Riemannian Geometry

Period of duration of course
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Course info
Number of course hours
40
Number of hours of lecturers of reference
40
Number of hours of supplementary teaching
0
CFU 6
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Type of exam

Oral exam

Lecturer

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Prerequisites

Basic knowledge of differentiable manifolds, vector andtensorial fields,differential forms, integration on manifolds.Suggested for Master students, or for Ph.D. students if they have not attended related courses.

Programme

- Riemannian Metrics

- Affine connections

- Riemannian and Ricci curvatures

- Parallel transport

- Geodesics and exponential map

- First and second variation of lenght

- Jacobi fields and conjugate points

- Isometric immersions

- Hopf-Rinow and Hadamard theorems

- Constant curvature spaces

- Bonnet-Myers theorem

- Negative curvature spaces


Time permitting: Sphere Theorems

Educational aims

The purpose of the course is to exmploy analytical tools and differential calculus to study the geometry and topology of manifolds.

Bibliographical references

- M. Do Carmo: Riemannian Geometry.

- P. Petersen: Riemannian Geometry.

- M. Spivak: A comprehensive introduction to differentialgeometry