Tensors and Riemannian Geometry: With Applications to Differential Equations

194 Pages·2019·3.17 MB·English

by  Ibragimov N.H.

Nail H. Ibragimov

Tensors and
Riemannian Geometry

With Applications to Differential Equations
Mathematics Subject Classification 2010
35QXX, 3501, 35L15, 53A45, 83XX

Author
Prof. Nail H. Ibragimov
Department of Mathematics and Science
Blekinge Institute of Technology
S-371 79 Karlskrona, Sweden
Email: [email protected]

ISBN 978-3-11-037949-5
e-ISBN (PDF) 978-3-11-037950-1
e-ISBN (EPUB)978-3-11-037964-8

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Preface
Tensors are very simple mathematical object from point of view of transforma-
tions. Namely, tensor fields in a vector space or on a curved manifold undergo
linear transformations under changes of the space coordinates. The coefficients
ofthecorrespondinglineartransformationareexpressedintermsoftheJacobian
matrix of the changes of coordinates.
Consequently,tensorcalculusprovidesacomprehensiveanswertotheques-
tion on covariant (i.e. independent on a choice of coordinates) representation
of equations. Hence, the tensor calculus, Riemannian geometry and theory of
relativity are closely connected with transformationgroups.
F. Klein (42) has underscored that what is called in physics the special
theoryofrelativityis,infact,atheoryofinvariantsoftheLorentzgroup. Indeed,
centralfortheNewtonianclassicalmechanicsisthe Galilean relativity principle.
It states that the fundamental equations of dynamics should be invariantunder
the Galileantransformationwhich is written, e.g. in the direction of the x-axis,
as follows:
x=x+at
and has the generator
∂
X =t
∂x·
In the Galilean transformation,the groupparametera is a velocity of a moving
frame, and the time t remains invariant. In Einstein’s special relativity, the
Galilean transformation is replaced by the Lorentz transformation
x=xcosh(a/c)+ctsinh(a/c), t=tcosh(a/c)+(x/c)sinh(a/c)
with the generator
∂ x ∂
X =t + ,
∂x c2 ∂t
wheretheconstantcisthevelocityoflight. TheLorentztransformationinvolves,
along with the space variables x, the time variable t as well thus leading to the
conceptof the four-dimensionalspace-time knownasthe Minkowskispace. The
Galileanrelativityisobtainedfromthespecialrelativitybyassumingthata c.
�
Indeed, formally letting (a/c) 0 in the Lorentz transformation, we have:
→
a2 a a2 xa
x x 1+ +ct x+at, t t 1+ + t.
≈ 2c2 · c ≈ ≈ 2c2 c c ≈
(cid:31) (cid:30) (cid:31) (cid:30)
Einstein’s theory of general relativity is aimed at replacing Newton’s em-
pirical gravitationlaw by the concept of curvature of space-times. According to
vi Preface
this concept, a distribution of matter causes a curvature, and the curvature is
perceived as a gravitation.
Furthermore, Riemannian spaces associated with second-order linear dif-
ferential equations were used by Hadamard (23) in investigation of the Cauchy
problem for hyperbolic equations (see also (14)) and by Ovsyannikov (55) in
group analysis of hyperbolic and elliptic equations. It is a long tradition, how-
ever,to teachpartialdifferential equations without using notationand methods
of Riemannian geometry. Accordingly, it is not clarified in most of textbooks
why, e.g. the commonly knownstandardforms for hyperbolic,parabolic andel-
liptic second-orderequations are givenin the case of two independent variables,
whereas this classification for equations with n >2 variables is given at a fixed
pointonly. Use ofRiemanniangeometryexplains a geometricreasonofthis dif-
ference and shows (28), e.g. that one can obtain a standard form of hyperbolic
equationswithseveralindependentvariablesiftheassociatedRiemannianspace
has a non-trivial conformal group, in particular, the space is conformally flat.
This book is based on my lectures delivered at Novosibirsk and Moscow
StateUniversitiesinRussiaduring1972–1973and1988–1990,respectively,Coll`ege
de France in 1980, University of the Witwatersrand (Republic of South Africa)
during 1995–1997,Blekinge Institute of Technology(Sweden) during 2004–2011
and Ufa State Aviation Technical University (Russia) during 2012–2013.
The necessary information about local and approximate transformation
groups as well as symmetries of differential equations can be found in (39).
I acknowledge the financial support of the Government of Russian Federa-
tion through Resolution No. 220, Agreement No. 11.G34.31.0042.
Nail H. Ibragimov
Contents
Preface v
Part I Tensors and Riemannian spaces 1
1 Preliminaries 3
1.1 Vectors in linear spaces 3
1.1.1 Three-dimensional vectors 3
1.1.2 General case 7
1.2 Index notation. Summation convention 9
Exercises 10
2 Conservation laws 11
2.1 Conservation laws in classical mechanics 11
2.1.1 Free fall of a body near the earth 11
2.1.2 Fall of a body in a viscous fluid 13
2.1.3 Discussion of Kepler’s laws 16
2.2 General discussion of conservation laws 20
2.2.1 Conservation laws for ODEs 20
2.2.2 Conservation laws for PDEs 21
2.3 Conserved vectors defined bysymmetries 27
2.3.1 Infinitesimal symmetries of differential equations 27
2.3.2 Euler-Lagrange equations. Noether’s theorem 28
2.3.3 Method of nonlinear self-adjointness 36
2.3.4 Short pulse equation 40
2.3.5 Linear equations 43
Exercises 43
3 Introduction of tensors and Riemannian spaces 45
3.1 Tensors 45
3.1.1 Motivation 45
3.1.2 Covariant and contravariant vectors 46
3.1.3 Tensor algebra 47
3.2 Riemannian spaces 49
3.2.1 Differential metric form 49
3.2.2 Geodesics. The Christoffel symbols 52
3.2.3 Covariant differentiation. The Riemann tensor 54
3.2.4 Flat spaces 55
3.3 Application to ODEs 56
Exercises 59
viii Contents
4 Motions in Riemannian spaces 61
4.1 Introduction 61
4.2 Isometric motions 62
4.2.1 Definition 62
4.2.2 Killing equations 62
4.2.3 Isometric motions on theplane 63
4.2.4 Maximal group of isometric motions 64
4.3 Conformal motions 65
4.3.1 Definition 65
4.3.2 Generalized Killing equations 65
4.3.3 Conformally flat spaces 66
4.4 Generalized motions 67
4.4.1 Generalized motions, theirinvariants and defect 67
4.4.2 Invariant family of spaces 70
Exercises 71
Part II Riemannian spaces of second-order equations 73
5 Riemannian spaces associated with linear PDEs 75
5.1 Covariant form of second-order equations 75
5.2 Conformally invariant equations 77
Exercises 78
6 Geometry of linear hyperbolic equations 79
6.1 Generalities 79
6.1.1 Covariant form of determining equations 79
6.1.2 Equivalence transformations 80
6.1.3 Existence of conformally invariant equations 81
6.2 Spaces with nontrivial conformal group 83
6.2.1 Definition of nontrivial conformal group 83
6.2.2 Classification of four-dimensional spaces 83
6.2.3 Uniquenesstheorem 86
6.2.4 On spaces with trivial conformal group 87
6.3 Standard form of second-order equations 88
6.3.1 Curved wave operator in V with nontrivial conformal group 89
4
6.3.2 Standard form of hyperbolic equationswith nontrivial conformal
group 90
Exercises 91
7 Solution of the initial value problem 93
7.1 The Cauchy problem 93
7.1.1 Reduction to a particular Cauchy problem 93
7.1.2 Fouriertransform andsolution oftheparticularCauchyproblem 94
7.1.3 Simplification of thesolution 95
7.1.4 Verification of the solution 97
Contents ix
7.1.5 Comparison with Poisson’s formula 99
7.1.6 Solution of the general Cauchy problem 100
7.2 Geodesics in spaces with nontrivial conformal group 100
7.2.1 Outlineof theapproach 100
7.2.2 Equationsofgeodesics inspaceswithnontrivialconformalgroup 101
7.2.3 Solution of equations for geodesics 102
7.2.4 Computation of thegeodesic distance 103
7.3 The Huygensprinciple 104
7.3.1 Huygens’principle for classical wave equation 105
7.3.2 Huygens’principleforthecurvedwaveoperatorinV withnontrivial
4
conformal group 106
7.3.3 On spaces with trivial conformal group 107
Exercises 107
Part III Theory of relativity 109
8 Brief introduction to relativity 111
8.1 Special relativity 111
8.1.1 Space-time intervals 111
8.1.2 The Lorentz group 112
8.1.3 Relativistic principle of least action 112
8.1.4 Relativistic Lagrangian 113
8.1.5 Conservation laws in relativistic mechanics 115
8.2 The Maxwell equations 116
8.2.1 Introduction 116
8.2.2 Symmetries of Maxwell’s equations 117
8.2.3 General discussion of conservation laws 119
8.2.4 Evolutionary part of Maxwell’s equations 122
8.2.5 Conservation laws of Eqs. (8.2.1) and (8.2.2) 129
8.3 The Dirac equation 132
8.3.1 Lagrangian obtained from theformal Lagrangian 133
8.3.2 Symmetries 134
8.3.3 Conservation laws 136
8.4 General relativity 137
8.4.1 The Einstein equations 137
8.4.2 The Schwarzschild space 138
8.4.3 Discussion of Mercury’s parallax 138
8.4.4 Solutions based on generalized motions 140
Exercises 141
9 Relativity in de Sitter space 145
9.1 The de Sitterspace 145
9.1.1 Introduction 145
9.1.2 Reminderof thenotation 147
9.1.3 Spaces of constant Riemannian curvature 149
x Contents
9.1.4 Killing vectors in spaces of constant curvature 150
9.1.5 Spaces with positive definitemetric 151
9.1.6 Geometric realization of thede Sittermetric 154
9.2 The de Sittergroup 155
9.2.1 Generators of the deSitter group 155
9.2.2 Conformal transformations in R3 156
9.2.3 Inversion 158
9.2.4 Generalized translation in direction of x-axis 160
9.3 Approximatede Sittergroup 160
9.3.1 Approximategroups 161
9.3.2 Simple method of solution of Killing’s equations 163
9.3.3 Approximaterepresentation of deSitter group 165
9.4 Motion of a particle in deSitterspace 167
9.4.1 Introduction 167
9.4.2 Conservation laws in Minkowski space 168
9.4.3 Conservation laws in de Sitterspace 170
9.4.4 Kepler’s problem in deSitter space 171
9.5 Curved wave operator 173
9.6 Neutrinos in de Sitterspace 175
9.6.1 Two approximate representations of Dirac’s equations in deSitter
space 175
9.6.2 Splitting of neutrinos by curvature 176
Exercises 177
Bibliography 179
Index 185
Tensor calculus has been invented by G. Ricci. He called the new branch of
mathematics an absolute differential calculus and developed it during the ten
years of 1887—1896. The tensor calculus provides an elegant language, e.g. for
presenting the special and general relativity.
TheconceptoftensorswasmotivatedbydevelopmentofRiemanniangeom-
etryofgeneralmanifolds(Riemann,1854)andbyE.B.Christoffel’stransforma-
tion theory ofquadratic differential forms (Christoffel, 1869). Subsequently, the
tensor notation has been generallyaccepted in differential geometry,continuum
mechanics and theory of relativity (see (5)).
The tensorcalculusandRiemannianspacesfurnishaprofoundmathemati-
calbackgroundfortheoreticalphysicsanddifferentialequationsofmathematical
physics.
Chapter1containsacollectionofselectedformulaefromtheclassicalvector
calculus and an easy to follow introduction to the index notation used in the
present book.
Chapter 2 includes a variety of topics on conservation laws from the basic
concepts and examples through to modern developments in this field.
Since the presentbookis designedforgraduatecoursesindifferentialequa-
tions andmathematicalmodelling, IprovideinChapter 3 a simple introduction
to tensors and Riemannian spaces with emphasis on calculations in local coor-
dinates rather than on the global geometric language.
The concepts of isometric, conformal and generalized motions in Rieman-
nianspaces,giveninChapter4,areusefulinvariousapplicationsinphysicsand
theory of differential equations.