Learning outcomes of the course unit
The goal of the course is to give to the students, by means of frontal class, the geometrical aspects of isometric actions, and an introduction to symplectic geomety and momentum map
The goal of the course is to give to the students, the basic tools of Riemannian geometry with special emphasis on the relationships that exist between local theory and global theory.
Course contents summary
LIE GROUPS, LIE GROUPS ACTING ON MANIFOLDS AND MOMENTUM MAP
An introduction to Lie group and Riemannian Geometry. Bi-invariant metric on Lie group. Proper action. Fiber bundle, Slice Theorem, stratifcation of the orbit space. Compact Lie group, maximal torus and Weyl group. Symplectic manifold, Darboux Theorem, momentum map and Convexity theorems.
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.
ALEXANDRINO, BETTIOL ''LIE GROUPS AND GEOMETRICAL ASPECTS OF ISOMETRIC ACTIONS, ANA CANNAS DA SILVA ''LECTURES ON SYMPLECTIC GEOEMETRY''
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.
The course counts 9CFUs which corresponds to 48 hours of lectures. The didactic activities is given by frontal class.
Assessment methods and criteria
Verification of the knowledges is achieved by an oral exam.