On the space of metrics with non-positive curvature

TL;DR

This paper investigates the topology of the space of non-positively curved metrics on certain manifolds, showing path disconnectedness in specific cases and introducing a new invariant based on geodesic string counting.

Contribution

It demonstrates the path disconnectedness of metric spaces on certain manifolds and introduces a novel geodesic string counting invariant for studying their topology.

Findings

Spaces of metrics on R x S^1 are path disconnected.

New metric deformation invariant based on geodesic string counting.

Potential for extending the invariant to broader topological studies.

Abstract

Let $\mathcal{M} (X)$ denote the space of complete Riemannian metrics with non-positive sectional curvature and with negatively curved ends, on a manifold $X$. We show that $\mathcal{M} (\mathbb{R} \times S ^{1}) $ and $\mathcal{M} (\mathbb{R} ^{} \times Y)$ are path disconnected, where $Y$ is compact and admits a negative curvature metric. The proof is very concise, using as the main ingredient Fuller index theory. Furthermore, we get a new metric deformation invariant based on geodesic string counting, and this gives a basic tool (likely to be very extendable) to further study the topology of $\mathcal{M} (X)$.