Title of the course: Totally geodesic submanifolds and curvature
Instructor: Dr. Alberto Rodríguez Vázquez
Institution: Université Livre de Bruxelles (ULB)
Dates: 16-22 September 2024
Prerequisites: Basic differential geometry
Level: Advanced undergraduate and graduate students
Abstract: The geometric objects that can be perceived by means of our senses are curves and surfaces. Submanifolds provide the natural generalization for higher dimensions of these objects and totally geodesic submanifolds are those with the simplest geometry. Intuitively, a submanifold is totally geodesic if it curves as the ambient space where it lives. The most basic examples are affine subspaces of Euclidean spaces R^n, or great subspheres of round spheres S^n. As we will see, totally geodesic submanifolds are intimately linked with curvature, and one of the purposes of this course will be to explore these links. Moreover, the existence of totally geodesic submanifolds is usually related to the abundance of isometries. Thus, the theory of totally geodesic submanifolds is particularly rich in spaces with a big isometry group such as homogeneous and symmetric spaces. Indeed, in certain cases we will be able to use Lie theory to reduce the study of totally geodesic submanifolds to a problem of algebraic nature.
Tentative list of contents:
Basics of submanifold geometry. Totally geodesic submanifolds. Some basic characterizations. Examples. Fixed point components of isometries.
Existence and uniqueness of totally geodesic submanifolds.
Positive curvature. Frankel theorem. Critical points of sectional curvature.
Totally geodesic submanifolds in homogeneous spaces.
Totally geodesic submanifolds in symmetric spaces.
Language: English
References/Textbook:
Berndt, Jürgen; Console, Sergio; Olmos, Carlos: Submanifolds and holonomy. Chapman & Hall/CRC Research Notes in Mathematics, 434.
Chapman & Hall/CRC, Boca Raton, FL, 2003.