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Viser: Riemannian Geometry
Riemannian Geometry
Sylvestre Gallot, Dominique Hulin og Jacques Lafontaine
(2004)
Sprog: Engelsk
Springer Berlin / Heidelberg
775,00 kr.
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Detaljer om varen
- 3. Udgave
- Paperback: 322 sider
- Udgiver: Springer Berlin / Heidelberg (Juli 2004)
- Forfattere: Sylvestre Gallot, Dominique Hulin og Jacques Lafontaine
- ISBN: 9783540204930
This book, based on a graduate course on Riemannian geometry and analysis on manifolds, held in Paris, covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results on the relations between curvature and topology are treated in detail. The book is quite self-contained, assuming of the reader only differential calculus in Euclidean space. It contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics about the geodesic flow and Lorentzian geometry have been added and worked out in the same spirit.
1 Differential manifolds.-
1.A From submanifolds to abstract manifolds.-
1.B The tangent bundle.-
1.C Vector fields.-
1.D Baby Lie groups.-
1.E Covering maps and fibrations.-
1.F Tensors.-
1.G. Differential forms.-
1.H Partitions of unity.- 2 Riemannian metrics.-
2.A Existence theorems and first examples.-
2.B Covariant derivative.-
2.C Geodesies.-
2.D A glance at pseudo-Riemannian manifolds.- 3 Curvature.-
3.A. The curvature tensor.-
3.B. First and second variation.-
3.C. Jacobi vector fields.-
3.D. Riemannian submersions and curvature.-
3.E. The behavior of length and energy in the neighborhood of a geodesic.-
3.F Manifolds with constant sectional curvature.-
3.G Topology and curvature: two basic results.-
3.H. Curvature and volume.-
3.I. Curvature and growth of the fundamental group.-
3.J. Curvature and topology: some important results.-
3.K. Curvature tensors and representations of the orthogonal group.-
3.L. Hyperbolic geometry.-
3.M. Conformai geometry.- 4 Analysis on manifolds.-4.A. Manifolds with boundary.-
4.B. Bishop inequality.-
4.C. Differential forms and cohomology.-
4.D. Basic spectral geometry.-
4.E. Some examples of spectra.-
4.F The minimax principle.-
4.G Eigenvalues estimates.-
4.H. Paul Levy's isoperimetric inequality.- 5 Riemannian submanifolds.-
5.A. Curvature of submanifolds.-
5.B Curvature and convexity.-
5.C Minimal surfaces.- A Some extra problems.- B Solutions of exercises.- List of figures.
1.A From submanifolds to abstract manifolds.-
1.B The tangent bundle.-
1.C Vector fields.-
1.D Baby Lie groups.-
1.E Covering maps and fibrations.-
1.F Tensors.-
1.G. Differential forms.-
1.H Partitions of unity.- 2 Riemannian metrics.-
2.A Existence theorems and first examples.-
2.B Covariant derivative.-
2.C Geodesies.-
2.D A glance at pseudo-Riemannian manifolds.- 3 Curvature.-
3.A. The curvature tensor.-
3.B. First and second variation.-
3.C. Jacobi vector fields.-
3.D. Riemannian submersions and curvature.-
3.E. The behavior of length and energy in the neighborhood of a geodesic.-
3.F Manifolds with constant sectional curvature.-
3.G Topology and curvature: two basic results.-
3.H. Curvature and volume.-
3.I. Curvature and growth of the fundamental group.-
3.J. Curvature and topology: some important results.-
3.K. Curvature tensors and representations of the orthogonal group.-
3.L. Hyperbolic geometry.-
3.M. Conformai geometry.- 4 Analysis on manifolds.-4.A. Manifolds with boundary.-
4.B. Bishop inequality.-
4.C. Differential forms and cohomology.-
4.D. Basic spectral geometry.-
4.E. Some examples of spectra.-
4.F The minimax principle.-
4.G Eigenvalues estimates.-
4.H. Paul Levy's isoperimetric inequality.- 5 Riemannian submanifolds.-
5.A. Curvature of submanifolds.-
5.B Curvature and convexity.-
5.C Minimal surfaces.- A Some extra problems.- B Solutions of exercises.- List of figures.
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