If I Al is the length of the vector, and if e is the angle it makes with the geodesic, then since a parallel of longitude is a geodesic (3 .13) (3 .14) ~ 0(o...) (o..) = ----j>,o.i '/,. I -I Al (t) C)}1., 8(t) III. C.2. ::. I ,,\ (t) :::. • >--' (o..) ~ 8(0..) I >- co..)\ Displacement Along a Parallel of Latitude. Theorfil!L_.l.:.4. Let coordinates f' ,?~ be ~hosen on a sppere in three dimensional Euclidean spa.ce with 14. ::; y.,' see Eisenhart, (1) cit., page 64. the colatitude~ y.,2- 14 the longitude. Let a vector~ be displaced by Levi-Civita parallel displacement along Then A (ci..) and related by A (t) 'l- 1 = 2 constant from the values of >. at ,x 1= a., '2- ~ a.. t to 'X-2.=t • respectively, are x.'J ChO..., [ ( t-tl) ~ (3.15) ~ [(t -o...) U-12., 1:.' 1 Proof: We may take ~z to be the parameter along the directrix. Then (3.16) Computing the matrizant, using (3.9) and (3.6) n t _.ll-o., [- r1R.. ~ i:4-t .n... ' -,. l . J =T - ( o(t-a.) {!..(rl,,,')<. , ~x ' •- ( t-tL) 1.. C,.b/4.,.. ..,_, + (3.17) 2. ( ( 0 - ( C,.6/4- [(t-u_.) U-4.X 'J - ~ [(t-"-) ~ ic 'J ~ - .,iv...x' ~ [( t-u..) w-,i., X- ' J )[./ Substituting (3.17) in (3.5) gives the theorem. III.D. A Tensor Expression for the Integral of Equations (3.3) Parallel displacement is essentially a geometric phenomenon. Therefore the results should be expressible in tensor form. We sha.11 do this by means of normal coordinate theory15 for spaces 15. For normal coordinates see Thomas (1), chapter 5, or Veblen (1) chapter 6. 15 r~ can be expanded in a Taylor series about~ <~> • for which Theorem 3.,2. If a contravariant_yector, indicated_py the column matrix /1. is dj.§_placed by Levi-Civita para.l_!el displacement alo!)g a directrix from 5 =a.. to 5 = t , if the coefficients can be expanded abo!tl_ ,:, (a.) in a Tailor serie.§, y, i ==- ~ \ "'5 ) r of connection fl and if A(.i.\ A Ct) are the values of .A at :t.("--); -x:. Ct) respectively, then (J.18) where I is the unit matrix, A.Jr, j Rressions for the normal t ensors ~ J [xJ )'., (o...) and ~tion ~_;= :\~ i:.-'(J lj J [x.J Proof: 16 , are the matric ex- , AY?.i L of they:Qace evaluated at are the funct;i.ons of y., from generic to normal coordinates with Q_!:igin are functions of s by reason of the directrix From (3. 5) and (3. 6) (3.19) A (t)= [ r-1 t rre ck\tg +l ~,e, k1cA sJ r ~ J./"J-r - . •• } ,,\("--) d J-r iJ.r a., "' 5 ~ We may now transform to normal coordinates x(a.) • We place a prime normal coordinates. I ~l with center at over terms which are evaluated in the 5 , of course, remains unchanged. comes (3.19) be- J r:._ c½""1-r -· ·• };\<a..) (3.20) /(t)= ~I-Jt r; b_f\L 5 + J t:r ' h fl 1 5 1 L J. -g a.. ti Ti° a.. O; 16. defining the transfQ..!m:: see Thomas,(1} pp. 102-105. d../J-' 16 Now we assume that the coefficients of connection, r~ , can be expanded in a series about the origin of the normal coordinates (3 .21) where r~ (o) and the different det'ivatives are all evaluated at the origin. But at the origin of normal coordinates, I f-1e (o)::: 0 (3. 22) are the normal tensors of the space where Aie, j (3. 23) - o . 0 The. law of formation of terms is readily seen. represent 11'v , ~t J(µ / ' •~ - / A .. • ~ ••• ':) Xi" J5 where A ••· has If in (3.23) we 1n, indeces by the number and integral composition by indicated product, (3.23) may be written ( 3 . 24) '/ (t) :::: [ I - [ 2. + 3 + 4 + •.. J + [ 2.. + '3 + 1./ + •.. J [ 2. + 3 + 4 + .. •J +- L,_2.-t3 + 4 -+ .. ·J [ Z. + 3+'{+ , .. J [ 2 +3 +tf+ .. · J-1-- + [2..+ 3 -t ··· J [ 2. +3-+··J [ z. + 3+ ...J [z. -t3+ • .. .J- , .. } 11 <o...) On changing back to generic coordinates, (3.23) becomes the conclusion of the theorem 17 (3.18) where the ~ 1., are given as functions of "J,, by the equations trans- forming the generic into normal coordinates. We note that the normal tensors are constants with respect to the integrations and could be written outside the integral sign if desired. . ' Further, since ~ :: J~ at J y.. te --,:.,i(a) , In general the functions -;t, (x) in a transformation of co- ordinates are scalar functions of 1, . a tion A'reJ·=A1ej. However under a transform~ CX) £ =- ;£ (x) , the normal coordinate functions undergo a linear transformation iii= u; J 1- .\().,) where the partial derivatives J J cl ~J ~ i~ (o.-) c);. are evaluated at ~(~) . Hence the integrands of (3.18) J are mixed tensors of rank two evaluated at )'., (a..) the normal coordinates. the origin of , The tensor indeces, of course, are sup- pressed in the matric notation. Theorem 3.; shows that the tenso r expression for A{t) ma.y be quite complicated. Hence it is worthwhile to seek some re- ar~angement of the result (3.18) which may make a more usable expression for A(t) Theorem 3 ._§.. matrix ~ Such a rearrangement will now be given as If a contrav9.:riant vector 2 _ _:i:,ndicated b;y: the column is dis2laced by Levi-Civita 2arallel di§_£lacement along a directrix to expanded in ~a Ta~lQ_r series @out r,(a.) and if ,.\ (a.) I= -t , if f11e Exes)] can :fil.!Q ,,\ (-1::) be 18 are the values of }I g ca., at and , res2ecti vely, thEfil 5 -=-"t- (3.25) is homogeneous of degree r in th~ where the matric term 'T1"-functions 'T, __ l;'.i~ [}:] To -(3.26) ~ given bx the table I 0 t tt ~r Are,· J ~jcx-J q___i_ LX.J 11 = 'T, -- T .. rr3 .I ::: - A t'lt j· t ' Q c:4,P [ x] J_ 5 S ~ [ X] ~ [ x] ~ 0 d, 5 ~ J ,- f ' Jt rr-: -- - A ~ j h,, f?, is d le ~ 1-f' /µ a, j f ' ~ [.x ] ~ Lt ] ~ Ii, [ x] cl ~ l'2 [ xJ h, f J. J 1.J' - d5 5 • Tt' "homogeneous of degree r in the functions means that multiplying each ~ 1. [xJ in '11 1'\, equivalent to multiplying Tn., o<.l\., ; by ~ ' [x] by a constant n o<. is ie, In., L ol '.)Lt x11 ::::. ol.l'l.,.. rrn,.. L'J iL~ JJ As in (3.24), the law of formation of the terms may be expressed T2- == - 2. (3.27) Ts = -s +2.·3 + 3 ·2... 1h = - t, + 2 · ~ + 3, 3 + 'f- 2. - 2- · 2. • 2.... and in general Tn = sum of all possible permutations of all integers ~ z.. such that the sum of the numbers in each permutation 19 i-S n, and each permutation is multiplied by (-1) u , where <J""" is the number of numbers in the permutation. The numbers, m, in the right members of (3.27) stand for terms of the mth degree in ~ 1- [x.J as follows (3.28) where there are m indeces R.,1,···,~ • Multiplication in the right member of (3.27) is integral composition. For example (3 .29) where the quantities in the first integrand are f unctions of para- 5 , and those in the s econd ar e functions of parameter meter Proof of theorem 3.6: s 2.. • From the absolute convergence of the Taylor series and the matrizant function (J.6), (3.18) is absolutely convergent. Hence (3.18) may be rearranged term by term to give Theorem 3.6. The results of theorems J.5 and 3.6 show that the tensor representation of ,.\ (t) can be quite complicated. A different approach to the problem will be taken in chapter VII which will supplement these results. III.E. ~he S2~cial Case of a Closed Directrix. Of all directrices, the closed directrix is the most inter- esting. Of the known theorems on Levi-Civita parallel displacement, / \ one o f the earliest and most fundamental is that of Peres, giving the results on displacing a vector about an infinitesimal closed curve. It will now be shown that this formula is but a special approximation to (3.25). 20 If a,_contravariant vector, indicateg~the column matrix is displaced by Levi-Ci.Yi ta parallel displacement ,X ' :s g_Qes from 1 completely about a cl9§.ed directrix 7,"-= 5 = a., 1Q 5 = Ir r',e [>r.] can be expanded in a in one_~ircui t, if Taylor series, and if )... (Cl..) , ~"Cs), where are the values of A and >- (t,) fore and after the circy1t, then to terms of second degree in be~ [xJ (3.30) where I is the identity matrix, is the matrix of the ff re.j 17 Riemann curvature tensor evaluated at 'j J, UJ is the es function of x which transforms to normal coordinat:i:-89- lj,., in transJ,(a.,) , forming to normal coordinates with cen!.§.r at ~ co..) • Proof: From theorem ,3.6, to terms of second degree in (3 .,31) \(tr)== LI - Alli Ir • 12. ~[xJ , f' ftµ ~jc~J ~ h:_ J.s } o .,_f .:ls )...ca_) Integrating the integral in (.3.31) by parts, since (3 • .32) 1/r ~ j (); Ir fr Ja, j ~ J_ 5 ~ 1/ j le J c,, t.( s Since the curve is closed, ' Ir,~ .:) J_ 5 ~_; [:x lo..)] = jj [){ (t-)J term on the right in (3 . .32) vanishes. , so the first Taking half of (3.32), inter- changing the dummy indeces gives for (.3.31) (3.33) A(lr)= s .1 -t- ( Aitc,-Ati ..i ) 1- 11,. ~.; bk. h.f' J.g J ,A(ll-) L V 2. a; J ·;:.f d.J Noting that (3.34) completes the proof of the theorem. 17. See Eisenhart (1) pp. 19-22, or any text on Riemannian geometry. Som€ authors differ in sign from the definition as given by Eisenhart. 21 li.rej = matrix (tc.~ n..i) , where tc~,e_; are the We note that components of the Riemann curvature tensor. Corollary 3.7.1 . I \ Peres formula . To terms of the second d~_r.ee in X ("s) , the generic coordinates of the curve, Proof: The transformation equations from generic to normal co- ordinates are ( 3 . .3 6) ~ ·\ "'i ) =- :t J. <s) - i, j ca..) + terms of degree two or more in x. Hence Ir to first degree terms in x. directrix is closed. ' Further fa... J:. \ot.) t • n.. d5 =Dsince the Hence to second degree terms in x, Substituting (J.37) into (J.30) gives corollary (J.7.1.) It is easy to show that if the directrix is a parallelogram (3.35) reduces to the forms given by Eisenhart and Thomas in their discussions of displacement about a particular infini tesima.l parallelogram. For the full significance of this result we need the theory of the matrizant as an analytic function on a Banach spa.ce. 22 The Matrizant as a Function on a Banach Ring. IV. IV.A. Definition of a Banach Ring. The Ring of Sguare Matrices. 18 A Banach ring is a complete normed linear space, or Banach space, with a multiplication defined . There may or may not be a multiplicative unit. A .set of elements y form a normed linear space N if the following conditions are satisfied. (1) A function of two variables called addition exists on NN to N, and is written + • In symbols, if 'j, €:. N -LI >113€:::f'/ (2) Multiplying any element ~I+ 111- - number -J'3 d... (3) yields another element and ~ z. & N then ~, f:. N on the left by a real O(.~,-= ~..,_ '- N. A function,called,norm, on N to the real numbers exists and is written II l;1) I\ • (4) '1,+ (~1-t' ~.) ~(1:),-t- l11-) + ½3 (5) ~,+\h :::: ~ .. -+'1, (6) o(, , (~,+~1-)=ollj,+ol~-z... (7) (o(, + o(l.) J ::. o(I (8) 11 + -i: :=:W\ (9) II o{ t:\ ll = 1°(1:/l'jll , 'j + ~2.. 1j ~iW e- N has a unique solution r.. t= N o( any real number ~ E: N (10) . //<jll ~O for all (11) if and only if /l~ll ==0 ~ =- D (12) ll1j 1 +y.,_II ~ II 'j, ii -t // 'j .,_, // (triangle law) for all 'j,/hb/J 18. Banach, S. (1) 23 A normed linear space is a Banach space, B, if in addition to the above 12 conditions, the Cauchy convergence criteria is That is, every sequence {x.Z1 of elements satisfied in the norm. of B converges if for any for all -m... 5 .t -? N I E there exists an N such that II 1'.-1>1- X, 1, /I< f A multiplication is a function of two variables on BB to B which is (4.1) (a) associative (~<;)z:-=: ;:t(~'c) (b) distributive with respect to addition e ( X+~) (X+~) =t:.-=- X 'r. f~'e-. . (c) modular /[ ~ ~II fc M •~ i l/ II ')II independent of ;t and where M is a fixed constant The least such constant is called the ~ "modulus" of the product = ~ X- -t- 'r.~ 1- j • As an illustra tion, consider the set of all square matri.ces of order n whose elements are real numbers. These matrices for m a normed linear space if the norm is taken as the maximum of the absolute value of any element in the matrix a 19 . In symbols , if A is matrix whose element in the c-th row and cth column is A~ Further, these matrices form a ring with ordinary matric multiplication as the ring multiplication. 19. For, in component notation Such matrices form a normed linear space with other norms also, such as the square root of the sum of the square of t he elements . However we shall consider only the one norm, ~ince it is easy to work with, and is used many places in t his .the~is. 24 ( A-t B) ~ =- A ( B (!_ ) ( = (A '~),,__ . ) e (. -=-- A:... ~ 131'>\.tt'.. L.. <:., A rt'™ + C3 n..,,, ) t 1>1~ t ( At B) =- t : :::. An..-m C, """'c.. + nu >t ~7'\~ 11-t ::: ~+Ii3 A ri c.. ( A-r;_ -t t3 ~ ) -= l ~ A~ -+ ( :, f3 ~ ==- {!., A -t- ~ 13 II A f311 -::; /I A ';.. P.i :'. II ~ n II A J/. II 13 I/ Clearly matric multiplication satisfies conditions (4.1) with modulus M-= n, the order of the mate-ices. Now let us consider two auxiliary Banach spaces with values in the Banach ring of squat'e matrices. Let B1 be the set of all square matrices of order n whose elements are continuous, real~ numerical, functions of a single real numerical variable range ct. ~ 5 = Ir • in the 5 Let B2 be the set of all square matrices of order n whose elements are continuous real, numerical functions of two real numerical variables 5 1 '{ in the range a..:: 1 J 'I = Ir • Then if {I /IN be the norm (4.2), and if * C~) be an element in B1 and ~ (~ 111) an element in B2 , we may take as the norms 11 ~ l~)\\ =- ;n.~ ( 4. 3) It t.(1 )IIN 11 ~ l 5 I rt) II = ln,0/j, j) ~ ls, >t) II N a.,~ 5 ~fr C(, f= S; 'l ~ Ir With the norms defined in (4.3) the sets B1 and B2 are easily seen to form Banach spaces. Now the mat ri·z an t (4.4) _n_: [AlsJ == T + s: Acs) J.5 + ... considered as a functional. apart from its geometric role is a function of two real variables, -<i, i t is a functional on B1 to B2 . • It is clear that .Jl.~ LAJ ;25 IV. B. The Diffe r ential Pro12e rties of the Matr-izant. Professor A.D. Michal 20 has proved the following theorems about the ma.trizant, considered as a function of the ma.trix A in B1to the Banach space B2 . Theorem :4.-.1· The matrizant function (4-.4) on the Banach spa.£J?. B1 to the _Banach S.:2§~ B2 satisfies the differential system in r Frechet differ-entials (4. 5) f JL : [ A >oA] .n.! LoJ 17 ] ,., J2. ; J A (s) J2.;, cl 1 = T where 6 1..12.";, lA '> J"A] is the first Fr,chet differential of _Jl: lA] r::i th inc rem en t aA , .1)..: [o] 1s the matrizant of the zero element of B1 , I is the unit of the ring.2 in t his case the unit matri~ . Theorem 4.2. The higher order Frechet dif ferentials of the ma- trizant with equal increments SA are given by (4. 6) J n -11.! LA :, oAJ =- k ! J,: ..n.;, [AJ cf A c~J J s, J,!• -11;~ J" A (-s,_) Jr .. ·.. sn. -1 [Ai cJ A Cs ) Jl. J",a d. rt ,..._ J 's,z _, ...J2. fn. n, Iheorem 4.3. 21 The entire analytic J 22 "t: ;,re., series (4.4) giving the matrizant function can be differentiated termwise_to give 20. 21. 22. Michal, A.D. (3) Michal, A.D. (4) An entire an alytic series i a a Banach space is defined as follows. A homogeneous polynomial of degree non a Banach space possesses the modular property, 11 P-ttt l<>II f M>t u i: 11 ..... M11 is the modulus of the homogeneous polynomial, -P~t~) . A series of homogeneous polynomials 2 -Pi <x) is said to be an ~nti re analytic serie s if the real numerical s e ries 2 Mc A~ converges for all values of ;.. . While it requires more than the modular property f or a function to be a homogeneous polynomial, it is easy to show that the series (4.4) is an entire analytic series in the sense of thifi footnote. For a more extended discussion of polynomials, see Martin (1) . 26 (4.7) f__n_! lA·> + -f JA}-:: ]"'1: JA (~) cl! -t r r• [_ J: ir-A-(-r,)clr, J"' 'An .. )d.'f,_ ... J"- ~-,A ls,)J-s, + i::; 2. J,,,t- A c~,)cl~, f :· cJ A (s._')o..'5 1. .. . I:" J /\,t , !' A t 'h) As A ls, ) J.. 3, J J"'S;-, A ( s/) cq·< ' + .. • + 3 1- • -- J r\., ,; _, oA l 1.: > cl s~ We note that (4.7) can also be obtained by substituting (4.4) in (4.5) Theorem 4.4. The following generalized Taylor Series Expansion holds for the matri·z ant for all continuous matrices A (5) !!!Q i A(5) . (4.8) o<> j :, _n!'s.. [ AJ cl A ( L ') J s.,. j 5.:-, .Jl_ -~/_-, [ A] J A Cs,) J)_3~ J. St.' It, .)~ Theorem 4.5,. where (4.10) a., ~ )l, 1=. e- t !::_ fr Jrv-t: A Cs)Js and where The matric exponential of a matrix 'T (4.11) -t is the matric exuonential of J~ A( 5 )d 5 is defined as the series 'T' E- where rp 1., is the matric product of i _fillual matrices 'T1 . The above theorems are assumed without proof in this thesis. 27 v. The Change in a Set of Parallel Vectors Produced by Warping _1he Space. In this chapter we discuss the effects of warping the space on a set of parallel vectors defined along a directrix. By a warping we mean such a transformation that the coordinates of all points remain unchanged, but the coefficients of connection are changed. Alternatively, a warping may be regarded as changing the coefficients, SiJ resulting change in , of the metric of the space, with the rll • Under a warping, the equations of curves, and in particular the e.quations of the directrix curve, remain unchanged . of parallel vectors, however, changes in general. The set Since any vector of the set is given by assigning a particular value tot in theorem 3.2, it suffices to consider the change in A(t) given by (3.5). The result is Theorem 5.1. ~et a contravariant vec~or, indic_a ted by the column matrg A ~isplaced by Levi-Civita parallel displacement along the directrix curve :i/-= X-i. ( 5) from "(Ir) J2,e the Y.alue.§_Qf J ::. ii- iQ. 5 -=-"t ). ,il 5 =o.. and J : : -t • 1tl }. ( tl.) and respectively. If the space is warped in such aw~ that the coefficients of connection {7 f?.. [ X n )j are given an increment O 'k [ X ('5 )J , if ). is displaced a:s before with the same initial value ACa.) fil!£Lif..l!Q!! at ! -= t then I (t) -,A(t) = f - J°"'r JL!,, c-rf2, cxcs >J %,tJ srl'e, [x c ,)J :Z:~ 4 ~ = ). Ct) 28 Proof: From (3.5) t J.1-lt \ (t)== Jl.o-- [-rtt---x-:g J 11ca.) ( 5.2 ) - J r12 b_fl. J A("-) 11 (t) == n;, [- rte #tt P-5 .ls Forming the difference, using (4.8) gives the theorem on identifying - rtt ~ As with A and - J'r1re. ~,e_ Jg with JA in (L;.• 8). ( 5 .1 ) is obtained by regarding Jl~~ Since in warping only as a function of - ,,-,re b:..k_ d-g • changes, our interpretation of (5.1) can r1e, be sharpened on considering _n__~ as a function of P,e, alone. In this context (5.3) Now 12 A( rre. 1 ;:: - rl'l .ls J. , j, is a linear function of f',e Hence it pos- / sesses Frechet differentials of all orders with increments given by Using the composition theorem for Frechet differentia.ls of functions of linear functions , (5.3), (5.4) ~ ~ Jl: [ r~ ~ J rI?. J = J "Jl: cA > J ' A <rR ) J ( 5 • 5) _ - Using "Jl. i- [- r~ ~ •, - J rre. .&::.n. J o., tl\- 5 '1-5 (5.5), (4.8) we have Theorem 5.2. wri.ttfil (5. 6) <; d Under the conditions of theorem_5,l, (5.l)can bf 29 V. B. Example . WarJ2!ng the Plane into a QQhere. As a,"1: example of a warping, consider the plane with pola r coordinates f ,e . Regarding it as the equatorial plane of a unit sphere with center at the origin, we map it onto the unit sphere by projection from the south pole. Hence the center of the plane maps into the north pole, the unit circle maps into the equator, etc. We assume that the points retain their old coordinates, f, e , under the mapping. We see from the geometry that the colatitude, 4 , is related to the co- ordinate f by (5.7) Figure 1. ~ = 2_ ~ -I f Using this relation, and the fact that on the unit sphere in spherical polar coordinates the metric is (5 .8) We obtain (5. 9) on the plane • ~ on the sphere k 2.. = d. t + f 1-)_0'"2.. i. 4 = 1--;:-)... J .. ,J. "f/ f k .. d a1.. -t 11 1- ')1I-'. /' (5.lO)on the plane r, :c (: (5.ll)on the sphere :) I'• = ( : 23 -n By direct computation of the Christoffel symbols , - t- (t-i )) Jt~ l.. 0 23. See note 13 on page 12 . 30 where f , e are considered the l-- 1 , ;x,z. coordinates re spec ti vely. Hence - I+ -4: fz. (5.12) ~ 0 0 t -t-f 1... J r2. = () - _ -y_ ~ I +f z. 0 l+f"I- A be displaced To fix our ideas, let a contravariant vector along the particular directrix with equations (5.13) a.., =0 -t =11 This directrix is half the unit circle on the plane, and half the equator on the sphere. For this directrix, from (5.14) on the plane r, = ( ~ ~ ) on the sphere rl rz_ (-o/ o) = ( ~ -~) 0 -- rz. . d r, =: ( -~~I) (5.10), (5.12) = 0 0 ( o o) c) rz. := ( ~I ~) Hence, in the plane, using (~--9) n. -J-r ( o- ,)~h - ( ~-~) (-t- fl,) (5.15) SL+,,_ L- rtt ~ J =- e "' i o :, ==- e Then ~ ( 5 .16) 1( ) - ~'tr' / - (-' 0 0) -/ and the successive Frechet differentials vri th increments (5.17) = ( o ') 5 rre.~k. ~$ -, 0 are from (4.. 6) 31 where i. E. = E., i-4 t, > '-I Hence the generalized Taylor series expansion for Ji: c-rR ;:-/ - er rre, 7/1 is (5.20) J _ -/ Cl) + (0 -I) + 2. / (Io) __Q_o [ - ( O + T( 'I ( I o) + , .. 11 3 ( o +- , 1r o -1 ) I ( 0 I ) -1 O 1( l. - ( o 1( -I o I 0 I I ) 3, - -I O If ( D I . ( - U;-v1r ~ 'iY This result checks exactly with the result by direct computation, which is easier in this instance. Hence from (5.2), (5.16), (5.20) (5.22) _n: c:~) 0 Al ) on the plane A(1't')= on the sphere ~(fr)= ...Jl;A(o)= (~~JA(o) ,\(o)= This is checked by direct readings. In plane polar coordi- nates the components of tvvo parallel vectors at opposite ends of 32 -a diameter through the origin are exact negatives of each other. Further, the equator is a geodesic on the sphere, hence parallel displacement along it does not change the components of A • 33 Directrix. In this chapter we consider two parallel vectors whose base points are connected by the directrix defining the parallelism. We discuss what happens to the second vector if the first vector is held fixed and the directrix is changed. is left unchanged. The space, of course, We shall assume that the new directrix is subject only to the condition that its end points be the same as the end points of the original directrix, the base points of the t wo vectors, and that the matrizant function .il.. [.- rtt ~n..J exist computed along the new directrix. We first give a tensor matric result related to (3.18) of theorem (3.5). Theorem__hl. !!!.?-~!!Jf Let the contravariant vector indicated by the column A be displaced b:y Levi-Civi ta parallel displacement along_ a direcm~ J;"= ,:,i( 5) the val ues of A at from 1::: a.. J =-a.. fil.1h t o 5 =-tr and 1 = Ir i,. ((l,,) , ,,\(tr) If A is di s- r es pec t i ve l y. plac~g alQng a new directrix with the same end points as the old, x/-= :r-·* c,s ) = ;r,.: (5)+ 1:.i(s) and if )-/'' (lt) > "l. i(s)=o ;s-=a.,) lr is the value of },. after displacement along ;l'.. l * Cs) then (6.1) ). If (lr)->,(tr)::: f]lr ( I - J /,.. [ An j ~J,[K] +A1e 1 j~ L[ X]!:/[-<J+-" • It La, · ] d <j cx1 k. f ,l.-g .. .. ,) 3 ())(. f (6 • 1) ( A 'l'>I ~ rL ~ ?J [ >C /3] ~ J. -r. [ xJ - <j -1 ,\' ?! [ x * 13] ""':1 _ 5' '>1 h., ,L,, ?>! ~ - ~ A3 ( xi<-J } + d 3 d 3 + A1t1 11.h. l 1:1 16 )f-ni ,A~ ,)(-'1 '1 "" h., ~)('-7'! T ~ }+", ) (1.- J _3 L A( j<jj-+Ah ~ j\/+··J -1;~7")(i-r-l11.+ " · ) J 3} ~(«.)+ 0 34 where Ail<--· t. , are the normal tensors of the space evaluated at ljieC xJ are the transformation functions giving the normal coordinates with center at x.' (a.) in terms of the generic co- ord inates. Proof: From (3.5) (6.:2.) >-*<1r)->-C1r) = 1.d: l- where .st;:: • r; 7; 1tJ-..IL: c-r,t ~ 1} >. (a.) is the matri·z ant oomputed along the varied directrix From (4. 8) where - cf rte c½R. = r tl-5 le h te.. <1-5 -r~ ch,*!? re ,,lg matrizant -5l. to yield _[)__~ . the increment in the kernel of the For clarity we write out the first two terms. , ( 6. 5).!.. J" 2.Jl: 2.! As in theorem 3.5, we transform to nor-mal coordinates with origin 35 at -XL(o..) • We expand generic coordinates. rte. about the origin and transform back into Osing (3.21) and (3.22) J. re r rt.* kj./4.,* k::: Ate j ( ~ • [ x (s ) J ~ (6.6) • [ Xt i >J J - R. j * J()(l\s )] ½lf- [ X~(r )]) + tl-5 J. i + l ! A R-j'l ( 'JJ. ~l 6-.~ - 1j * ,; lj x-t -6: ~) "'s J-.s whet'e A l'l,i ... .L are the normal tensors evaluated at -;i; i( a..) , ~,; == ~,; [ x ( r )J are the parametric equations of the original directrix and ~ .i ::: Lj .:H- \ x* l 's )J are the equations of the varied directrix in normal coordinates transformed to generic coordinates. Using (3.6), (3.21) and (3.22) ( 6. 7) Hence using (6.6) and (6.7) in (6.J) and (6.2) proves theorem 6.1. VI.B. I Two Lemmas for the Existence of Frechet Di~ferentials ,;, "1r r r J-,,.,n. o ....I (... (),., ► - rt ♦, rs * I J s The theorems of chapter IV permit us to compute the Frlchet differentials of the matrizant J L.JL [ A ·) JA 1~J with increments cf A Usually, however, we do not think of a displaced vector as a function of -rre. ~k. rectrix. but as a. function of the space, or of the di- In chapter V we discussed the Frlchet differentials with increments cfr~ produced by warping the space. we shall now prove two theorems which will guarantee the existence of the differentials with increment Theorem 6.2. ~i ( 5 ) p roduced by changing the directrix. If the function ? I., ( 5) is a function on the real 36 numbers to the vec,tor space 1=~ a. '=. 3 s lr of sets of n continuous functions differentiable almost everywhere, and if ~ is regarded d. 5 as a function of x .Qll E1 the norms of (6.8) and f" 1 to , a second vector space, and if £ '2.. are respectively E'z.. l \ 'JL."(s )j I ¼~ ,:Cs )j 1 0--:::-gf:: lr j =- li'l0 [ I ~ (s ) l J ~ s !: Ir- j /I )'., ~( ~) /Ir::: "'YYl¥ ~ l =l1 • " 1 7v J Cl 11 <j L ( s ) I) ~ i L'-::; I ) ' • '1 I\, a_ L / possesses a first Frechet differential with increment ¼ i. (5) Ts .x and for any ~ 1!1 given by £, (6.9) The Frichet differential is given by the Gateaux diff er- Proof: ential provided the Gateaux differential is additive and continuous in :c and provides the principal part of the first difference. The Gateaux differential is (6.10) which is obviously additive in 1:, D :: J...'-b J-g , vt'3 the differential is obviously the whole of the first difference. Since it is additive it will be continuous in~ if it is continuous in =t near the origin of £, But from (6.8), if . /f=r:11 t.... E. then Since the first difference ~ ll ! ; II t.... £. • Hence where E.. is arbitrarily small, 1; is continuous in z and the Fr/chet differential exists. I 4 l It is necessary to include 1~5 ) ological curves 'z: t 3) for which l 'l> I in (6.8) to allow for pathis small but \ clb I is very °' s 37 large. This guarantees the continuity of For if 1/x.JIE, otherwise would not exist. then given any M no matter how large and JC-" ts) it would be easy to find a curve and Theorem 6.!.2. J., 'l: d--g in =r.... were just max /1-'(s)J , c no matter how small such that //~.:II ~ E. would not be continuous in ,::, The function rl'l '7ifl which , but . on the n Banach spaces B1 of sp.uare matrices of order n and then vector spaces E2 to the Banach SQ.§£..f. B1 possespes a first Fr~chet differential with c. increment in E2 given by (6.11) Since Proof: r~ is an analytic function of the coordinates x exists and is the ordinary differential, Jrn.. ~.l • By J xt. I j., le_ cl 1-z. theorem 6.2 f [x, Cb] exists and is given by 1 ; . Hence is a sum of multilinear function s the theorem folsince rri J'r/?.,L><it-1 ½ t; lows from the formula for Frechet differentials of multilinear functions. vr.c. The Fi rst __Frtchet Differ ential J' ,. \ [ x > c I lrJ V..Te now use theorem 6.J to prove Theorem 6_-..it.• matrix ~ Let a contravariant vector, indicated by the column be displaced 12.Y_Levi-Civi ta parallel displa£ement a.long the directrix z,"== ~,. c5 ) be the values of ,),(Ir)= from A at 1 = _n_:; }.(~) where Jl.~ "-i 5 -=.A. lr to s=- fr • Let ),. C6...) , )... (tr) respectively, and related__£X is the matrizant function . ILthe directrix is changed to a new directrix with th e same end ooints 38 :z i. = x *._Cs) = ;t i. (s)-t- ?: with equations then L (~) / Frechet differential with increment ( 6 .12) Proof: r. c,(Lc 4 > ho(1 rc,o{ c1. ~.) _n_-:, [- ri posses_ses a first given by r.. J' A C><:/c 11r J = J.,.,1r n:, c-r1e Jy---cl?. n,J A(tr-) I d-v 1.. 7-s~ J Jt; From theorems 4.1 and 6.2 we know that the Frechet differwith increment ential of .Jl.~ [- r,e, J,,t:RJ J..5 that o'[r,ehte. ''l:] exists. ~ A r-r,e J,y.k.7,, cl s J may regard JJ...:; we exists, and as a function on ) E1 to B2, that is, a function of x defined by _n_: [xJ = ...IL; CA c-rJJ I Then by the composition theorem on Frechet differentials, JI Ir _n_<I,., (6.1.3) c><,"r-J exists and is given by 1 J'Jl.~ [)<'>~] J .Jl.: [A> J'A ["':,~JJ - 5 1 ...11..1r - 0., c- rre.o\.5) ~1e. • - cr'r h. ,e J -tol:s = J'..1)_ lr-[_-r w ~n... _ cJP-!::_ -::z::.?'h-&~- f1.e. ~ ] A5 ) ox.= d..J J:s (e, By theorem 6.1, (6 .1.3) is ( 6.14) I Ir ('\ 1,- [-rt hk.1-sJ (- ~ ~-n<.,4£. J1L., [ x:itJ :::: Jo. .Ir, JL./.lJ re d.3 Jx?><cJ..,..d,, _(l.~ [- f (° ;:;,~ tributing the middle factor. )A., 1 J.v Taking the second term and inte- grating by parts, ( 6 .15) - ]a., Ir_[)_: r~ ~ .Jl_ ~ 14 ::: - { Jl_: ft_ r. L JL: j : t.,.. Ir J -+- j ,?(. e.-£.J (6.14) can be broken into two parts by dis- The right member of -t- /7,L {1~ )_ _fL 4 r. l _IL-:: )4.; a., pl.A.., ir- .Jl '.: (I., + J/#1r ....n..t ,4.... J 11,e .h:. .,. __ "r.,L JLp., k o;,:= d..5 rt ~ 1. J... ;::-4., h 39 Now since the varied directrix has the same initial and end points as the initial directrix, t:Ut.) = ~(1,-) =-0 . term on the right of (6.15) vanishes. (6.16) Hence the non-integral Further (,- d.. _[)_4 J./4.., and (6.17) - f1. L~ ) J-y_kc~J _iL-4., [-[l-l-J.,,,J. lsJ ~ te. ~ As Substituting in (6.15) gives ( 6 .1s) - J1r_Q a., l,./4- r 1e. J.-1: k. JL!' ~ ~ Ir 1 _n_: [ h:.e auhd f1" 2 l: le..+ Pl. J..f.f- f11e. 2 If. _f1fl r. t?. µ__, 11, 1 k~ l ..Jl! k ~ J Using (6.18) in (6 .14) ( 6.19 ) o'Jl.:: cx,1-J =]4,1,--11..t {arn._ or.( -tr'tf1a -Pt?.r-e}:z:.te.b_{ ...!)_: if..u h.1- oi 1e .:A. .4, But the matrix fot>m of the Riemann curvature tensor with one contravariant and three covaria.nt indeces in generic coordinates is ( 6. 20) o 11... £,/'t::: o_rI t - .\ r ox,e_ d):le v_.{ ;- r.e. f1,e - rR r .t Using (6.20) and (6 .19) gives the theorem, since 40 (6.21) l I >.(1r)::: J I _n_a.lr Aca. ) because J' >. (a.) ~ 0 VI .D. The Second Fre{chet Diffe_rential cf 2.),. [ x:, ~ I &J Theorem 6.4 has gi-ven us the expression for the first Fre'chet differential of A . of We now seek the sec ond Ft'e'chet differential A with equal inc rem en ts Theor em 6.5. 1: We begin with a lemma. • Under the conditions of theorem 6.4 (6.22) Jl.. .v Jv eo{t. ?/ r · Proof: I?, ( 1") z: le <-r) _n_; Following exactly the same line of reasoning as i n theorem 6.4 we have ( 6. 2.3) J IJl ; L x ~ :q ::: Jn.'r JL: ( - J~ f7xlk 1/- 3::1!: le._ f1 d.,Lur: k ) _()_ 'l J..u, /4,, cd_,4,, te., Inte grating the second term of the right member of (6.2.3) by parts ( 6 . 24) - j ?{'Y nL r --1 r1 / ',e, ;J., cl r: 1e. .-fln, ok, =. _ 4, - [ n 'l"' -1L ,Q,J r c te, k + J11.,r ~ _r,__,,. fl ~ k.. ..1l 4.- k ~ .. ,v le. }t + te- n ,tJ ] -1 L n_ ,,, i'( Jr cl rre Je k l -1LA-> ~ ?t.-'c dx,L ,,;...v Evaluating the non-integral terms on the right in (6 .24) l1'A- }t, +Jt' ..il 't- r1 'i;k._e!.__Q4-c4_, 11. /4->fe_ IM,, 11. ,,., since _J)_ ,4, = ..L ) the unit matrix, when s = 1"· Using (6 .20), (6 .24) and (6.25) in (6.2.3) gives theorem 6.5 Using theorem 6.5 we now consider . Theorem 6.6. Under the conditions of theorem 6~, A (tr) p_ossesse;, 41 I a second Frechet differential witq egual i!!~r§ments give,n by 'l. (6.27) Proof: Since Ac~) is independent of the directrix, (6.28) 61..~[ X>i:>~]-::. cl z. [_fl__: ,).(tt.JJ-.:: cf1.._IL;;_ [.><'.i ~ '.>~J.\Ca..) Further J2...JL~ ==- ; ' £ s 'Jl: 1 ::: d (6.29) 1 1lr_JL~ rcd.,"- 2.. u o(' J-L, 0., r. o(L_n_:::;- k Ir Now Jl.. 4,, .R.cJ.. c(z.. are elements in the Banach spaces B2 )B1 of square 1 matrices. E1 W 04,, and respectively. ~ ol.. are elements in the vector spaces E2 , (\ lr and . Clearly J ~L~ is a sum of multilinear functions on B2 B1 E2 E1 B2 to B2 . Hence if the various factors have Frtchet differentials with increment z, then the Fre'chet differential of J '__n_t with increment z exists. I and l,- BUt theorems 6. 5, and 6.2 guarantee the existence of cf _Jl/42., [x: , :e] 1 • ,, o J re -1, ol t_ =o(,3 since R. a1. d.. J" "< [.x; '2:] • J' i:. = 0 and o ,~ c( _, -= 'Z, ,~ 2.. - -- () X. o(3 ~ IL is an analytic point function whose Fre"chet differential is the ordinary differential. / Hence by the theorem on the Frechet diffe r- ential of a multilinea r function 42 (1) .(.2) J Ir Ir- a, _j)_,4) (3) P1,.-__n_1,- J ti.) (4) ,4.., For convenience, the four terms on the right in (6.JO) are numbered (1) - (4) Using theorem 6.5, terms (1) and (4) together become ( 6. 31) J 1,-f Ir Ir Jl_ t IV Cv R. ri/1, ;-A 1.- hA ;9z.. .JL t ).1- (c ):t- -.: ,,a., , d. '.1 o( 1.. J.,c,°' 1 C: o<z.. _[)_ cu ~ )-4..J + )4 (i) (ii) (iii) (iiii) where the terms in (6.31) are numbered (i) - (iiii) for convenience. The terms with ~ with ~ indeces are functions with parameter /4.I. Those indeces have parameter~ . Interchanging the order of integration and certain dummy indeces in (i) shows (i)= (ii). Combining (iii), (iiii) with (2) of (6.30), we may write (6.J0) as (6.32) J ?... V_fl_(µ [x:, t]-= / J Ir _fL.IJ.,Ir R_ 2..' a, + J...,lr_1l ~ [ c) o(, ,(?.. ~.<.I - µ, 1:,~~ l:. o(z_ J.4_ I J .JL 4.1 a,, 4, _A..,_, {?_ /314.t. ;a,/32. J-,t,f3, ~ -__Jl_A, ,M.,2.. + µz_ r -t- [I<t 3 /Ce.<,«., - /c cl.,o<,_ "'3] to(' 1= .Cz.~ o(3 __n_:-t4 + 43 Hence on using (6.32), (6.28) and (6.27), we have theorem 6.6. VI.E. Dagger Differentiation. On comparing the expressions for \' 2. we see that o fr _QI)., ,1Jl1r c, a.. cx,-lJ and cf "l.. Ir Jl..a.. [x, ~J is almost but not quite obtained by formally differentiating the expression for 0 1-11.t with respect to an arbitrary parameter. We shall now define a formal notation, called "dagger differentiation", and symbolized by "- whose rules will from J'_n_ by formal operations. give S"...Q Theorem 6.7. Dagger Differ-entiation. If the operation called de.noted ''Dagger Differentiation"" by the symbol 11 J " has the following properties ( 6 • 33) J _n_; R, ~, .Jl_: i ::: 1:. ct_<- )1,1' ::. pd. q - 0 i- I z_ ol. J t _fl_~ r,4_, ct_ o{z_ I :::z. o(3 L and if the ordinary formulas for the derivatives of products and sums hold for dagger differentiation, then lr •I 1,- J JL 0-, [ )() ~ } -:::.. _n_~ Ir ~1..Jla.. [)('.>1:] Proof: Setting (6.34) f J ..n.. ~ tr 1r O 1 _fL 1,- ti, -..11..t,.i~ - ,i=a,, ,4., 'l-o(,o<i. (), of (6.33) gives (6.12). t= Ir in (1) ko(' d.,4.1 c o( t. Ir J,( ,4, j(_ 4,, --11.1,1., J _o( , ((, r/. Io(. 2. (,,vy.,_ (fa, ~ '·} p..,(l_., c,(2. ~:::= ,4.., _j)_,,_, ~ + (1) By (6.33) 44 Ir 1 1,- 1 1,- ~ ~o( I 1:: 0( i_ __J)_-:; )a, -t R. o(, oi 2. µ.,, a., _fL,a., tJ.; 1 IrJl,4-' lr p.; 4rx, Ro{, .J.1... t (2) ~c/.7-._n: ,,4, + (3) c/,.4.., Ir (4) -11/3, (5) whose terms are numbered (1) - (5) for convenience. Numbers (1) and (5) from (6 .34) a r e Note that terms with ~ indeces are functions of parameter ,4;-1 and those with ~ indeces are functions of parameter ,4,L. On interchanging the order of integration in the first of (6.35) ) terms (1) and (5) add together to give (6.36) 2.. ! lr 1 11, Ir JL A-1 Ro(,~1... %.,{/ ~ o{z_ k, JQ.-4,/_fl~',., R.A;9.,_, )-y.~, :z:;Sz.JL1t.kL tl4z.,, I Further, term (4) vanishes. term in (6.32); We identify (6.36) with the first terms (2) and (3) with the second and third terms of (6 .32) to give the theorem. vr.F. The mth _Frichet_Differential Having found the first and second Fre'cb.et differentials we seek a. recurrence formula for the mth Fre'chet differ entials with equal increments~. we define the sequence of P functions As a preliminary 45 (6.37) The first in this sequence of P functions has a.lready been defined in (6.27). We extend the properties of dagger differentiation defined in (6.33) to include (6.38) IJ Po( , ... o<~ - po<, "' c<-,,, c/. nt+I r::. A'c°"t -- 0 ~ <><'m-1-1 With this extension we have Theorem~. differential Under the conditions of theorem 6.4, the mth Frechet 61'11., ~ [x) -:z: I Ir] with equal increments l: exists and is given by the recurrence formula J™'>. [X'.>'cj/,-J-=: _n_'.::, (6.39) .n.;,Ir'{, ... (nt,)•·· wh ere ~ 'I, .. . (M..)" • t ).(tt.) is the mth dagger differential of _!Lj;" defined in (6.33) and (6.38) Proof: As in (6.28), since ~ (a__) is independent of the directrix, Om~ (1,.)::: s-r-i___n_; [l<'.,l] ,A (4-) We shall now show that The proof of (6.-4,l) is by induction. to be true form= 1, 2. true form. Theorem 6.7 shows (6.41) We assume it true for m-1 and prove it By considering the defining rules (6.33) and (6.38) of dagger differentiation, it is clear that the formula for the m-1 I,- lr lf · · •( rt,. -1), .. p, dagger deri va ti ve of .Jl...., is not a simple one. However ..Jla.., is the sum of terms each of which is homogeneous and of degree m-1 in z. A typical term is (6.42) fL-J)_~ P o1. ... o1. • hot 1 l;. ~1: . . l; "-~; J 4.1 n :• Po< , . .. o J,1::~' t1 o1-,•+1. ol1>1n4,'-)4 ~ of4 L ,_.,_ 'ol. nv 't. •·· ~ ....l l- tJ..o 'Z.. t- 1 J , tJ..o J +I --' d.A, l... I The term with the most integrals is I We shall show that the Frechet differentials of (6.42) and (6.43) with increments z are indeed given by the dagger derivative. Further the method is general and applies to any other term in the l,- ,t . .. (m -1) -- - 11- expansion for .Jl~ . Consider ( 6. 42) . SL ~> Po< 1 , .. _o(~ are elements in the Banach spaces B2 , B1 of square matrices of order n of chapter IV. ~ ~ is also 'e:-< an element of the vector space E1 while ~o< d. ;zd.., an d t{j,L, are in the vector space E2 . Clearly (6.42) is a multilinear function on B2 :s1 E2 E1 Bz B1 E2 E1 B2 to B2 . of Hence the Frefchet diffe renti·a l (6.42) exists if the Fre'chet differential exists for each factor. Since Pc<i-- •"'i is a differentiable function of x, exists given by the ordinary differential I J P«, ... ot J [ x ; e] 47 Theorems 6.5 and 6.2 give the formula for d dv;i,~ [x • ~ J ~ ) 1 • 5 :z:. o(. = <f' J e o< = D J.--4.., Let us write S' (6.42) for the first Fre'chet differential with increment z of the expression given by equation (6.42). I Then by the formula for the Frechet differential of a multilinear function J'(t.,,,'-12.)= ( 6) (' Ir Ir- Jp,, __IL,o._l D f o(1 •·• ~ J✓,r_.J, ~I t a{ ol.. 7° .. ~ _; µ, 1/4.' «; f) 4-, Jt,,,4 ~ Jfa(,HI• --d. ""- J .t li' L 8itJ ••.q. t4'L. where the terms have been numbered (1) - (6) for c onveni enc e. We wish to compare these results with the dagger de rivatives. Hence we compute (6.42) d,ri {}-/4; ~ omitting all terms involving since these terms vanish. ~ ~ The terms are numbered (1) to compare with terms numbered (1) - (6) in (6.45). indeces have been omitted for clarity. or 11 - The obvious (6) l.? 48 (6.46) continued. (3)' ]JL p J.=t:- 't.··· ~ )A.,jJL pcl.'b O"'-: JL .14 + ( 4) t J .JL p Z, ~ ~ Cl . . . t ~ A- J .JL" p t • • z: JL k + 11 J.-.,t '2, ... 'le p<.4.,, 1 } .Jl. p J.:e't··c-..JLh-+ (5) 'J -11.. p ()4., ,:).A_, I ., (6) J _Q f_r;,,-r, tu, On comparison, (3) l: c• " I . ,:;<-<I., = (3) t, . From theorem 6. 5, (1) will have . two terms, one i nvo 1 ving J..a JL.,..Ir (r I term is identical with (l)~ (1) involving + i, k J J)_ p ~ Z:: • .. ~ _fl_. dA-, I It- Ro1 o( I t., j l.. .,«, ~ ,:1-:r ~ o( ,_ _n_,_,,.,t; clt on using (6.33). This . The other term of r~ 1='6 we will combine with (2). Further from theorem 6.5 (4) will consist of three terms. a 4~1 d. t l+The one invo 1 ving j 4-,z. -1ltA-, J'\..,ot, o1.z. # '2:: z. .Jl. -z.. ~ is identical 4 4 with (4) '- . The term involving -rr ~ 'i is combined with term (2). The third term involving+ r~zt is combined with (5). Similarly (6) consists of two terms, one identical with (6) and one which is combined with (5). But the three terms (2), the original (2) plus the extra term from (1) and the extra. term from (4), add together to give (2) ~ . The three terms (5), the original (5) plus the extra term from (4) and the extra term from (6), add together to give (6) i The method is clearly perfectly general. . Hence For example (6.43) is a sum of multilinear functions on B2 B1 E2 E1 B2 B1 E2 E1 ... B2 to B2 , with the Fre'chet differential computed by the product rule. Every term involving ,, 1._o(, o VV¥-_ 1M, [x,~] is the same as the corresponding 49 1 term involving 4"" i- • ~ a. term involving r)' Each term involving J l?.a. o(L is combined with I from the J .Jl. , the differ-ential of the hr IM., matrizant appearing just to the left of a ter-m involving - f\ h-,.. on the right to give f'lot.,d..z.. ~~.~L • The remaining terms of the differ• This method can be applied to any ter-m. ~' (6 . 42) resulting from = in (6.43) and with from the differential of the matrizant M.lk!.' entials of the matrizants .Jl... are _()_t- equality • 1 (6.42) t we note that the required a regrouping of the terms S1 (6.42), but that it requir-ed no regrouping of terms arising from any other term in J • always true, so for each term T of J" -m-1 Ir Jl.a.- . -m-t This quality is Ir JL a.- we have Hence J / 6ni-l __n_;- -= ( J1n -I _a_~ ) 1' --=- ( 6 • 8) 4 ( Jl~ Jt'· • (rn-1) ••• o/,) i = _n_;- '- .. ,~)· ••~ which completes the induction . Under the conditions of theorem 6.4 the vector CorolJ_ary 6.8.1. A (ir) possesses the generalize9: Taylor series ex12ansion oO • ·-z, JL At [x> 1,]} /I (et.) l::: I ' A(lr) = £.L -ti Proof: The proof is irnmediate from theorems 4.4 and 6.8. VI.G. The E~licit Fot'm!J_lae for and 'L ,3nlr o ..--JLo.., V f,. . [x~z:.] 1 J --17...a., LxJia.J 1' S Jl.1-l Lx>~J It is clear from the recurrsion formula (6.39) that successive Fre"chet differentials of __o_:, with equal increments z will be more and more complicated. J~_J)._t and However, it will be of interest to compute Jv.Jl: for futur-e reference. Further the recurrsion 50 formula is simplified by the fact that the original and varied directrices have the same end points. The first Frechet differ- ential for the more general case where the original and varied directrices have different end points was given in theorem 6.5. It will be of interest to compute the second as well. We give the first result as Under the conditions of theorem 6.4 the third and Theorem 6.,2_. fourth Frechet differenti.§:1S of JL: [)(J with equal increments z are given by ( 6.49 ) <;' 3 o + J Ir (), l,- _ ? _fLa.., [><~'i:] - c.... f'l 1,_JL ~ 0 ro(,o(1.o(a"'I JIr J -o(I !!2::_ d,A., o., r\ Ir -11.....,,o_., o< i:. i.t. O I ro(,o<2,c( « ,/. 3~ 3 « <><, ~~ t: 11 4.., I 'l-11.....()., t:i4..- ,J., i.r:.-<3 J2. a., I. t:Mv-+ + + 3 Jcvlr-_fl~ le o<, d.1. c¼~J 'c J.2. ~ , t,,-4, JL:~ R. A!3 -,_ t:2../3, t= /31.. _f)_:2. JA, and (6.50) J-w?... + 51 ( 6 .. 5O) cont ' d . + [ If ~ , J1., l,- 1,- h""1 _{/_,4., {( J., -<2. "J:i_ I. f Jl.2.Jl.. ~ i!, 2. /1-"l I Ja., "2-2., rJ. 1 /I R h A/3:z. J • .4. J Ii; -2. J. A.J f t 'c. I . 11 • ~ , )4, s .-a3 A:z. ,,, :JLA3 II.. l..lf r, Y2. ~ J ~ ~,. µ,3 U3 A. <f Ja.,:JL.-4.,,, tr.J.Jz. h ' i ~Ao- h'f µ'( Proof: The formulas (6.49) and (6.50) follow immediately from the dagger differentiation laws. It is noticed that many of the terms resulting from the dagger differentiation are identical. Combining these like terms gives the peculiar arrangement of coefficients in (6.50). For 52 For the first of (6.52) is ( not writing the curvature tensor terms or the terms summed against it) I µ_ J A., A., I. j )2.2.. /43 J. ( 6 • 5-3 ) Jra.,1.,- JA ,Ir Jl. Irt ..I/... A.t ' o,.;t ' a., ..fl. ,a ., ti'-<1- " a, .J2. A _fl_ .3 A.2, 3 a.., Cl'-(!, Interchanging the order of integration with respect tot and with respect to ( Ir A- 1 gives 1~t Jl_A,"t ),o._ J ~, (r 6.54 ) ]a., Jlt µ I a., A-1 JL,4,2.. # J,4.'-_f)_.42.. 2. a_ ,,-4,..3 fi-3 _n_a.., 1. ""'"4--3 which on renumbering the dummy parameters and inserting the curvature tensor terms gives the same result as the last of (6.52). Similarly for the other two terms of (6.52). The second result listed at the beginning of this chapter is Under the conditions of theorem 6.4, the second _!~em 6.10. I Frechet differential ,z.. 'r' e _fLx. [x, ~] (6.55) dI~ (r) 7:, l'r) z:fa(r) .JL; o(. 0 ~ ,8 + l'o{ ('I'") 1:."" (t) r/6 Cr) z. 1.3 ( t) ..fl. ; is given by 53 (6.55) cont'd. - 2. r~ (T) loo( (r) r/3 ('1) "2: ,.s (l\) + _(L; +- ..Jl.; rc1. Cl\) ctC>t) r13(ll) 'l:.,a(11l + + .JL; cl r,d't) z: olC'J-\) ~/3(ll..) J-y_,13 Proof: From theorem 6.5 (6.56) o'".Jl: cx>'r.J - J' 1r.Jl:1 = S' [Jr JL 1' R.cJ. o( ht-"'' z:. <J.LJL ,a., k + _/2_; ~ Cti) i: o(c,t)+ 't .,L.4.)t ,<), I T'r:J. (r) - '- :c~(r) Jt; 5 t" I Clearly J Jl.ri lx> t] is the sum of mul tilinear functionals on B B 2 1 I E1 E2 to B2 . All the factors possess Frechet differentials, so the right member of (6.56) is given by the formula for the derivative of a sum of multilinear functions . By direct computation the t'ight member is 2. J,,,t Jl.: fL.,c(, 2 4..L, l:.-'-1.. k, J?'/Jl'.JL;' fr;,, f.l .LJs,~ /31.. n A-z.h + (l) +Jr- .J1.,: Pol,~2- "'-3 t4-o1, ~ <><._ r -<3 _n_: k (2) I 'I ( 6. 57) J;.., 1\ -t J1/ 1" L J. v-v-, 'l"' _IL R, R_ o<, o( 2. J.-t-o<, µ_, t:-- ~1.. n ~ ....J L )t I 1/-..4.- /"1 / JL""'!-_ {),i.~ +- JL...?t Z. ( 3 ) (4) (5) (6) (7) (8) 54 (9) (10) (11) + _fl__ ; c) f1o1. J r<( c>t) =t: o( ( 11 ) ~ 0 X, (3 1= "" er) i!: /3 ('J-) f\ 1l ) (12) ..Jl. ; (13) ()"X, ;3 where the different terms are numbered (1) - (13) for convenience. In the computation of (1), (2) and (3), the contributions due to r 4/ the para.meters s in .JL.,4, and ..Jl?t a.re in cl uded in (2), but the contr-ibutions due to parameters 't-' and are given in (4) and I'{ The remaining terms are straightforward. (5). Adding all terms (1) - (13) gives (6.55), the theorem. VI.H. Applications to a Closed Directrix. By means of theorems 4.4, 6.5 and 6.6 a shorter and more meaningful proof of corollary 3.7.1 is possible. we give a lemma first. Theorem 6.11. If a vector 1 indicated__gy the column matrix )\. , is displaced __QY Levi-Civi ta Qarallel. displacement §:bogt J! closed directrix, z " =- ~ •-' (1s) , if in making_ the circuit to j -= Ir , and if v L = ,x ~ (s )- x'ca.) then_ttie first and second Fre'chet differentials of A[ " tlr-J with increment v (6 . 58) 5 g__oes frQJl! s = t<- exist and at'e given gx 55 where R point ~.:(a..) • , the Riemann curyature tensor is evaluated at the Consider a closed dil'.'ectrix Proof: 1- i:(it.) o(,o(?.. ~i·=µic 5 ) through the point =- x." (Ir) , parameterized so that 5 goes from a to b as the parameter traverses the directrix. such an increment 1:.L that Then give the directrix l i • Then from theorems ,M, ( 5 )+e\-s)=XCs). 6.5 and 6.6. (6.59) We now shrink the original directrix This implies that ~d..,rJ..2. ).,t.,iti ) to the point II u} c~) - u.,: (a.,) 11~ 0 • Lxc~l,3 ➔ de °'-,c(i 1/(<1.)J. Hence • u., L(a.) • 'l. i ( ~) ➔ 1r 'cs) , k.,q Ir In the lim1 t -D..,.a., -=- I , - .14 • =0 • , .z::_ 1.--= v '- . The only non-va...~ishing term of (6.59) is the fi rst of Jz.. A which is J°1..A [x:>v- llr] :=. R.. ot ,o{z.. f Ir d.v"'' (µ 1Fo1LJ.,: d.5 ..) which proves the theorem. Theorem 6.12. If a vector ~ is displaced by Levi-Civita parallel displacement about a closed directrix -,:.,; =- ~«:ls> , and if f.!:.Qm ~ to 5 goes Ir as th~_p_g_rameter ·12oint traverses the directrix, anq are the values of .A at before and afte_!: ~'Co..) gj,_sJ2_lacement, then the n,rigcipal t.fil:m of the~eralized Taylor series e::irn,ansiog for the difference )..(1r)- .>-. (~) • I l l 1: cf z.. A[x;,rJ 56 Proof: Corollary 6.8.1 gives the generalized Taylor series ex- Since the directrix is closed, J,,_,lr ,r"'z. ~"• J 3 -= la-1,. ~"'-,x,,,,.J..r. . .:1-s v<s .) Using (6.58) gives the theorem. pa.:r1sion. Corollar~ 6.12.1 / ' Peres Formula. If the directrix is an in- fini tesimal curve in the sense that If '\l" i II _(6.60l gives the change ,\ (1r)- .>.(o...) The chang_g is the second Frechet diffE;.J:fill!i§;l of ;\ Proof: • with inc_rement is infinitesimal, then ir • From the definition of the generalized Taylor series expansion, each term is of higher degree in ing terms. z than the preceed- Hence if 1/ zll = I/ vii is infinitesimal, the only con- tributing term is the first non-identically vanishing term given by (6.60). (6.60) is the equation usually discussed in texts like Eisenhart and Thomas under "Parallel Displacement About an Infinitesimal Closed Curve", although frequently a parallelogram rather than a general curve is taken. It is a matter of simple quadrature to reduce (6.60) for the parallelogram to the form given by these authors. 57 Parallel displacement is essentially a geometric phenomenon, and is independent of the coordinate system. vector If the contravariant A is displaced along a given directrix, the value of A at any point of the directrix is independent of the coordinate system. That is, the components of A in one co.o rdinate system are related to those in another system by the ordinary contravariant tensor transformation law (7.1) A ls) :::: >:Lxl -i; l]= B (5) ), [X ("s)j or (7. 2) .-\ c-s) =A( ~ ) I ( ~ ) '1.1\fhere ACs>, 13 Cs) are the matrices of the partial derivatives (7.3) c))lrL) A ( s )= ( ~=;:c. ) B <s) == ( a ;x; A.) c) ')('..c where the direct and inverse transformations are given by ) It is seen, that as written, 8cs)= B[~tJJ and A(s)= A [xl}J 23 are functions on the vector space E2 regarded as defining AL£ ( 3 )J to B1 24 . Of course (7.1) is and might more fully be expressed as (7.5) and (7.2) as (7. 6) Now since ), ( Ir) and (7. 7) ~ (a.) A(tr):: __n_t >. (o..) 23. See theorem 6.2. 24. See section IV.A. are related by theorem 3. 2 as 58 it is worth while to enquire into the tensor characteristics of .J)_ 'r. . and __fL We seek the r-ela tion between _fl ~ [ .1<} =- _n_: [- f1o1.. kol... I 5 ] (,.. 11,, [ £] :::: Ir Jl(,l., - [ - .1.--;- tC ri:c. ~ 16 J "'s . One approach from a different point of view has already been made in theorem J.5. Theorem 7.1. If a vector ~ displacement algn.g a directrix if A(o..) , ,\ (1r) is displaced by Levi-Civita_parallel 1/· -:: ;ic: ',( s) from the values of ).. .at 1 -=- o.. > Ir 12.Y_the equation ,\ (t,-)::::- ..n..~ ~ ( 11. ) Proof: s -= lr and respective1_y are_related then the matrizant fµncti2:9. _n__~ is a ~ i n t tensor with base pQints of order one at •x.( lr-) "s =- <!t. to '1- ( A-) , X. (&-) and covariant of order one at , contravariant -x. ( a..) . The following equations hold in the x and x coordinate systems. (7. 8) (7.9) where the bared t e rms ar-e the functions computed in the bar coordinates. Using (7.5) in (7.9) gives - lr (7.10) B (tr) ),. (tr) = _[)_°" LxJ B (a.) >. c"- ) On multiplying on the left by A(lr) == 1 8- (1,-) (7.11) Since (7 .11), (7 .8) must hold for arbitrary vectors }. ( o..) (7.12) n lr [ )( j - ' l.. o.., -- A l t,-) JL.tr[ x ] f3 cc.. ) - But (7.12) are the defining equations for a two point tensor, base -59 ,points )c. ( o_. ) , 1G Ctr) , contravariant of order one at X(lr) , covariant of order one at X( A) , which proves the theorem. It is clear that the parameter limits may be anything in the range of definition for Theorem 7.2. J . Hence we have by the same proof If a contravariant. vector ), is displaced by LeviI ' Civita 2arallel dis:Qlacement along a directrix --:K. t= ~'c r) from ! = a... and if A (~) , A Ct ) !2 5 ;::.lr are the values of ).. at 1 =Jt and res12ecti vely, are related by A(1"') =- -1L~ ).. ( >t ) , then _fl_.~ for1~ a two ;2oint tensor fieJ:d with base point_s at ¼( t') , ;z. Ci-t.) , 'J.. ( t-) coy_ariant at ~ Clt) for al__1 77. /1.J such that contravariant a. ~ }1 1 t- '=- lr .That is (7.13) ,t =.,... Note that the theorem is valid even if .JL'11.11 ::: I ~ / and _JL 7t r- i. However, 0 '}-' -1l 11. _l_...J2... 7-'~ -::~ [x ~=cJ or 1'( > r , since u... __[)_ n. = (-,1 -' 1.. ?. ,,_. I >t. ) -/ does not have such general tensor pro- perties, as is shown by the following theorems. Theorem 7.J. Let a contravariant vector A be dis:2laced by _1,evi- Ci vita parallel displacement along x '·-= x. i. (s) from 1 = CL to ~-:. Ir- ,1tl -n__t ~ -g -=. o... be the matrizant function relating the values of • at Ir and s -=-lr .l2l 1-llr-)=- _n_a... .-\Cit). Let the directrix be varied to a nex directrix with the same end t?Oints fet'ential of JL~ 1-,(a.,) , ;c (tr) with equal incr-ements .. and with egua- z. two point tensor with base points ')'. (tt) , X( lr-) fot' all z in E 1 contra- variant of order one at --y, (t,..) and covat'iant of ot'der one at x C"-- ) if and onl;[ if the tt'ansformation -£ = f" (>) is l i ~ - 60 Proof: The theorem will be proved by induction on the order of the Frlchet differential. (7 .14) l,- The right member of -1.,- .Jl_~[)(.] = A(tr)_f)_a., rx:J Bea.) is a trilinear function on B1 B2 B1 to B2 , where B1 is the above defined Banach space of square ma.trices of order n whose elements are continuous real numerical functions of one real numerical variable, and B2 is the Banach space of square matrices of order n whose elements are real numerical continuous functions of two real numeC'ical variables. Hence if A , 6 , _fl_}: nu each have Fre'chet diffet"entials, the differential a' JL/: [x; 'e] will be given by the formula for the derivative of a trilinear function. A,B But are matrices whose elements are differentiable functions of I x and x, hence theiC' Frechet differentials are the ordinary differ- entials. 1 1 Hence cf A [ i';i) and J' r3 Lx;:eJ exist. But since Z=(A-)= "t.(1r)=O, because of the end point conditions on the di r ectrices, <f 1 A[.i>i/lr] = J'B [x>=e:/.t.J=O. 1 function of x, 0 £ (rj Further since -:i (x.) is a differentiable exists and is given by the ordinary differ- ential, in matric notation (7.15) Hence using the composition theorem for Fr~chet differentials (7.16 ) , I Ir ) I - /r a -1la., Lx·i:z:J = A (tr- rJ .JL~ LX- .iJ =- A (tr) J I Ji ( i c~ ; -6 ) ] 13(a.) t [ X j B i': J B(o.) But T3i =i if and only if the transformation is linear. expand · :i ('tt ~) we obtain For if we 61 (7 .17) x (x-i-~) == x (,:) -t-/3-2 + higher order terms involving d 1..xo<. z::.~'c, i cl ):.f3Jx. 'll' by definition. Hence (7.18) ib B~ '-- .( + c) higher order terms involving~ ~~~¥ Jxf1c))(¥ Hence (7.19) if and only if the higher order terms vanish, ie, the transformation equations be linear. Hence, if and only if the transformation is linear, then ( 7 • 20) CJ t ..Jl. !: t )( z } .:: A Llr) 6 JLf [ X j i ] B(~) 1 j Clearly, if the transformation is linear, and if we make the induction hypothesis, then in equations (7.14), (7.16), (7.20) /,- ':'e can replace ...fl(µ by J equations will be valid. ~-I lr _n__l'- Ir I • lr by J ~_n_a- and J" Aa..- , and all Hence J" JL /: [x:, iJ is a two point tensor. This completes the induction. (' "l. Ir Conversely, let us assume that o ~~ is a two point tensor, without assuming that { () Ir J ' -1. 1,..~ is a two point tensor. Then we can write where Then 1 +A J'Jf.; ex> J 1i: t:t'>-=e. >J B+ A 0 J2.;cx ; Jj Lx ~i.,JJ 8 -+ A J 2..Jl.): [ x ., ~ i ~ :J B 62 Hence since is a tensor, the last four terms of (7.22) must sum to zero. Of these f our terms, the second, is homogeneous of second degree in z, and the othe rs are >if not identically zeroJ of at least third degree in z. Hence the second term is linearly independent of the other two, and must vanish independently f or all z. I d i vanishes. lx '> l) But (7.23) vanishes only when But this implies that e. be independent of x, or that the transformation be l in ear. Hence, i f Jz..Jlt [)(',r.;~J is to be a tensor, the transformation i-=- ~ (x) must be linear. .An holds if cf i, JL;: This completes t h e proof of is G. tensor. identical type of ar gument theorem 7.3. z ==- i (x.) is Similarly, if and only if the transformation • t' linear, then 5 i. ...Jl1t [x; eJ is a two point tensor. For differ- entiating the results of theorem 7.2 (7. 24) cf 1 -1l;L-x-~r.J - 1 5 Acx)~l r--J-1i;LicJ B llt)i- A (~J JLt [ x] 5 13 [xj 'e llf] + 1 A/:t) o'Ji ~ [.x,; a'-i L)Q'>-eJ Bt,i) If x i t'.tx) is linear, then 1 \::. 1 J A::: J' 'f3 -=- CJ since A and B are matt:'ices of constants, and fut:'ther, o'-i"c x.i~J='Bi!,.==-~ transformation is not 1-inear, then l 'Bt-.,.,)~ O unless vt-=a.- . o1 A(1-) ¾, O • If the unless 1°'= tr , 63 In a manner similar to that used in theorem 7.3 it is then easy to show Theorem 7~. Under the conditions of theorem 7.J, the Fr6.chet differentials of the matrizant tfi..Jl.~ [x·,ic.J are two point_tensor fields if and only if the trap_§formation of coordinates ~-= i'(x) is linear. The situation is entirely different when the change in the parallel vectors is caused by warping the space as in chapter v. In this case we have Theorem 7.,2. ' -t J i._J)._a, cr~>J r1.,J Under the conditions of theorem 5.2 is a two_QQint tensor field, contravariant of order one at "/., (t) and covariant of orde r one at Proof: ;t::(a...) • , the coefficients of connection in the bar co- rl'?... I or-dinates a.re Frechet differentiable f unctions of ment t§ rri • l11c, with incre- For, the equations of transformation of the co- eff icients of connection are, in component notation (7.25) Clearly the Fr/chet differentials are given in rnatric notation by (7.26) (7.27) and (7. 28) J '- Fl?. [f'.t ~J/7-e J exist and 64 Now differ-entiating the equation corr-esponding to (7.12), and using the composition theor-em for- Frechet differ-entials (7.29) JL! [ r ] == Alt) (7.30) tJL~ L r'>orJ t .J)_tl- [rj G (A.) = Aet) o ' _if_:[i=r )cf ' r1[ r j J"rJ] 13co..) ::: A lt) o'Ji..£ [ r i cf r J B c~) Since A, B ar-e independent of Pre. ~ ' (3 , the differ-entials both vanish, hence no ter-ms involving o'A or- 0 16 in (7.JO). J' ' A , occur However, (7.30) is the defining equation of a two point tensor, contravar-iant of or-der- one at of order one at ~(~) . J(.(~) and covariant This proves the statement of theorem 7.5 for the first Fre"chet differentials. The proof for all differ- entials is by induction, using a step by step proof which is entirely similar to that used for- the first Fre'chet differential. The Classification of Riemannian _§paces by Their P~!~· VIII. VIII.A. The Concept of the D~gree of a Space. It is a common and u·s eful practice to cl·a ssify plane curves by their degree. The degt'ee of a curve, when it is expressed parametrically, may be defined e,s the order of the lowest order derivative which vanishes for each coordinate equation and each parameter value finite or infinite degree. 24 . Plane curves may be of Similarly we seek to classify / Riemannian spaces by means of the Frechet differentials of parallel displaced vectors with changes in the directrix as increment. Spaces of zero degree are spaces for which the first (and hence all higher order) Frech.et differential of an arbitrary vector vanishes for any and all directrices and variations. Spaces of degree one are spaces for which the first / Frechet differential of some vector for some directt·ix and some variation does not vanish, but for which the second (and hence all higher order) Frechet differential does va.nish for any and all vectors, directrices and variations. In general, a space of degree mis one for which the mth Fr t chet differential does not vanish for some vector directrix and variation, a.nd the (m + l)s t I Frechet differential vanishes for arbitrary vectors, directrices and variations. A space of I infinite degree is one for which no Frechet differential vru1ishes for arbitrary vectors, directrices and vat'iations. Of course the parametrization must be in terms of an essential parameter. ':j~t 3 > x == t 3 - sis only a first degree curve, even though the second and third derivatives do not vanish for this parametrization. 66 Because of the dependence of the displaced vector on the matrizant function, this problem is equivalent to classi/ fying spaces according to the vanishing of the Frechet differ- entials of the matrizant. VIII.B .. We now seek the se conditions. The Conditions for the Vanishing of the fr {chet • Ir Differentials cfLJL~ [X','e] Theorem 8.1. Under the conditions of theorem 6.~., the ne£f£- sary and sufficient condition for thLvanishing of J 1JL~ Lx>tJ for a.11 directrices an open region and variations e i:-=-~'(s) ;l:.i=- ;t:: t( l ) in D of the space of definition is that R. ,ej =D 2-.t all points in D. re, i = J, .. . ,Jt., Proof: From 6.19 Clearly, if R. .i. ,c1.2- vanishes, , so does (8.1). On the other hand, if (8.1) vanishes for all directrices and variations, it vanishes in particular for the directrix and variation (8.2) J'..i (i ) arbitrary except in the neighborhood of ;r,i( s ) -,:,fl (J) =- "J,t(s 0) L =f, ... ) fe-1,fl+}"' ,r -x-" (L) -t- ( 1 - L) k l,V\""-F =E.+s-so :L- E. ~ s !:: s" -= E. s. f:: £ ~ ~d-t S l s- so l~ Zt. /"5-f.,/ ~ Z L. ;:i 0,) 67 We illustrate (8.2) in Figure 8.1 a, b, c L+2. £. Fig. 8.1 a Fig. 8.1 b Fig. 8 .1 c With these directrices and vac-is.tions, ( 8 •~"J ) \ I Ir (J _(lo.. [X','c.] _ - J So+ 'F"-£ .)0 2 C. 1.,- JLA- R_ol. oi. I l. # .,l j.A._, 1 ~ol.i_ n .-0... I. ---1 L Q... ~ (not summed on 1 ) If we expand the matric terms in a Taylor series about we have 50 68 where O (E. 3) represents terms of degree 3 and higher in E. Since we may take (8.5) & as small as we please, 0 _j)__t R,flj (so) J2.! -=-0 But -n.. ;: and JLJ, 0 since __n_ ~ ....D..t." = I have inverses . _a[· and _n_;: respectively, (This result follows since -1l~ _f)_ 0 f~ is the matrizant describing the displs.cement of a vector from '1 o to l.r and back to Jo , which leaves the vector un- changed.) Hence, ( 8 • 6) /?_, 1?.j [ X I3o] =- {) But 'Jt( so) is an arbitcaary point. where. Hence R. re.j vanishes every- The openness of domain D guarant ees the existence of the curves in Figure 8.1 in D for sufficiently sme,11 E. This proves the theorem, since ~i • can have any values 1, ... , n. Under the conditions of theorem 6.4, the neces- Theorem 8d. sary and sufficient condition foy :the vanishing of the s~coBg Frfichet differential, and variations ~ le~) oz. Jl.t [x:, 'e] for all directrices ~ I-'(~ ) in an open region D is that (8.7) Proof: If 5 'JLi" c x ) eJ ft ,i,s =- t> every,Nhere, then from theorem 8 .1 vanishes and hence so must ~2.J2.; [ x '., cJ . The proof of the necessity of the condition is similar to the proof for theorem 8.1. The two special functions for 1a <5) will be 69 sketched only in Figure 8 .2 and not described analytically. E Fig. 8.2 Let the directrix be given by (8.2), and l et ,(8.8) I :f. = / o Cs) l_'- = ,, ('$) z.3 = 0 I Then from theo rem 6.7 (8.9) J7=Jl: [x:_;~J-= fa-1r__n_}:, Rr1.,<J.z:. ~°'' r.t>(z:._n_~ J-v + We expand the matric terms in Taylor series, obtaining (8.10) dl._[}_~ [Xj'.!}::: • O 0 + 70 + Use is ma.de of the skew symmett'y of in o< , > «z.. in the ft o( 1 rJ.'l.. fit'st term. We consider the integt'als of the fit'st terms of the three series in the right member of (8.10). We are interested only in order of magnitude with respect to E s3 • The first integral is of order £z.. , the second of order order This can be vet'ified by direct computation Ot' by £q • and the third is of noting that the order is unchanged by changing the cut'ves slightly as long as a tet'm is not dropped or made infinite. Hence in the integrals in the second and third terms of (8.10) the function s/~ / ' may be replaced by the step function ll(1,)=-D s s- 1:-'-Cs)= E.. $0 ~ ~'( s )= O sd+E L.. L... s ~ so-tE.. s The third integral of (8.10) is then of the same order as (8.11) and the second is of the same order as (8.12) 71 and the first, with no approximation, is (8.1.3) JJ ., "so+E. - E. k = - E. 2. All the deleted terms will be of larger degree in than the first term in the series in which they occur . . Clearly then the lowest degree in c appearing is the second. may take E. Since we as small as we choose, the coefficient of E. -z.. in (8.10) must vanish, to wit, Hence as in (8.5), (8.6), since _n_;: 1o Ir is ~rbi trary and _J)_ ! o , have inverses, everywhere. But the variation could have been chosen similarly, but differently to give any permutation of the subscripts on ~~, a L • Hence everywhere (8.16) for all values of k, j. This completes the proof of theorem 8.2. Theorem 8.~. The necessary and sufficient condition for the ~ n g of J 3 _n_: '1:- '- ( 5) for all directrices -------------· - J:,'( 5 ) and vari·e .tions in an open region D of a Riemannisn space is for all k,j = 1, ... , n and all points in D. 72 The proof of theorem 8 . .3, while along the same lines as the proof for theorems 8.1 and 8.2, is rather more difficult. We define for future use as values of ~ the curves (8.1 8) E. E. E. Note that each function / '"C"i) 1 i: = 1, ···, m,... is the same as every other function except in the range 50 :: 5 ~ so+ c. "'ange, L ,J ;, c-.:)-= (J-!o)i ~ 1 £ i:-1 - . and in this Th e proce d ure w1·11 b e t o c h oose • particular varied directrices so that ~" (3) = / I?. Cs) or zero. The matric terms in the differential will be expanded and factored out of the integrand. we particula.rly note at this stage those integrals involving J., i c,) / ~r) which ai:-e summed tl-5 R. ' 73 a gainst a t e rm which is skew symmetric ink and j because of the skew symme try of R. ~J in k, j. For such integrals any contribution of l / ~l-.;,.) f J·Cs) will cancel that of over the region 1. +E. ~ s = s'o+ 2. E. ,,LI J'cr.> f tee~) ds But from the dagger differ- ~s • entiation formul~s (6.33) and (6.37) for the different Frichet cl !~ dif ferentials, the only place 7g appears is when summed against the first index of ftte.j some t e rm in the P1e,o1. . .. 13 s e ries. -l:..« The othe r indeces are always summed against a term i~n only appears in an expression Hence • 1 -s since ( l. :a-.,( =t. /3 _ ~.a~~ ) ks in c1,,. f3 or d.s • If R. d. /3 . ~ o(' = / vanishes ove r , 13 i:.. = / J< i ) , "cs) and are all skew symmetric , P~ ;3 .. . l!' the parenthesis term 3 +c.~ sf:: 50 +Zt. as noted above. 0 i These functions, / (5 ) all diffe rent, are so chosen that the degree in & of any single or iterated integral of any combination of them can be computed at a glance. (8.19) ('1 2 t.t-s., .J. i,( -S ) J j - £ T So and hence is of order Z E + so (8.20) J :So - t+ = 3 0 E. z. cli_L('O £. 2. i=I ::: S"/r. E z. i. = 2... = (...L -rl..) ?>1 +1 2. i. for each / JCs) 4 § Jg and hence is of order E.?.. for all For example ==m l • Simila,rly, I., - - cz.. i+j "J j • Further 74 is of order £ ~ as is easily seen by direct computation, or by changing the curves by letting the portion in the range 5 .,+ t. f 1 =s.;+ z. £ approach pet'pendiculari ty. the degree of the integt'al in E. , This does not change but makes it easy to inte- The rule for computing the degree in e grate at sight. of an integral is given by Lemma 8.,2.l. In the proof of theOt'fil!! 8.3, an iterated integral consisting of C( (8.18) and derived functions '( integrations, f3 I '-cs) defined in functions c).. t·('f.) with J. s last Qerformed integration will be of degre e limits ol. -t- ,/8 on the a., Ir .!!l E., • Proof: From the above discussion, if the integral does not involve ~ d... 1 {, tls we may vary the c.. pe rpendiculat'ly from f '- l-s) so the curve j 'ls) drops to zero at so+ c. and is zero for all This does not change the degree of the integral larger 5 in E. , since the integrands are all positive. that each f Then it is clear introduces a term which on the final evaluation of the limits is of order c" • But each / ,. is divided by so the net contribution to the degree by each / • ( 5 ) is one, independent of i. Each integration similarly increases the degree by one. When J.. I 'c 5 ) occurs in a term, the net contribution to the J. 5 degree over Sti ~ 5 ~ f + E. will be zero, since J../ i n ) is of degree 0 one less that / iC s) . d.s From an above argument, in the region 1.,+ E ~ 5 ~ so -t Z. E. the contribution will cancel that from another term. This proves lemma 8.3.1. From (6.49) we now consider J 3 _n_:[ x)eJ for the particular directrix given by (8.2 ) and variations ~~ chosen from (8.18). 75 We expand the matric terms in (6.49) about s =~ (8.22) _Qt po£1cJ.._o(3 _n_r == .JL~o + _fl_ !,- P"-,ix1..<><3 D obtaining 0 Cs.)JL!c + .JL. So _/ . ~'( CL ~ - (-;s-s.) + J. 5 ( ) JL J., J 1.x~¥ 'L. So lc<,CX:2,cJ.?,o<'i-(so) 1 +2! JL 1r P. L -<, ... o1., so c,., cJ..-g-z.. ("s-so) + +.!... JL~ P-<,··· d.s(ru)-11[" ~-<''( k-<,,-('t'-r)2. + 2. ! ~o ol 5 AS :, ~o +··. (8.2.3) _J)__~ Pot, ol.1."-3c<'l Jl.; = We consider a sample term in an iterated integral in some detail, since it is more complicated. We discuss the term which may be written Now we expand the matric function of s.. about 50 • (8.25) _[Lt fc.o1_,<1.z.(~,)J2{~ r?. 13, 13,_ (s.,_)JLa-' t. =- Jl~, feo<,~,_ (s,) JL i [ ftt3,fa2,(s.) Pt3Jl... + ±, Pp, ~,._p313., ~":, • is ... + ..!... Pi f 3 1 f.3; [r.J J.,,.f3-1 cs,. - s"/--t Ts2.- h .a~ Cs.,,-1.,; + +z., f3,/?>1./3a ~ • d-.s l, (s.,,-rJ + 5,:.. ,1.. 76 Similarly, expanding the mat~ic functions of about 5 0 51 in (8.25) we get The matric terms are all expanded in a similar way about So , giving for the expansion of a _fl_~I.,.. [x~~J 3 (1) (2) (3) (4) (5) (6) (7) (8) 77 (8.27) cont' d . J.4."'• olz. J Js, J. "l-/3, cr1z.J + + + 3 .J2..io1,- fL,.,d.z. R. /3,/3i.. .JL sa: [ J tr 4, ~ s, a, d. h Si. 1,1, (9) (11) (12) (15) (16) (17) +·. () 78 (8.27) cont 1 d. (20) (21) Note that (8.27) is the sum of several series. Each series is started with a double plus sign, and the first few explicitly written terms of lowest order in £ are written with a single plus sign. lr 3 • Since rJ '-1La.. [x>i.J is to vanish for all directrices and variations ~~ , it vanishes for the directrix (8.2) and variations taken from (8.18). We compute all the integrals in (8.27) and collect terms of the same degree in £ . Since f 1 is arbitrary, the coefficient of E~ E.. 1 •• • ) s"; .. , must vanish. In pa rticular we see that term (1) is the only term of degree three, and that no terms of lower degree exist. (1) must vanish. Hence We write (1) in detail Ir So J1,- J e °'1 o(z. ol3 I (8 • 28) ~ .JL~ Pel"' oi. J (s) Aa.. -d_S =c. =t:. .:?ls .)o I L O OJ on expanding the summations in (8.28) , the terms fall naturally 79 into groups where the indeces 1,2,4; ... ; n-2, n-1, n. groups. c,1_,) o<.._ ) o<.. 3 There are have the values 1,2,3; (~) such mutually exclusi",e Without loss of generality we may discuss the first Expanding the summation, this group in (8.28) be- group only. comes (8.29) r" SL L- J (A) "So (B) (C) l.r J -11 1., where J. Ir j_ I __!:, r. I i=- 2. d.;r + (D) J~ J_J...:s r: :z::' cl s Jla., (E) _[2_ .lo p/12. I,\., til5 ,.,, Jo '~" fr =t. I I f _il.~o j stands for the sum of all terms obtained by giving the indeces 1, 2, 3 all possible (six) permutations. The terms have been l ettered A - E for conveni ence. Now because of the skew symme try of P a...1r~ in "l;. I- J is only ove r from the f > l := l, z. ' (8.30) l:. I l'l.::: terms If we choose variations ~ 3 ~a , terms A vanish D and E vanish. and a.., lr , ( I I, 2.. 7"' '3 I We choose four sets of variations • of (8.18) as follows! f I~ , t s> I f 1 J II I / 3 1 '7 80 With these choices the statement that (1) of (8.27) or (8.29) vanishes for these four vari~tions yields four homogeneous linear matric equations in four unknowns with non-vanishin g determinant. Since 0 0 P, z. I vanish. P,z. z. __n_! Hence each matric term SL ~ r r, ~L ~ _f2_ .;,° , JL ~o p -Z.. 1 2. , must JL !,o has an inverse, this implies (8.31) when any two of the three indeces k, j, 1 are the same, for might have been any pair of indeces. l 1 z.. \ We now assume t- 3 -=\ o , but ~ =- O for L L >3 • Then since we have shown terms B, C, D, and E of (8.29) to vanish, we consider t e rm A, which must also vanish. We evaluate the integrals for the three choices of e~,~=11 l, 3 i. (.. I, -i ' ::: (8 • .32) i I' z_ t:- 7.. -- t 7. 1 3 l.' ~ -1- 3 I" t i_ ' ?, ,~ Recalling that the integrals from 1o+ £, to s o+ 2 E. all cancel, the integrals in (8.29) are, for these three cases, (,- th ' 1 a., 1 1.,- 1 1r ~ Cl, i e2.t 3 el5 p._ ~ -- ii iii E, 3/t., f, ·y3 E 3h t ½. J. 'e"l. r:.' 'i:.3 J-:s J5 -- E, 313 £ i/z olg -- 6 /2.. 2. £ oh-' J..5 t;; I ~ ?.. 3/ 3 5 £ 3/ 0 81 Equating (8.29) to zero for these three variations gives three homogeneous linear matric equations in three unkno~ns with nonvanishing determinant. Hence each matr·ic ter>m vanishes. Since -n._ ~ has an inverse, we conclude that (8.33) From (8.31), (8.33) and since we could perform the same analysis with any other triplet of components, and since is skew symmetric in Ii J , Pt?o'l indeces. PRj t is skew symmetric in all three For adding P,'" 3 +- Pi 1 z.. =- a p-z., n t- 1-=;_ ~ I -= Q which are valid from (8.33) gives ? l,n +- P2.3r -=- D which shows skew symmetry in (8.34) P1c.J- l 12. L of R<-J.t • Hence is skew symmetric in each pair of indeces k;>R J)j· l Now since Pre s l is completely skew symmetric, for a.11 values of x, 3 in the space, all derivatives of -1 L 3 I o1., o1,_o1.3 --'fL\e..,! are completely skew symmetric in o< , o<.1., ol3 • Hence any term (', Ir-:) which is summed against ·an expression t ~~ ~ ~ 3 must vanish. Hence all the terms in the series (1), (2), (3), (4), ... of (8.27), and in the series (5); (6), (7), (8), ... vanish giving no new necessary conditions. The term of lowest degree in is of degree four. c remaining is (9), which (9) is the sum of two terms which involve • 82 In the first term, ~ appears cl :s in the first of two iterated integrals, and in the second term~ qifferent iterated integrals. appears in the second. By choosing the directrix as in (8.2) then choosing a similar directrix slightly differently par·a materized, (9) will be the sum of two terms_ with different coefficients for the two cases. Since (9) must vanish for each case, we con- clude that each term in (9) vanishes separately. We write the first term in detail. (8.35) As in treating P.,z,o1..,_d...3 that, for a particular we choose different sets of , x.o<'. , we obtain from (8.35) ~ such 27 linear homogeneous matric equations in 27 matric unknowns with nonvanishing determinant. Hence each matric term vanishes and (8.,36) for all values of k j , lm. Condition (8.36), however, implies that (8.37) For writing (8.36) in component notation for 1 - k, j = m, (not summed on k, j) gives (8 • .38) That is, the matrix R,ej must be nilpotent of order two. 83 Let us now choose coordinates which at x(~.) are orthogonal. This is always possible, since picking n mutually orthogonal directions at a point is equivalent to picking a self polar tetrahedron with respect to the quadric Ji::i ~l· 1 / in an n-1 dimensional projective space. In general , these coordinates will not be mutually orthogonal everywhere in the spa.ce. In this coordinate system, at iCC5.) , and (8.39) then (not summed on r.) Since /cJi.c.. te.j is skew symmetric in r,c, so is R11e.j • Then (8.38) may be written for those terms for which c = r. But if the sum of squares of real terms vanishes, the terms themselves vanish. a.t ;tC's.) • But Hence ~(~") can be taken as any point of D. Hence (8.42) holds at every point of D, and is a necessary condition for the vanishing of J" 3Jl.~ Lx:, roJ. But R ,,.,__,11., k~-= o for all points is the necessa.ry and suf- ficient condition for the vanishing of / (,- . cf '..IL., [x:,'l.J , which is a sufficient condition for the vanishing of f 3 o lr JLa., [ x 1 ~J . Hence the necessary and sufficient condition for the vanishing of 5 3-11; [>: , z] for all directrices and variations is 0 fl., le -o J. - This completes the proof of theorem 8.3. at all points of D. In general, the same condition will hold for the mth I Frechet differential. For, from the dagge r differentiation law, and the same type of argument as in theorem 8.J, at least one / necessary condition for the vanishing of the mth Frechet differential for all directrices and varia.tions is of the form where there are [ m;i J factors in (8.44). 7n +I 2. est integer conts.ined in ••• == e; j for order [ -rn;'J 2 f' t... • [ -rn;'] In particular (8.44) holds = d = . . • = f and ff. ie_; is nilpotent of Now there exists an integer p crn;i J ~ 2. p +I • If nilpotent of order 2 R. 1ej· f -+I • is the great- such that is nilpotent of order L7>t : 1J it is Then, if q -= 2. f' , let and ( not summed on k, j). On choosing orthogonal coordinates at ~ cso ) > Jj : 12. J is symmetric in n., ol • Hence (8.46) can be written, 85 For r ~ c (8.47) reduces to and hence (8.49) and (8.45) imply that is nilpotent of or-der R.~;, 2.P . Continuing the pr-ocess leads to the conclusion that (8.50) R. ltj =- D Hence we have proved, since sufficiency is obvious fr-om theor-em 8.1, Theorem 8~. ( -m., CJ lr :Jlo- [ x :i eJ directrices The necessary and sufficient condition that given by (6.39) of theor-em 6.8 vanish for ill ):',<s) and variations the space of definition of (8.51) J::; ~ l. "(~) in an open region D of is Rl'lj = D evern1here in D for all k, j = 1, ... , n and all m = 1, 2, ... VIIr.c. The Classification of Riemannian ppaces ~X.. Their Degr~e. From theorem 8.4 and the fact that we have Theorem_Ll. Ever-y flat ~iemannian space is of degree zer-o and all other- Riemannian spages are of infinite degree. IX. AJ2plications of Matrix Theorems to Displacement About a Closed Directrix. When the directrix is a closed curve, the displaced vector is located at the same point as the original. the effect of a rotation of A(a.) into >i (Ir) • Hence tr Jl....:i, has This geometric fact leads to If~ in an-dimensional Riem.._ag_n_!~~R.§:ce, a Theorem 9.1. contravariant vector, indicated bl the column matrix A !§. g_isplaced about ~ _closed directrix, xi = ;e '- Cs) from s = a.. to s=-lr are re~ted ~ , :;r.Lca..)=:):i:(1,-) and if the values of >.c~)J).(1r) (9.1) , then the determinant of the matrizant function_!:§, Qlus one. Proof: In smbols, Since the 1-ength of an arbitrary vector is unchanged under parallel displacement, in component notation, the square of the length is (9.3) ' • lr~ ~,-~ >,'- (1,,) )/(1r) = j i:j _n_,._, ~ - trJ' JL()., (l ~ \ .t A <.a..)/ <.a.. ) ~ k Q. ). '2ctt.) Ai (a..) Hence Taking the determinant of both sides of (9.4) (9. 5) I '!i I • I .12,i" i i. = 1~ I and (9.6) 1-11; J -i.. =- I hence (9.7) I _n_: I =-~I 87 But parallel displacement does not change the angle between two displaced vectors. Hence, an orthogonal ennuple, or set of n mutually orthogonal vectors, displaces into a new set with the same orientation. Hence the plus sign only holds in (9 . 7) which proves theorem 9.1. We seek next the vectors which have the same direction to within sign before and after displacement. Iheorem 2•~• Let a contravariant vector ~ be displaced by narallel displacement about a closed direck!2£ ~ i. =;:. ' ( $ ) • Then if n, the dimension of the Riemann space, is odd, there are exactly 1,.3,5, ... , n-2 .Q!. n linearly independent vect~ which have th§ same direction,.i_to within sign~_ before and after displac~!!_l,§nt...._ =I.... f n j.s even, there ar~ _exactl:r o, 2, ... , n .§_Uch ljnearly inde~endent vect~. We regard the components A'- as the n Proof: a point in n-1 -12.~ dimensiona.l projective space. coordinates of The matrizant in (9.1) describes a linear trensformation or collineation in the space. A fixed point, Ao , will be a solution of the linear homogeneous equations in n n unknowns) The necessary a.nd sufficient condition for the existence of such points is that the determinant of the coefficients vanish. where I is the unit matrix. This condition is an equation of degree -n in f , hence has n roots, real and complex. Since complex roots occur in pairs, there are exactly 1,3, ... , n-2 or 0,2,4, ... ,n-2, n real roots as n is odd or even. po( on substitution in(9.8) gives a vector , into the vector length, f~ f.,. A~. which displaces Since parallel displacement preserves must be ~ I , or complex. are preserved, the elementary divisors be linear. Ac1.. F.very root, Further, since angles 25 of the matrix must all Hence each real root of (9.9) yields a distinct "fixed" vector Aot. and r distinct real roots yield r linearly , independent vectors. Since complex roots occur in pairs, this completes the proof of theorem 9.2. That linearity of the elementary divisors follows from the preservation of the angles between two displaced vectors is proved by the following contra.diction. If there is an elem~ntary divisor which is not linear, there will be fewer than n linearly independent vectors which assume their original directions. These, say n-m, vectors determine an n-m dimension space, S-;i_-m., and by subtraction determine an m dimensional space, S-m- . , All of the vectors of S-in are orthogonal to 5-n-m.., • Since angles are preserved, each of the vectors vector which is in S-m. • V-m- in S-ni displace into a Similarly for' the vectors Y11. --Y,,,,... in S11-m . Hence the transformation determined by the matrizant function must be the direct sum of two transforme,tions, one in the space S n- m and the other in the orthogonal space Sm • Since every collineation has at least one fixed point, the collineation 25. Bocher, M. (1) 89 in S1"1. has at least one fixed vector.. But this contre.dicts the statement that there were only n-m fixed vectors, and hence contradicts the assumption that at least one of the elementary divisors was not linear. Under the conditions of theo_r,~m 9.1, there are an ev~n number of the· vectQ!§_ of theorem 9.2 y(hich return to their original direction by~.12.Qsite sense. Proof: It is shown in theoc-em 9.1 that l_fl.!\=+I . Let the coordinate system be so chosen that the real vectors of theorem 9.2 lie on some of the coordinate axes. The relation (9.1) may then be written (9.10) >.i(lr)::, f,.~ (u,) x '-+I (/;-) =-/',',-ti (tl...) where the first 1 vectors are those returning to their initi'al The j. vectors returning to their initial direction but opposite sense have components o,• .. , o) 1, 0 > • .. , o etc. The --n-i-j linearly independent real vectors which span the remaining space do not 90 return to their original direction. determinant I w ! = + I theorem 9.1. However the ~-~-J rowed from the same type of reasoning as in Since ' I .JL: / = (-l)j (t- 1) '- I w I-= +I (9.11) This completes the proof of theorem 9.3. j must be even. On using theorem 9.2 and 9.3, there follows immediately Theore!Il 2•A• Let a contravaria,!!t vector ). be displaced abou,:t a closed directrix by para.lle~L displacement in a Riemannian spac~. Then there are §Xactu 1,3,5, ... , n-2, n Q! O,z,4, ... , n-2, vectors which return to their Qri_gi!!!l-1 value as n is odd or even. Clearly if two or more vectors return to their original direction and sense, any linear combination of them also does. Similarly any linear combination of vectors which return to their original direction but opposite sense also returns to its original direction and opposite sense. The two groups of vectors are quite distinct, for Theorem hl• The vector§__Qf theorem 9.2 which return to their griginal direction and sense are or~hogonal to those which re}!!!'.!! to their original girect~pn but .QP.P.osite sense. Proof: Let >., be a vector which returns to its original direction and sense and >. 2.. be a vector which returns to its original direction but opposite sense. between A, and (9.12) But this implies ~~ Then, since the angle is unchanged by parallel displacement, 91 A, and and hence ~z. are orthogonal. If such "fixed" vectors exist at a point of a directrix, they exist all along the directrix. If f Theorem 9.~. In fact is a contravariant_vector which ret~£!!.,e ~o it§_o,rigl:nal direction either with or without the same sense after paralle;t_gj:§.pla.cement a.bout a. closed directrix in Riem~ian space..1, then evesy vectp_r parallel to ). . on being 9-!spl·aced about the_di rectrix retur!}§.. to. its 2,.rj._ginal direction with OJ: with9ut its.origin~~~~~-sespect!veli. If Proof: is parallel to ).. U-) )\ la.) at some point Jt. (r:.) then (9.14) t- (C) = _n_;:_ )-_(ll-) is displaced once about the directrix and the curve If /\ <.c) parameter s goes from c to d, then J.. .n~ >,. (c) ::: _Q~ JL:, >- <.11.) = .ilo.- A(o..) - _J)._~ SL: A lo..) = ± -11t A("<- ) = ~ >-. (.c.) where the positive sign occurs if sense, and the minus sign occurs if ~(.o.) ~<.~) returns with the same returns with the op- posite sense after being displaced about the directrix. Use is made of the fact that (9.16) ~ ..Jl_~ == JL tr . 92 X. Extension of_J;he Theorx to Tensors of Types Otre£._Thap Contravaric.nt Vectors. The entire theory of the previous chapters clearly extends immediately to tensors of types other t~ n contravarie~t vectors. For the sake of completeness we indicate how this is done and prove some of the theorems corresponding to earlier theorems. Theorem 10.1. If the contrava.riant ~§.!!.§.2.!: t t'._; of rank two i,s di~J2;J;?-..£ed bYJ,evi- Ci.Yi ta :Q_?.ralleJ: displac.em~nt alo~ directrix ;r;'=1-'c5 ) from 5 =o... to s=lr and if a re the values of t ~·J at and t'-'_;(tr) t"J·ca..) x=a.. !filQ 3 = f.r respectively, then (10.1) (not SUirl!!l~d on a , b) t,.. n... where ( 10.1 ) is written in cgmpone!}t notation., ie __n_;_, C [-r.e. J _, ~ f7-J _,.. a.ls J is the component in the rth row and cth column of the matrix Proof: The equations of parallel displacement for a contra- variant tensor of rank two are (10.2) We may now integrate equations (10.2) by successive approximations in exactly the same way as equations (3.1) were integrated. The verification of the result (10.1) involves interchanging orders of integration and certain theorems in combinatori'-a.l analysis. The proof, while simple, is tricky, and becomes r:-athe r 93 complicated when extended to tensors of order greater than two. A different proof proceeds as follows. Consider the particular tensor (10 . .3) rr •. ":1 = .A l. j ),l the direct pr odu ct of Ai ls) ,uJ l ~) two contravariant vectors. along the directrix x- 1..' = ;t ,·l r)we have that On displacin g 'r i:-> 111 ~-> , mus t satisfy - - r~ J,y, le m;e dg 1---rl I -nij - r~ 7't le ~ n. mi?¾, ). r , which is the same differential equation as (10.2). But from (3.5), (10.4) has the unique solution at 5 =tr (not summed on a, b) Since ti.S satisfies the same differ ential equations as T i..i , the theorem is immediate. In an entirely simila.r way, Theorem 10.2. If a coQ_j;;ravariant tensor of ord~r Q, -t ij'• .. I?._ , is displ§-ced J:2y Levi-Civita parall~ dispJ.B-.£~,!_Ilent 2,long a directrix --,..'-::::. -;x:.'Cs) from s=a.. 1Q s=lr , then ( 10.6) t~_; . .. R.(Ir) =_rr Ir~ _Q o.-1r.;l( . .. __Jr\ Irle. /3 -tol.,'( .. ,f., {J.., CJ(.. where there are ;g_indeces L(A,, (<1.) i, ~ .. . te. j rJ., -,;, . .. /3 (not ~µmmed on a 1 b) 94 Now clear 1 y, t ( j ... k [ n Air] ....JL_ considered as a function Ir of the ma.trizant ..Il.a.., is a multilinear function on the Banach space B2 of square matrices of order n whose elements are real continuous numerical functions of two real numerical variables to a new Banach space whose elements are sets of elements from B2 . I Hence by the composition theorem for Frechet differenti'a.ls , Theo~J. O.~. The Fre'chet differentials of 8:!}¥ order of t i j ... ~c.-) g_iven in theo_r_§__m 10.2 exist and are given bz. the appropriate I formulae by means gf~_{l0.6) and the th~ems on Frechet differentials of a multilinear function and the compositio!!_theorem. / Theorem 10.3 is clearly valid for both the Frechet differentia.ls resulting from warping the space l\nd from varying the directrix. The details for covariant and mixed tensors are only slightly different from those for the contravaria.nt tensors. For a covariant vector, ){ , the equations of parallel dis- placement are (10.7) which can be written in the matrix form ol..u.. ::: µ fJ h J (10. 8) c.i s A-g Equation (10.8) differs from (3.3) only in sign and order of matric factors. ( 10. 9) 7 we have V r l.r 0,., - Therefore, if J2... &- [ /1.·J ~ J,J d.5 ,t., trans posed 95 Theorem 10.4. matrix µ If a covariant vector..,_r§J2!'_e~~nteq_ _by th~ _!O:?£ is displaced by Levi-Ci vi ta parallel displa£._e~,.! Jc. ~-::: ~ i( t ) f!.2m 3 = a.. in a Riemannian space_ _§.lon~ectrix 12 -g=lr , and i f µL a..) ,f<- llr) are the values of )l ~ s=-41 lr respectively, then (10.10) Ir ~ U,-) = A l~ ) V- a., Jl.~ [ f'h. ~~~] . Further where v o.,lr is the transpose of th~e matrix Theo~-1,Q_. .5. All the theorems on the Fre'chet differentia.ls of a coptr-avariant vector under p§.£..§:J.)-~;L displacement have their counterparts in theorems for a covariant vector )-{ • theorems are obtained by interchanging trans- fQ.r. -r,a posigg all matrices in the theorems for These A and replacing A ~ f( We state the further theorem Theorem 19...!.£• If a mixed tensor t ~~ ·_j is displaced_ b:[ Levi-Civi~a parall~l displacement along the directrix fr-om ~ ~ t(.. !Q 5 = f.r- values a.t 5 -==-Q.. (10.11) ~ , then if t ~-.-.J. (tL) , t ~:·. :~ Clr) ,, ~ =x 'C~ ) , ~th£ 5 =- Ir respectively, then t Ri: ...· .. ;)._ (tr)_- ..Jl a.,1r .: . .. J nl. c,_1r.;p t m1- "·.. . /l.p co..) U a..,1r ~ . . . u a.,1r J...n. '?I\ 'r <- and all the Fre'chet differentials of t ~.':.~ U-) exist and are given by their obv1,QusJ9rmul§&. 96 XI. Bibliography. Appel,Paul (1) "Traite de Mechanique Rationelle 11 Paris.(1926) vol.5 pp 136-137. Banach, Stephan (1) "Theorie Des Operations Lineares 11 Warsaw. (1932) Bocher,Maxime (1) "Introduction to Higher Algebra" New York. (1938) Chapter XXI. Dienes, Paul (1) "Sur 1 1 Integration Des Equations du Deplacement Parallele de M. Levi-Civita" Rendiconti Deo Circolo Matematico di Palermo t. XLVII (1923) pp 144-152. Duscheck-Mayer (1) "Lehrbuch der Differential-Geometrie". Band II. Riemannische Geometrie. Berlin(l930) pp 123-125. Eisenhart, L.P. (1) "Riemannian Geometry" Princeton (1926) pp 65-67. Frazier, Duncan and Collar (1) "Elementa.ry Matrices and Some Applications to Dynamics and Differential Equations" Cambridge (1938) pp 53-56. 97 Levi-Civita, T. (1) 11 Nozione di Parallelismo in Una Varieta Qualunque e Conseguente Specificazione Geometrica della Curvatura Riemannian" Rendiconte deo Circolo Matematico di Palermo t. XLII (1917) pp 173-205. s. Martin, R. ( 1) 11 Contributions to the Theory of Functionals 11 Ph.D. Thesis. California Institute of Technology Pasadena (1932) Michal, Aristotle D. (1) "First Order Differentials of Functionals With Arguments in Topological Abelian Groups" Revista de Ciencia.s, Lima, No 453 Ano XLVII pp 389 - 422 (2) 11 Matric a,nd Tensor Calculus with Applications to Mechanics, Elasticity and Aeronautics" New York (1947) (3) 11 On a Non-Linear Total Differential Equation in Normed Linear Spaces" to appear in Acta Mathematica.. (4) 11 The Frechet Differentials of Regular Power Series in Normed Linear Spaces" Duke Mathematical Journal vol 13(1946) pp 57-59. Peres., J. (1) "Le Parallelisme de Levi-Oivita et la Courbure Riemannienne" Rendiconte delea R. Academia dei Lincei Serie V. vol XXVIII 1 pp 425-428. Semestre (1919) 98 Thomas, T.Y. (1) "Differential Invariants of Generalized Spaces" Cambridge (1934). Veblen, o. (1) "Invat·iants of Quadratic Differential :Eot"ms" London, (19.33). VVhi ttaker, E.T. (1) "Pat"allelism and Teleparallelism in the Newer Theories of Space". Journal of the London Mathematical Society. vol.5 (1930) pp 68-80. Whittaker and Watson (1) "Modern Analysis" American Edition. Cambridge (1945) pp 221 . - i