Differential Geometry and the Calculus of Variations

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Page iii

Copyright page
Page v

Preface
Pages v-vii
R. Hermann

1 Introduction
Pages 3-5

2 Tangent Vector-Vector Field Formalism
Pages 6-10

3 Differential Forms
Pages 11-20

4 Specialization to Euclidean Spaces: Differential Manifolds
Pages 21-27

5 Mappings, Submanifolds, and the Implicit Function Theorem
Pages 28-33

6 The Jacobi Bracket and the Lie Theory of Ordinary Differential Equations
Pages 34-45
Jacobi Bracket

7 Lie Derivation and Exterior Derivative; Integration on Manifolds
Pages 46-62

8 The Frobenius Complete Integrability Theorem
Pages 63-72

9 Reduction of Dimension when a Lie Algebra of Vector Fields Leaves a Vector-Field Invariant
Pages 73-80

10 Lie Groups
Pages 81-97

11 Classical Mechanics of Particles and Continua
Pages 98-109

12 Differential Forms and Variational Problems
Pages 113-121

13 Hamilton-Jacobi Theory
Pages 122-141

14 Extremal Fields and Sufficient Conditions for a Minimum
Pages 142-151

15 The Ordinary Problems of the Calculus of Variations
Pages 152-169

16 Groups of Symmetries of Variational Problems: Applications to Mechanics
Pages 170-231

17 Elliptic Functions
Pages 232-240

18 Accessibility Problems for Path Systems
Pages 241-257

19 Affine Connections on Differential Manifolds
Pages 261-271

20 The Riemannian Affine Connection and the First Variation Formula
Pages 272-283

21 The Hopf-Rinow Theorem Applications to the Theory of Covering Spaces
Pages 284-290

22 The Second Variation Formula and Jacobi Vector Fields
Pages 291-299,301

23 Sectional Curvature and the Elementary Comparison Theorems
Pages 302-317

24 Submanifolds of Riemannian Manifolds
Pages 318-341

25 Groups of Isometries
Pages 342-361

26 Deformation of Submanifolds in Riemannian Spaces
Pages 362-369

27 First-Order Invariants of Submanifolds and Convexity for Affinely Connected Manifolds
Pages 373-377

28 Affine Groups of Automorphisms. Induced Connections on Submanifolds. Projective Changes of Connection
Pages 378-385

29 The Laplace-Beltrami Operator
Pages 386-393

30 Characteristics and Shock Waves
Pages 394-400

31 The Morse Index Theorem
Pages 401-419

32 Complex Manifolds and Their Submanifolds
Pages 420-426

33 Mechanics on Riemannian Manifolds
Pages 427-430

Bibliography
Pages 431-433

Subject Index
Pages 435-440