Scheda programma d'esame
ISTITUZIONI DI GEOMETRIA
|ISTITUZIONI DI GEOMETRIA||MAT/03||LEZIONI||63|
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KnowledgeThe aim of the course is to provide the students with a solid knowledge of the most important differential geometric tools, with an eye toward their use in all areas of Mathematics, as well as in the application of Mathematics to other fields. In particular, the student who successfully completes the course will acquire a solid knowledge of: - differential geometry in the large; - vector bundles, vector fields and flows; - basic Riemannian geometry; - differential forms and de Rham cohomology.
Assessment criteria of knowledgeIn the written exam (3 hours) the student should solve up to three exercises related to the content of the course, showing his/her ability in using the material taught in the course. During the oral exam the student must demonstrate his/her knowledge of the course material by answering questions on specific topics and providing proofs of results presented in the course.
- Final oral exam
- Final written exam
Delivery: face to face
- attending lectures
- individual study
SyllabusThe course covers the basics of differential geometry: smooth manifolds, smooth maps, partitions of unity, tangent vectors, vector bundles, tangent and cotangent bundles, tensor bundles, sections of vector bundles, vector fields and differential forms, the flow of a vector field, Lie brackets, orientation. The second part of the course will cover the basics of Riemannian geometry: connections, covariant derivative, parallel transport, Riemannian metrics, isometries, Levi-Civita connection, geodesics, exponential maps, Riemannian distance, first variation formula of arc length, minimizing properties of geodesics, an introduction to Riemannian curvature. The last part is devoted to de Rham cohomology and sheaf theory: integration and external differentiation of differential forms, orientation, Stokes theorem, Mayer-Vietoris sequence, Poincare' lemma, cohomology of euclidean spaces and spheres, Poincare' duality, Kunneth theorem, sheaves, Cech cohomology, de Rham theorem.
Bibliography- M. Abate, F. Tovena, "Geometria differenziale", Springer, Milano, 2011.
Ultimo aggiornamento 14/11/2016 17:27