Title of the course: Isometric actions and homogeneous spaces
Instructor: Prof. Alberto Rodríguez Vázquez
Institution: Université Libre de Bruxelles
Dates: 1-6 September 2025
Prerequisites: Basic differential geometry
Level: Advanced undergraduate
Abstract: The course will provide an introduction to the theory of isometric actions and homogeneous spaces. We will begin by considering simple examples of isometric actions and studying the slice theorem along with some of the key consequences. Next, we will explore homogeneous spaces—specifically, Riemannian manifolds that admit a transitive isometric action. We will derive relevant properties of homogeneous spaces, with a special emphasis on their Riemannian geometry, and examine specific classes of these spaces, such as naturally reductive homogeneous spaces and symmetric spaces. Finally, if time permits, we will discuss aspects of the structure and geometry of cohomogeneity one manifolds.
Language: English
Textbook:
M. Alexandrino, R. Bettiol: Lie Groups and Geometric Aspects of Isometric Actions (2015); A. Arvanitoyeorgos: An Introduction to Lie Groups and the Geometry of Homogeneous Spaces.