Differentiable Manifolds. Germs of functions. Derivations and tangent space at a point of a differentiable manifold. Applications, local immersions, immersions and embeddings, submersions. Vector fields. Local flows and vector fields. Poisson bracket and Lie derivative. Vector bundles. Fiber bundles. Riemannian manifolds. Differential forms. Groups of deRham cosmology. Connections. Curvature. Connections in the tangent bundle. Levi-Civita connection. Parallel transport and geodesic curves.
G. Gentili, F. Podestà, E. Vesentini, LEZIONI DI GEOMETRIA DIFFERENZIALE, Bollati Boringhieri, Torino, 1995.
S. Helgason, DIFFERENTIAL GEOMETRY, LIE GROUPS AND SYMMETRIC SPACES, Academic Press, 1978.
W. M. Boothby, AN INTRODUCTION TO DIFFERENTIABLE MANIFOLDS AND RIEMANNIAN GEOMETRY, Academic Press, 1975.
Learning Objectives
The course aims to provide knowledge and technical skills in Differential Geometry of manifolds and vector (and fiber) bundles. The knowledge of the topics covered in the course and the technical skills provided are needed, or in any case of great help, to learn a large variety of advanced subjects in mathematics, like for instance in algebraic geometry, differential and riemannian geometry, real and complex analysis, mathematical physiccs, differential topology.
Prerequisites
The knowledge of basic material of analysis of one and several variables, of linear algebra, of general topology and differential geometry of curvees and surfaces are necessary pre-requisites.
Teaching Methods
Lectures and training session.
Type of Assessment
Written and oral tests
Course program
Atlases and maximal atlases, differentiable structures. Differentiable manifolds. The Riemann sphere. Germs of functions. Derivations. Tangent space at a point of a differentiable manifold. Applications, local immersions, immersions and embeddings, submersions. Vector fields and tangent bundle. Local flows and vector fields. Poisson bracket and Lie derivative. Vector bundles. Weak and strong equivalence and structure theorem of vector bundles. Metrics along the fibers of a vector bundle. Riemannian manifolds. Differential forms on a differentiable manifolds. Construction of the classical Groups of deRham coomology of a differentiable manifold. Covariant differentiation and connections along the fibers of a vector bundle. Curvature of a connection. Connections in the tangent bundle. Levi-Civita connection. Parallel transport and geodesic curves.