Mathematics

FRANCESCO BEI

General objectives:
acquire basic knowledge in Riemannian geometry.

Specific objectives:

Knowledge and understanding:
at the end of the course the student will have acquired the basic notions and results relating to the Riemannian varieties, connections and the different notions of curvature, the geodesics and fields of Jacobi, completeness and spaces with constant curvature.

Apply knowledge and understanding:
at the end of the course the student will be able to begin the study of advanced topics of Riemannian geometry, and to solve complex problems in this area.

Critical and judgmental skills:
the student will have the bases to analyze and appreciate the analogies and connections between the topics covered and the most varied themes coming from differential, algebraic topology, from algebraic and complex geometry.

Communication skills:
ability to rigorously expose the contents in the most theoretical questions present in the written test, and in the eventual oral part of the verification.

Learning ability:
the acquired knowledge will allow to face a possible master's thesis work on advanced topics of differential / Riemannian geometry, but also of complex analytical / differential geometry.

Bachelor in mathematics

vector fields and integral curves, distributions, tensors, differential forms, elementary theory of Lie groups;
- Riemannian and semi-Riemannian manifolds, space forms, homogeneous spaces, covariant derivatives, parallelism, geodesics, curvature, Jacobi fields;
- first Cartan's theorem, classification of constant curvature spaces, basic geometry and topology of nonpositively curved spaces, Synge's theorem.

-Do Carmo, Riemannian Geometry
-Gallot-Hulin-Lafontaine, Riemannian Geometry

lectures and exercise sheets

Strongly recommended

Written examination

Exercises and questions concerning manifolds, metrics, connections, geodesics, etc, etc