Alexandrov spaces are a generalization of complete Riemannian manifolds with a lower sectional curvature bound. However, they may exhibit behaviors different from Riemannian manifolds due to the topological and metric singularities that they carry. It is then of fundamental importance to investigate whether one can extend a given property in the Riemannian setting to the Alexandrov setting.
In this project we address some of these properties and explore how Alexandrov spaces behave with respect to each property. In the one direction, we focus our attention on the topological features. The primary objective here is to understand how far the topology of Alexandrov spaces are from that of the smooth manifolds. In the other direction, we examine Alexandrov spaces with positive curvature in the presence of symmetry and the goal is to classify them. To this end, we need to find obstructions and recognition tools, which, in particular, rely upon our understanding of topological behaviors of Alexandrov spaces investigated in the former direction.
Publications
We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the \(C^{1,\alpha}\)-sense to a \(C^{1,\alpha}\)-continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.
Related project(s):
43Singular Riemannian foliations and collapse79Alexandrov geometry in the light of symmetry and topology
We prove that there exist ????????(3)-invariant metrics on Aloff-Wallach spaces W^7_{k1,k2}, as well as ????????(5)-invariant metrics on the Berger space B^{13}, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.
Related project(s):
79Alexandrov geometry in the light of symmetry and topology
We show that Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature.
Related project(s):
79Alexandrov geometry in the light of symmetry and topology
We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing k-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are. This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro-Wilhelm our approach partly generalizes recent work by Berndt-Olmos on the index conjecture.
Related project(s):
79Alexandrov geometry in the light of symmetry and topology
Team Members
Dr. Masoumeh Zarei
Researcher,
Project leader
Universität Hamburg
m288.zarei(at)gmail.com