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Indian Institute of Science Education and Research Bhopal
MTH 513: Introduction to Riemannian Geometry (4)
Pre-requisites: MTH 405 and MTH 508
Review of differentiable manifolds: vector bundles, tensors, vector fields, differential forms, Lie groups
Riemannian metrics. Definition, examples, existence theorem; model spaces of Riemannian geometry
Connections: connections on a vector bundle, linear connections, covariant derivative, parallel transport, geodesics
Riemannian connections and geodesics: torsion tensor, Fundamental Theorem of Riemannian Geometry, geodesics of the model spaces, exponential map, convex neighborhoods, Riemannian distance function, first variation formula, Gauss' lemma, geodesics as locally minimizing curves; completeness, statement of Hopf-Rinow Theorem
Curvature: Riemann Curvature Tensor, Bianchi identity, scalar, sectional and Ricci curvatures
Jacobi Fields: Jacobi equation, conjugate points, second variation formula, spaces of constant curvature (if time permits)
Curvature and topology: Gauss-Bonnet Theorem, Bonnet-Myers Theorem, Cartan-Hadamard Theorem
Suggested Books:
Texts:
- J. M. Lee. Riemannian Manifolds, An introduction to Curvature. Graduate Texts in Mathematics. Springer (1997).
- M. P. do Carmo. Riemannian Geometry. Birkhauser (1991).
- S. Gallot, D. Hulin, J. Lafontaine. Riemannian Geometry. Springer (2004).
References:
- I. Chavel. Riemannian geometry, a modern introduction. Cambridge University Press (2006)
- S. Kobayashi, K. Nomizu. Foundations of differential geometry, vol. -I, Wiley Interscience Publication (1996).
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