(in Polish) Struktury geometryczne na rozmaitościach 1000-1S22GSM
Smooth manifolds carrying additional structure are basic objects in many branches of mathematics and physics, including Control Theory, Riemannian Geometry, Lagrangian and Hamiltonian Mechanics, Field Theory, General Relativity. In the seminar we want to introduce/recall some basic differential geometric notions, including tangent and cotangent bundles, vector and tensor fields, fiber bundles, jets. Later, we would like to dig deeper into one or more specific topics according to the preferences of the students.
Some proposals of these specific topics are:
-foundations of Riemannian geometry, Levi-Civita connection, Riemannian curvature
-sub Riemannian geometry, normal and abnormal geodesics and the problem of smoothness of length minimizing curves
-Geometric Control Theory, emphasizing results about accessibility such as Frobenius, Chow’s-Rachewski and Sussmann theorems
-Theory of product-preserving functors and its relation with Weil algebras
-structure of jet bundles and geometry of partial differential equations
Main fields of studies for MISMaP
Type of course
Mode
Prerequisites (description)
Learning outcomes
The intended outcome is that a student will acquire a foundation in the differential geometry used across a variety of modern fields of contemporary research.
Assessment criteria
Evaluation based on a given seminar talk and active participation in the classes.
Bibliography
Lee, Introduction to smooth manifolds
Lee, Riemannian Manifolds: An Introduction to Curvature
Kolar, Michor, Slovak, Natural Operations in Differential Geometry
Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications
Saunders, The Geometry of Jet Bundles
Spivak, A Comprehensive Introduction to Differential Geometry. Volumes I-V
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: