Cohomogeneity One Action On Manifolds and Alexandrov Spaces
Time: 2017-09-27
Published By: Ningbo Lu
Speaker(s): Masoumeh ZAREI （BICMR）
Time: 10:00-11:30 October 11, 2017
Venue: Room 29, Quan Zhai, BICMR
Abstract:
Studying cohomogeneity one actions is initiated in a project proposed by Karsten Grove to advance the theory of positively curved Riemannian manifolds. In fact, the general knowledge and understanding of Riemannian manifolds with positive sectional curvature is still limited. In particular, although only a few obstructions are known, the list of the known examples is rather short. Since all known examples have a fairly large amount of symmetry, it seems natural to look for new examples with large amount of symmetry. One of the natural measurements for the size of the isometry group of a Riemannian manifold is its cohomegeneity, i.e., the dimension of the orbit space. For example, having cohomogeneity 0 means that the isometry group acts transitively on the manifold, i.e., it is homogeneous. A cohomogeneity one action can be considered then as the second largest symmetric action.
In this talk, I will give an introduction to the cohomogeneity one actions on a space X, where X is a smooth or topological manifold or an Alexandrov space. I will review some basic definitions, facts and examples and give some words on the equivariant classification of such spaces. In the end, I will talk about some of our results on the cohomogeneity one actions on topological manifolds and Alexandrov spaces.
Studying cohomogeneity one actions is initiated in a project proposed by Karsten Grove to advance the theory of positively curved Riemannian manifolds. In fact, the general knowledge and understanding of Riemannian manifolds with positive sectional curvature is still limited. In particular, although only a few obstructions are known, the list of the known examples is rather short. Since all known examples have a fairly large amount of symmetry, it seems natural to look for new examples with large amount of symmetry. One of the natural measurements for the size of the isometry group of a Riemannian manifold is its cohomegeneity, i.e., the dimension of the orbit space. For example, having cohomogeneity 0 means that the isometry group acts transitively on the manifold, i.e., it is homogeneous. A cohomogeneity one action can be considered then as the second largest symmetric action.
In this talk, I will give an introduction to the cohomogeneity one actions on a space X, where X is a smooth or topological manifold or an Alexandrov space. I will review some basic definitions, facts and examples and give some words on the equivariant classification of such spaces. In the end, I will talk about some of our results on the cohomogeneity one actions on topological manifolds and Alexandrov spaces.
