Manifolds with lower sectional curvature bounds and Alexandrov geometry
Karsten Grove (University of Notre Dame)
MSRI: Simons Auditorium
The aim of the talk is to provide a survey of the main tools, results and open problems concerning manifolds with a lower (sectional) curvature bound. It is well known that local bounds on sectional curvature can be described geometrically via local distance comparison to constant curvature spaces. For lower curvature bounds this comparison is global, as expressed in the Toponogov Comparison Theorem. This together with critical point theory for distance functions paved the way for studying manifolds with only a lower sectional curvature bound, resulting in Finiteness, Structure, and Recognition Theorems. There are several equivalent versions of Toponogov’s Comparison Theorem, some of which make sense in a general metric space. Moreover, such a metrically expressed lower curvature bound is preserved by the process of taking a Gromov-Hausdor limit. An Alexandrov space is a finite Hausdor dimensional, inner metric space with a lower curvature bound. It turns out that, despite their general definition, Alexandrov spaces have a surprisingly rich structure and are natural objects in their own rite. Their applications and significance to Riemanian geometry stems from the fact that there are several natural geometric operations that are closed in Alexandrov geometry, but not in Riemanian geometry. These include taking Gromov-Hausdor limits, taking quotients and taking joins of positively curved spaces. All concepts alluded to above will be explained and discussed, as will examples, some of the main results, and fundamental open problems.