Fundamentals of Differential Geometry

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Springer Science & Business Media, Sep 21, 2001 - Mathematics - 540 pages
The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differen tiable maps in them (immersions, embeddings, isomorphisms, etc. ). One may also use differentiable structures on topological manifolds to deter mine the topological structure of the manifold (for example, it la Smale [Sm 67]). In differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a Riemannian metric, ad lib. ) and studies properties connected especially with these objects. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. In differential equations, one studies vector fields and their in tegral curves, singular points, stable and unstable manifolds, etc. A certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings.
 

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Contents

CHAPTER I
3
CHAPTER II
22
4 Manifolds with Boundary
39
3 Exact Sequences of Bundles
52
4 Operations on Vector Bundles
58
2 Vector Fields Curves and Flows
88
4 The Flow of a Spray and the Exponential Map
105
2 Lie Derivative
122
3 The Covariant Derivative on a Riemannian Submersion
383
4 The Hessian and Laplacian on a Riemannian Submersion
387
5 The Riemann Tensor on Submanifolds
390
6 The Riemann Tensor on a Riemannian Submersion
393
PART III
395
CHAPTER XV
397
2 Covariant Derivatives
407
3 The Jacobian Determinant of the Exponential Map
412

4 The Poincare Lemma
137
8 Darbouxs Theorem
151
5 Lie Groups and Subgroups
165
APPENDIX
173
3 Reduction to the Hilbert Group
180
CHAPTER VIII
195
2 Sprays and Covariant Derivatives
199
5 More Local Results on the Exponential Map
215
Curvature
231
3 Application of Jacobi Lifts to 7expY
246
5 Taylor Expansions
263
CHAPTER XIV
270
3 More Convexity and Comparison Results
276
6 The Flow and the Tensorial Derivative
291
2 Growth of a Jacobi Lift
304
5 Rauch Comparison Theorem
318
3 Totally Geodesic and Symmetric Submanifolds
332
2 Alternative Definitions of Killing Fields
347
6 Parallelism and the Riemann Tensor
365
Immersions and Submersions
369
2 The Hessian and Laplacian on a Submanifold
376
4 The Hodge Star on Forms
418
5 Hodge Decomposition of Differential Forms
424
6 Volume Forms in a Submersion
428
7 Volume Forms on Lie Groups and Homogeneous Spaces
435
8 Homogeneously Fibered Submersions
440
Integration of Differential Forms
448
2 Change of Variables Formula
453
3 Orientation
461
4 The Measure Associated with a Differential Form
463
5 Homogeneous Spaces
471
CHAPTER XVII
475
2 Stokes Theorem on a Manifold
478
3 Stokes Theorem with Singularities
482
CHAPTER XVIII
489
2 Mosers Theorem
496
3 The Divergence Theorem
497
4 The Adjoint of d for Higher Degree Forms
501
5 Cauchys Theorem
503
6 The Residue Theorem
507
The Spectral Theorem
511
Index
531

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