Riemannian Geometry

Prerequisites:  Analysis in Rⁿ, Fundamental Theorem of ODE, some knowledge of Differential Geometry, EDP and Covering Spaces

Riemannian metrics. Levi-Civitta connection. Geodesics. Normal and totally normal neighborhoods. Tensor of curvature. Covariant derivation of tensors. Jacobi fields and conjugated points. Isometric immersions; Gauss, Ricci and Codazzi equations. Complete Riemannian manifolds; Hopf-Rinow theorem, Hadamard’s theorem. Constant curvature spaces. Variations of arc-length; applications. Rauch’s comparison theorem; the Bonnet-Myers theorem, the Synge theorem and other applications. The Morse index theorem. The minimum point place. Other topics.

References:
CARMO, M. – Geometria Riemanniana, Rio de Janeiro, IMPA, Projeto Euclides, 1979.
CHEEGER, J., EBIN, D. – Comparison Theorems in Riemannian Geometry, Amsterdam, North-Holland, 1975.
JOST, J. – Riemannian Geometry and Geometric Analysis, Berlin Heildelberg, New York, Springer Verlag, 1995.
O’NEILL, B. – Semi-Riemannian Geometry with applications to Relativity, New York, Academic Press, 1983.
PETERSEN, P. – Riemannian Geometry, Graduate Texts in Mathematics, Springer-Verlag, 2006.

* Standard program. The teacher has the autonomy to make any changes