Index theory and deformations of open nonnegatively curved manifolds

TL;DR

This paper employs index theory to demonstrate that the space of nonnegatively curved metrics on certain open spin manifolds has complex topological structure, revealing nontrivial homotopy groups in infinitely many degrees.

Contribution

It introduces a novel application of Hitchin's index-theoretic technique to study the topology of metric spaces on open manifolds, including the concept of homotopy density of cylindrical-end metrics.

Findings

The space of nonnegatively curved metrics has nontrivial homotopy groups in infinitely many degrees.

Homotopy density of metrics with cylindrical ends is established as a new tool.

The approach extends understanding of the topology of metric spaces in differential geometry.

Abstract

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.