AG Differentialgeometrie (Winter Semester 2014/15)
- Lecturer: Prof. Dr. Wilderich Tuschmann
- Classes: Seminar (0126600)
- Weekly hours: 2
Talks this week
|Seminar:||Wednesday 11:30-13:00||SR 2.59|
|Lecturer||Prof. Dr. Wilderich Tuschmann|
|Office hours: by appointment|
|Room 1.002 Kollegiengebäude Mathematik (20.30)|
Speaker: David González (KIT/Universidad Autónoma de Madrid)
Title: Some restrictions on positively curved Riemannian submersions.
Abstract: The main difficulty when studying positively curved manifolds is the small number of known examples. At the present state of knowledge, Riemannian submersions are necessary in their construction: starting with the correct manifold with nonnegative sectional curvature as total space, one searches for some submersion that would guarantee a positively curved basis thanks to the well-known O'Neill formula. However, this is not so easily done, pointing out to the possible presence of restrictions for the existence of such Riemannian submersions from an arbitrary nonnegatively curved manifold. In this talk we bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and its conjugate radius. This is joint work with Luis Guijarro.
Speaker: Marco Radeschi (KIT/WWU-Münster)
Title: Metrics on spheres all of whose geodesics are closed.
Abstract: Riemannian manifolds in which every geodesic is closed, have been studied since the beginning of last century, when Zoll showed the existence of infinitely many metrics on the 2-sphere all of whose geodesics are closed. Among the many open problems on the subject, a conjecture of Berger states that for any simply connected manifold all of whose geodesics are closed, the geodesics must have the same length. The result was proved in the case of the 2-sphere by Grove and Gromoll. In this talk, I will show recent work with B. Wilking, where we prove that the Berger conjecture also holds for spheres of dimension >3. If time permits, I will discuss current work on how to extend the result to S^3, and to projective spaces.
Speaker: Jesús Núñez-Zimbrón (UNAM-Mexico)
Title: Circle actions on Alexandrov spaces
Abstract: Alexandrov spaces are metric spaces which have curvature bounded from below in the sense of comparison triangles. This class of spaces has been studied from the point of view of isometric group actions: Berestovskii showed that finite-dimensional, homogeneous metric spaces with a lower curvature bound are Riemannian manifolds. Galaz-Garcia and Searle studied Alexandrov spaces of cohomogeneity one (i.e. those with an isometric action of a Lie group whose orbit space is one-dimensional) and classified them in dimensions at most 4. We will present a classification of the (closed) Alexandrov spaces of dimension 3 admitting an isometric circle action. Together with the work of Berestovskii and Galaz-Garcia, Searle, this completes the classification of Alexandrov spaces of dimension at most three, admitting an isometric action of a compact, connected Lie group.