Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications

535 Pages·1954·16.7 MB·English

DIE GRUNDLEHREN DER
MATHEMATISCHEN
WISSENSCHAFTEN
IN EINZELDARSTELLUNGEN MIT BESONDERER
BERUCKSICHTIGUNG DER ANWENDUNGSGEBIETE
HERAUSGEGEBEN VON
R. GRAMMEL. E. HüPF. H. HüPF. F. RELLICH
F. K. SCHMIDT. B. L. VAN DER WAERDEN
VOLUME X
RICCI-CALCULUS
AN INTRODUCTION TO TENSOR ANALYSIS
AND ITS GEOMETRICAL APPLICATIONS
BY
J. A. SCHOUTEN
SECOND EDITION
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1954
RICCI-CALCULUS
AN INTRODUCTION TO TENSOR ANALYSIS
AND ITS GEOMETRICAL APPLICATIONS
BY
J. A. SCHOUTEN
EMERITUS PROFESSOR OF MATHEMATICS IN THE UNlVERS!TY
OF AMSTERDAM
DIRECTOR OF THE MATHEMATICAL CENTRE AT AMSTERDAM
SECOND EDITION
WITH 16 FIGURES
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1954
ISBN 978-3-642-05692-5 ISBN 978-3-662-12927-2 (eBook)
DOI 10.1007/978-3-662-12927-2
ALLE RECHTE,
INSBESONDERE DAS DER ÜBERSETZUNG IN FREMDE SPRACHEN
VORBEHALTEN
OHNE AUSDRÜCKLICHE GENEHMIGUNG DES VERLAGES
IST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS
AUF PHOTOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN
COPYRIGHT 1954
BY SPRINGER-VERLAG BERLIN HEIDELBERG
URSPRÜNGLICHERSCHIENENBEI SPRINGER-VERLAG OHG. IN BERLIN, GÖTTINGEN AND HEIDELBERG I954
SOFTCOVER REPRINT OF THE HARDCOVER 2ND EDITION I 954
This book is dedicated to the memory of
DR. GREGORIO RICCI CURBASTRO
in life Professor of Mathematics in the
University of Padua,
who laid the foundations
of tensor calculus.
Preface to the second edition.
This is an entirely new book. The first edition appeared in 1923 and
at that time it was up to date. But in 193 5 and 1938 the author and Prof.
D. J. STRUIK published a new book, their Einführung I and li, and
this book not only gave the first systematic introduction to the kernel-
index method but also contained many notions that had come into
prominence since 1923. For instance densities, quantities of the second
kind, pseudo-quantities, normal Coordinates, the symbolism of exterior
forms, the LIE derivative, the theory of variation and deformation and
the theory of subprojective connexions were included. Now since 1938
there have been many new developments and so a book on RICCI cal-
culus and its applications has to cover quite different ground from the
book of 1923. Though the purpose remains to make the reader acquainted
with RICCI's famous instrument in its modern form, the book must
have quite a different methodical structure and quite different applica-
tions have to be chosen.
The first chapter contains algebraical preliminaries but the whole
text is modernized and there is a section on hybrid quantities (quantities
with indices of the first and of the second kind) and one on the many
abridged notations that have been developed by several authors.
In the second chapter the most important analytical notions that
come before the introduction of a connexion aredealt with in full. The
theory of integrability and PFAFF's problern are treated here and do
not get a chapter of their own this time as they did in the first edition
because ScHOUTEN and v. D. KuLK's book on PFAFF's problern of 1949
can be referred to. A special section pays attention to CARTAN's sym-
bolism.
In the first edition the chapters IV-VI contained the theory of
affine, riemannian and WEYL connexions and in these chapters pro-
blems of projective and conformal transformation and of imbedding
were also dealt with. In the light of modern developments this mixing
up of quite different topics could not be tolerated. So there is now a
chapter III dealing with linear connexions, curvature and normal Co-
ordinates, also with respect to anholonomic systems of reference and
in different notations. Then there is a chapter V on imbedding and
curvature, in the last section of which the theory of higher curvatures
VIII Preface to the second edition.
is developed. Chapter VI is dedicated to projective and conformal
transformations of connexions and in this chapter subprojective con-
nexions and concircular transformations of a vn find their natural place.
Between chapter III and these latter chapters it has been necessary
to insert a chapter IV on the theory of LIE groups. This theory not
only gives a very beautiful example of speciallinear connexions but it
also forms the necessary foundation for the following chapters, espe-
cially for chapter VII.
This chapter VII on the theory of variations, deformations and
movements leads in §§ 4, 5 to the theory of the holonomy groups that
is used in § 6 to give an application of CARTAN's method.
The last chapter in the first edition dealt with theinvariant decom-
position of tensors and in 1923 this was a new and interesting topic.
But now we can refer to D. E. LITTLEWoon's book of 1951 and it does
not seem any Ionger justifiable to treat such a special purely algebraic
problern in a book on Rrccr calculus and differential geometry.
Instead a chapter VIII has been added containing miscellaneous
examples, mostly taken from recent literature. Here the harmonic Vn
(§ 1), the many different connexions for hybrid quantities and their
properties with respect to imbedding and transformation (§§ 2-8) and
the spaces of recurrent curvature (§ 9) are considered. The choice of
these subjects was entirely free. Of course this chapter could have
been made many times longer. But there must be a limit to the size
of a book of this kind and this is also the reason why all those topics
contained in the following long list, however interesting they are, could
at most be mentioned very shortly:
1. properties in the large as for instance properties of compact
manifolds;
2. theory of geometric objects; fibre bundles etc.;
3. theory of VrTALI;
4. geometries of FrNSLER, CARTAN, A. KAWAGUcHI;
5. extensors of CRAIG and generalizations of M. KAWAGUCHI;
6. linear elements and surface elements of higher order;
7. connexions of higher order;
8. projective and conformal geometry with supernumerary coor-
dinates either in the tangent spaces or in the whole space;
9. path spaces; K-spreads;
10. natural families of curves;
11. non-linear connexions;
12. special geometries for small values of the dimensions;
Preface to the second edition. IX
13. contact transformations and their connexions;
14. treatment of differential equations by means of linear connexions
belonging to them;
15. spaces with a fundamental tensor of rank < n;
16. G. KRoN's applications of tensor analysis to electric circuits.
In the first edition there was an extensive literature list. With its
196 titles it covered nearly all the literature on many dimensional
differential geometry. The literature list of the Einführung of 193 5-1938
with its 488 titles represents only a selection of the literature at that
time. This is even more true of the literature list presented here which
is by no means exhaustive though it contains about 1400 titles and refers
to about 35 0 authors. By selecting the titles in such a way that the
reader interested in some topic will always find at least a few titles
that can lead him to more references, the author has tried to retain
something of the encyclopedic character of the first edition.
The author owes many thanks to the Mathematical Institute of
Amsterdam University and the Mathematical Centre at Amsterdam
for the valuable help of inany of their collaborators. In the first place
I mention Dr. NIJENHUIS (M. I. and M. C.) who has given much
time to the study of literature and the collection of exercises and who
proposed valuable improvements to the text in nearly every chapter.
Mr. BARNING (M. 1.) and Mrs. V. RoOTSELAAR (M. 1.) did important work
by studying literature and checking formulae. For the correction of
proofs I had the valuable help of Mr. BARNING (M. I.) and Mr. VER-
HOEFF (M. C.). Mr. Woon of the Clarendon Press, Oxford, has kindly
given advice on the language. This is the third book with which he
has helped me and I am very grateful to him for all the English idiom
he has taught me. Last but not least I have to thank my wife
Mrs. ScHOUTEN-BYLSMA who spent much time on the administration
of all literature references.
I wish to express my best thanks to the publishers for their most
agreeable collaboration.
Epe (Holland), im Juli 1954. J. A. SCHOUTEN.
Contents.
I. Algebraic preliminaries (1).
§ 1. The En (1).
affine space (1) - En (1) -affine group (1)- Ga (1)- minor (1)- summation
convention (1) - Af, A~. (1) - point space (1) - allowable coordinate trans-
formations (1) - rectilinear coordinates (1) - homogeneaus linear group (2) -
Gko (2) - centredEn (2)- point transformations (2)- * (2) - dragging along
of coordinate system (2) - !5): (2) - KRONECKER symbol (2, 7) - «5f (2) - gene-
ralized KRONECKERsymbol (2) - kemelletter (2) - running indices (2) - fixed
indices (2) - kemel-index method (3) - flat submanifold (3) -linear submani-
fold (3) - Cf (3) - rank (3) - B~ (3) - point (3) - straight Jine (3) - plane
(3)- hyperplane (3)- coordinate axes (4)- coordinate planes (4)- coordinate
Ep's (4) - net (4) - lie in (4) - contain (4) - translation (4) - parallel (4) -
p-direction (4) - improper Ep_1 (4) - Ep_ 1 "at infinity" (4) - direction
(4) - reduction (5) - tjp-parallel (4) - section (4, 5) - spanning (4) -
join (4) - projection (4, 5) - screwsense (5) -sense (5) -sense of rotation (5) -
opposite (5) - inner orientation (5) - outer orientation (5).
§ 2. Quantities in En (6).
quantity (6) - components (6) - kind (6) - sum of quantities (5) - manner
of transformation (6) - scalar (6) - contravariant vector (6) - contragredient
(6, 7) - contravariant basis (7) - dead indices (7) - living indices (7) -
covariant vector (7) - covariant basis (8) - transvection (8) - reciprocal sets
(9) - strangling (10) - tensor (10) - affinor (10) - valence (10) - co- and
contravariant tensor (10) - cocontravariant (10) - mixed (10) - unity tensor
(10) - A~ (10) - intermediate components (11) - equiform (11) - pseudo-
tensor ( 11) - pseudo-scalar (1 1) - pseudo-vector ( 11) - class of pseudo-quan-
tity (11)- tensor density (12) -tensor .d-density (12) -weight (12)- W-tensor
(12) - w (12) - G8 a (12) - .connecting quantities (13).
§ 3. Invariant processes and relations (13).
addition of quantities (13) -isomer (13) - general multiplication (13) - con-
traction (13) -transvection (14)- saturated indices (14)- dummy indices (14)-
free indices (14) - mixing (14) - round brackets (14) - symmetric (14) - sym-
metric part (14) - altemation (14) - square brackets (14) - altemating (14) -
altemating part ( 14) - concomitant ( 15) -invariant ( 15) -rational integral ( 15).
§ 4. Section and reduction with respect to an En& in En ( 16).
decomposition with respect to a rigged E,., (1 6) - section ( 16) - reduction ( 1 7\ -
rigging (17)- projection (18)- E,..-part (19).
§ 5. Rank, domain and support of domain with respect to one or more indices (20).
,u-rank (20) - ,u-domain (20) - support of domain (20) - double brackets [[)] (20).
§ 6. Symmetrie tensors (21).
symmetric tensor (21) - symmetric multiplication (21) - divisor of a symmetric
tensor (21) - irreducible symmetric tensors (21).
Contents. XI
§ 7. Multivectors (22).
multivector (22) - p-vector (22) - bivector, trivector, quadrivector (23) -
altemating multiplication (23) - divisor of a multivector (23) - simple multi-
vector (23) - n-vector {23) - ~···"M {25)- <;~ .1 {25) - cx: (25) - @:"•···""
(X} 1••• 11
{26) - e.1, ••• .1,. {26) - identifications {27) - pseudo-scalar (27) - tables of
alternating quantities in E 3 (28).
§ ·s. Tensors of valence 2 {28).
Cocontravariant tensor of valence 2 (28) - trace (29) - spur (29) - eigenvector
{29) - eigenvalue (29) - characteristic equation {29) - first canonical form of
the matrix {33) - elementary divisors (33)- co- and contravariant tensors (34) -
s-vector tensor (34) - adjoint (34, 38) - symmetric co- and contravariant ten-
sors (34) - index {34) - signature (34) - rank tensors (34) - definite {35) -
indefinite {35) - semi-definite (35) - co- and contravariant bivectors {35)
blades {36) - classification of trivectors {36) - reduction number {37).
§ 9. Introduction of a metric in an E,. (40).
length (40) - distance {40) - indicatrix (40) - fundamental tensor (41) -
euclidean R,. (41) - minkowskian R,. (41, 42)- time-like (41) - space-like (41) -
±-region {41) - nullcone (41) - unitvector (41) -angle (41) - duration {41) -
isotropic Ep (42) - full isotropic Ep {42) -null Ep (42) - perpendicular {42)-
cartesian coordinate system {42)- minkowskian proper rotations {43)- LoRENTZ
transformations {43) - minkowskian reflexions {43) - improper minkowskian
rotations {44) - vector {44) - raising and lowering of indices (44) - I"•···"n
w - w
{45) - i,1, ... .1,. {45) - i, i {45) - g {45) - principal multidirection {46) -
theorem of principal axes (46) - theorem of principal blades {46) -orthogonal
coordinate transformations (47ff.) - rotations (47) - reflexotations {47) -
principal angles (48) - reflexional orthogonal transformations {50) - proper
and improper rotations {50) - tfp-parallel (51) - tfp-perpendicular (51).
§ 10. Hybrid quantities {51).
quantities of the first and second kind (51) - hybrid quantities (52) - weight
and antiweight {52) - hermitian (symmetric) tensor (52)- hermitian alternating
tensor (52) - hybrid bivector {52) - ±-invertible (53) - index {53) - signature
(53) - ±-definite {53) - indefinite {53) - semi-definite {53) - the auxiliary
E 2 ,. {53)- principal E,. (53)- first and second set of invariant E,.'s {54) -her-
mitian fundamental tensor {54)- R,. {54)- ordinary R,. {54)- kets and bras {54)
- unitary perpendicular vectors {54) - norm of vector (54) - unitvector {54) -
unitary cartesian coordinate systems (54) - unitary orthogonal group (55) -
fundamental figure in .R,. {55) - nullcone in R,. (55) - the auxiliary R 2,. {55) -
principal multidirections of a hermitian tensor (56) - theorem of principal axes
{56).
§ 11. Abridged notations (57).
collecting indices {57) -representativeindices (57) -the notation of GIVENS (57)-
VITALI's method (57)- CRAIG's extensors {58)- idea:tvectors (58)- systems of
"direct" calculus {58)- skeleton of a formula {59)- the method of the radius-
vector in E,. and R,. {60) - CARTAN's method {61) - abbreviations ad hoc {61).