Alexandrov's Geometry and Applications

Abstract

Award: DMS-0406482

Principal Investigator: Anton Petrunin

The principal investigator proposes to continue his research in

Alexandrov geometry and its applications, studying collapse with

lower curvature bound (joint with W.Tuschmann and

V.Kapovitch). This part of the project can be thought of as an

attempt to generalize and refine Gromov's Betti number theorem.

This research can be divided into three main parts:

(i) Finding new topological invariants which are finite on any family

of ``similar'' manifolds which the PI has described.

(ii) Using the gradient push to limit the number of bundles with the

same fiber and base which can admit given lower curvature and upper

diameter bounds

(iii) Showing that simply connected spin manifolds with non-zero

A-hat-genus can not be almost non-negatively curved.

The principal investigator also proposes to continue his study of

positive functions of curvature which give a bounded integral

along any positively curved manifold with lower curvature bound,

upper diameter and lower volume bound. This should clarify the

nature of curvature tensors of Alexandrov spaces.

The PI would like to show that the curvature tensor for

Alexandrov spaces is well defined as a measure valued tensor in

any distance co-ordinates. If true, this should give a solution

to such long standing questions as whether a convex surface in

Alexandrov space is an Alexandrov space.

The principal investigator proposes to compile a collection of

exercises in modern geometry. The idea is to find problems which

could be solved in one step. However solutions are supposed to

be non trivial and would also lead to a discovery of important

ideas in modern geometry. PI has already gathered some number of

such problems which can be viewed online and many people had been

engaged in this project. This project is oriented toward

students and young scientists.

Riemannian manifold could be considered as a simplified version

of space-time. The author considers an approach to studing

Riemannian manifolds. It is based on considering extremal

metrics, in an appropriate sense, for example how Riemannian

manifolds collapse to lower dimenssional objects. This method

makes possible to get new resuls in the main stream direction of

Riemannian geometry: how to make conclusions about global

structure of space basing on local properties.