Laurea magistrale in Matematica

In particular, the course will provide:

• theoretical training objectives: consistent use of a rigorous mathematical language; assimilation of abstract concepts, theorems and their proofs related to differential geometry and its application to other parts of Mathematics;
• applied training objectives: the student will learn computing techniques in an abstract situation to solve problems; the student will be able to solve standard exercises and new problems, in which it will be necessary to develop new strategies and apply the concepts learned or develop simple proofs similar to those seen in class.

Metric structure of a Riemannian manifold. Isometries.

Linear connection and covariant derivative. Hessian and Laplacian. Parallelism. The Levi-Civita connection. Parallel transport. Riemannian curvature tensor and its properties. Sectional, Ricci and scalar curvature.

First derivative of the length functional and geodesics. The exponential map. Metric completeness and the Hopf-Rinow theorem. 

Isometric immersions: the second fundamental form, Weingarten endomorphism and the Gauss, Codazzi-Mainardi and Ricci equations. Mean curvature and minimal submanifolds. 

Manifolds with constant sectional curvature. 

Second derivative of the length functional and Jacobi fields. Some fundamental theorems  (Hadamard, Myers, Synge-Weinstein).

Time permitting: the Hessian comparison theorem and applications (Tompkins theorem, Cartan's theorem, Preissman's Theorem).

IMPORTANT: all didactic information, and the material, can be found on the Moodle page of the course. Students are kindly asked to register there.