The notion of differentiable manifolds is the basis of several mathematical theories, both pure and applied. It is certainly preparatory to the more sophisticated notion of algebraic varieties and schemes in algebraic geometry, but it is also fundamental in the theory of dynamical systems, in General Relativity, in analytical mechanics (varieties of configurations and phases) as well as in the study of the so-called statistical varieties. It often happens that doctoral students lack basic knowledges and/or awareness of differentiable manifolds necessary for the specific studies in which they are engaged. The aim of this course is to fill this gap. The treatment will be elementary and will therefore not interfere with the more advanced courses of colleagues in the geometric-differential area. Many exercises will be proposed.

Linear Algebra, Analytic Geometry. Elements of General Topology. Analysis I and II.

The program will cover a subset of the following topics: Recalling the notion of differential of a differentiable vector valued function of more real variables; topological manifolds as locally R^n topological spaces. Differentiable structures. Examples: the spheres S^n , Toruses, projective spaces, real Grassmannians. Tangent and cotangent bundles. Tangent vector fields and differential forms. Vector fields as globalizations of systems of ordinary differential equations. Integral curves, prime integrals. The Lie algebra of vector fields and Frobenius' theorem. Pull-back of differential forms. Lie groups. Invariant vector fields on Lie groups. Lie algebra of a group. Examples with classical groups. Introduction to Riemannian and symplectic varieties.

On site

Oral presentation

P.D.2-2 - March

Suitable references will be recommended during the course. Much of it will be a more modern re-reading of Boothby's classic book "Introduction to Differential Manifolds and Riemannian Geometry"