Riemannian geometry
Prerequisites: Analysis in Rn, fundamental theorem of ODEs, some knowledge of Differential Geometry, PDEs and covering spaces.
Riemannian metrics. Levi-Civitta connection. Geodesics. Normal and totally normal neighborhoods. Curvature tensor. Covariant derivation of tensors. Jacobi fields and conjugate points. Isometric immersions; Gauss, Ricci and Codazzi equations. Complete Riemannian varieties; Hopf-Rinov theorem, Hadamard theorem. Spaces of constant curvature. Arc length variations; applications. Rauch’s comparison theorem; Bonnet-Myers theorem, Synge’s theorem and other applications. Morse’s index theorem. The place of minimum points. Other topics.
References:
CARMO, M. – Riemannian Geometry, 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.
CARMO, M. – Riemannian Geometry, 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.
* Basic syllabus. The teacher has the autonomy to make any changes.