F. Warner, Foundations of differentiable manifolds and Lie groups.
R. Bott-L. Tu, Differential forms in algebraic topology.
M. Do Carmo. Riemannian geometry.
Lecture notes.
Learning Objectives
The course aims at providing the students fundamental knowledge and understanding in Differential and Riemannian Geometry. One of the goals is to let the students develop advanced technical skills and critical thinking, needed when modelling and solving mathematical problems in Differential Geometry and other fields of mathematics and its 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 do 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.
Further information
Office hours: see
the web page of the instructor
Type of Assessment
Intermediate written examinations: a selection of problems is proposed during the course. 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
Differentiable manifolds, submanifolds. Vector fields. Distributions. The Frobenius theorem. Tensor calculus.
Differential forms on a manifold. De Rham cohomology on a manifold. Poincaré lemmas.
Morse functions. Structure next to a critical point. Morse inequalities.
Riemannian metrics. Levi Civita connection. Parallel transport. Geodesics. Jacobi fields. Variation of the energy. Theorems of Hopf Rinow, Hadamard and Bonnet-Myers.