Author | Gallot, Sylvestre. author |
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Title | Riemannian Geometry [electronic resource] / by Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2004 |
Edition | Third Edition |
Connect to | http://dx.doi.org/10.1007/978-3-642-18855-8 |
Descript | XV, 322 p. 58 illus. online resource |
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 Levyโs 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