**MTH 613: 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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