Students with a knowledge of Riemannian Geometry at the master level.
IEMath-GR (Instituto de Matemáticas de la Universidad de Granada)
January 7-18, 2019 (two weeks).
We will give the rudiments of Finsler Geometry using techniques which are familiar to those who have worked with Riemannian Geometry, aiming to provide the basic tools which allow a further study: associated connections, geodesics, Jacobi fields and curvature tensors.
We will begin at the vector space level, giving the definition and characterizing the pseudo-Finsler metrics (or pseudo-Minkowski norms) in terms of its indicatrix, proving that the pseudo-Minkowski norm is Minkowski if and only if the indicatrix is strongly convex.
The next step will be to introduce the anisotropic calculus in order to handle the tensors which depend on the direction. Having at our diposal the anisotropic calculus we will introduce the associated anisotropic connections to a pseudo-Finsler metric and then the Chern curvature tensor and the flag curvature. We will compare the anisotropic connections with the classical connections over the tangent bundle and then will derive the Bianchi identities. Next, we will define geodesics and Jacobi fields, showing their variational properties. This will lead to introduce the exponential map, proving the Gauss Lemma and the existence of a neighborhood where geodesics are the only minimizers of the length.
The notion of Finsler spacetime and cone structure will be introduced, showing the relation between both notions and giving a good amount of examples. Killing and conformal fields will be introduced, and the role of the latter for cone structures will be stressed. Some basic concepts of submanifold theory, generalizing the Gauss and Codazzi equations are introduced. The framework of Einstein-Finsler equations will also be treated.
As an extra activity, the students of the course can attend to the Finsler Meeting, celebrated along this second week.
Basic Course “Finsler Geometry from a Riemannian viewpoint” 10 hours from Monday to Friday, in blocks of 2 hours: 1,30 theoretical and 30 minutes of practical exercises.
Advanced Seminar consisting in a 6-hour course entitled "Applied Finsler Geometry and spacetimes".