The course is an introduction to the basic concepts and methods of differential geometry and topology. We expect that students understand the local/global tension, become confident with the language of manifolds and bundles as tools in non-linear geometric problems, get an insight in functorial approach, and understand metric geometry in the framework of Riemannian geometry,
Course Prerequisites
The contents of the courses Linear Algebra, Algebra 1, Geometry 1 and 2, and of the Analysis courses of the first two years of the Laurea in Mathematics curriculum
Teaching Methods
Lectures
Assessment Methods
The exam is oral. The goal is to check the understanding of the students around the topics of the lectures, the clarity in the exposition and also the ability in applying the notion in concrete problems. The student is required to have a good understanding of the notions treated in the classes, which will be investigated either by asking questions on specific topics or by proposing some problems that can be solved by means of the notions presented in the course. The grade will be obtained by considering the broadness and the deepness of the understanding, the clarity of the exposition and the problem solving skills.
Texts
Notes by Gian Pietro Pirola. Frank Warner: "Foundations of differentiable manifolds and Lie groups". Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin. Bott Tu Differential forms and Algebra Topology, Graduate Text in mathematics 82. Springer Verlag, Milnor, Topology From differentiable viewpoint, The University Press of Virginia, Charlottesville. Boothby, William M.: "An introduction to differentiable manifolds and Riemannian geometry". Pure and Applied Mathematics, No. 63. Academic Press, New York-London, 1975. Th. Broecker and K. Jaenich: "Introduction to differential topology". Milnor, J.: "Morse theory". Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963.
Contents
Differentiable manifolds: tangent and cotangent spaces, vector fields and differential forms. Tangent and cotangent bundles: hints on vector bundles and related constructions. Vector fields, flows, Lie brackets. Coordinate fields, distributions : the Frobenius theorem. Topics in differential topology. Sard’s lemma. Whitney immersion theorems. Exterior differential, de Rham cohomology, cohomology of forms with compact support. Poincaré lemmas. Mayer Vietoris sequence. Poincaré duality. De Rham Theorem. Riemannian geometry: riemannian manifolds and Levi-Civita connections, curvature, geodesics, completeness, the theorems of Hopf-Rinow and Whitehead; Jacobi fields.
