Mathematics | University of Parma

Professor
Academic discipline: Geometria (MAT/03)
Field: Attività formative affini o integrative
Type of training activity: Related/supplementary

48 hours
of face-to-face activities

6 credits

hub: PARMA

course unit
in - - -

lectures timetable unavailable

exam calls

Learning objectives

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.

Prerequisites

Differential Geometry

Course unit content

Riemannian Geometry

Full programme

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.

Bibliography

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.

Teaching methods

class

Assessment methods and criteria

Oral exam

Other information

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RIEMANNIAN GEOMNETRY cod. 19482