Differentiable Manifolds. 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.
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 the students with fundamental knowledge and understanding in Differential Geometry of manifolds and fibre bundles, and in Riemannian Geometry. One of the aims is to let the students develop advanced technical skills and critical thinking, needed when modelling and solving mathematical problems in Differential Geometry and in other fields of mathematics and is applications. Special attention is paid to help the students develop communication skills necessary for teamwork. The course covers topics, introduces scientific problems and provides learning skills that are needed, or strongly suggested, to pursue a master degree in mathematics or in any scientific subject, and for training to research.
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: presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training sessions: training of the students to modelling and solving a wide selection of problems in Differential Geometry of manifolds and fibre bundles and Riemannian Geometry. The training sessions are conducted so to:
-- help the students develop communication skills and apply theoretical knowledge;
-- encourage independent judgement in the students.
Moodle learning platform: online teacher-student interaction; posting of additional notes, exercise sheets, and copies of past tests.
Remark: the suggested reading includes supplementary material that may be useful for further master studies in mathematics and for training to research.
Type of Assessment
Intermediate and final written examination: a selection of problems is proposed. The tests are designed to assess the ability of the students to apply their skills to advanced problem modelling and solving. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the originality and effectiveness of the methods adopted.
Oral examination: a number of questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Each student can choose to skip the written examinations. In this case he will be assessed with an enlarged oral examination, extended to the themes of the written 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.