Learning outcomes of the course unit
The goal of the course is to give to the students, the basic tools of Differetial Geometry with special emphasis on Riemannian geometry mainly relationships that exist between local theory and global theory.
Students must demostrate knowledge and understanding in a field of study characterized by the use of advanced textbooks and scientific articles and to possess the ability to find and use data to formulate answer to well-defined problems. In addition, they must demonstrate that they have developed those skills allowing them to study by themselves without a guide.
theory curves and surface
Course contents summary
The first part of the lectures is an introduction about general facts of differential and Riemannian geometry.
The second part of the course is devoted to the local theory of a Riemannian manifold mainly the existence of a geodesic between two points and the existence of closed geodesics.
The third part of the course give us a relation between local theory and the global theory, when you think to a Riemannian manifold at whole.
Riemannian metric, Riemannian distance, a group of isometries, properly discontinuous actions, Riemannian submersions, integral and volume form
Affine connection and Levi-Civita connection, parallel transport, geodesics, the first variation formula, Gauss's lemma, the existence of a convex neighborhoods.
Curvature, sectional curvature, scalar curvature, Ricci curvature, Riemannian Laplacian, Killing fields, harmonic forms, Hodge theorem, techniques of Bochner
Jacobi fields, conjugate points, focal points.
Theorem of Hopf-Rinof, Hadamard theorem.
Manifolds with constant sectional curvature, A Theorem of Cartan, classification of space form.
Homogeneous Riemannian manifold, O'Neil's formula, symmetric spaces
Second variation formula, Theorem of Bonnet-Meyer and theorem Weinstein-Synge.
Index (Focal) Lemma index, Rauch comparison theorem, Comparison Theorem of Berger-Rauch and corollaries.
Morse index theorem, cut points.
Existence of closed geodesics, Theorem of Preissmann.
Manfredo do carmo, Riemannian Geometry, Birkauser
Cheeger-Ebin ''Comparison theorems in Riemannian geometry, North-Holland
Chavel, Riemannian Geometry: A modern introduction, Cambridge Univ. Press, Cambridge 1984.
Sakai, Riemannian Geometry, Translations of Mathematical Monographs vol. 149.
During the lessons we develop all the topics of the program. We also give some class of exercises to provide students the solutions to homeworks that have been left during the class.
Assessment methods and criteria
The exam consist of an oral exam at the end of the course. The aim is to evalue the level of knowledge and understanding acquired by students during the class. Moreover, the students should also be able to explain correctly the arguments exposed in the class to another person.