Spaces of nonnegatively curved surfaces

TL;DR

This paper characterizes the topological structure of the space of smooth, complete, nonnegatively curved metrics on surfaces with positive Euler characteristic, revealing it is homeomorphic to well-known infinite-dimensional spaces depending on the regularity parameter.

Contribution

It determines the homeomorphism type of these metric spaces under various regularity conditions, extending understanding of their topological structure.

Findings

Space of metrics with infinite regularity is homeomorphic to the separable Hilbert space.

For finite non-integer regularity, the space is homeomorphic to a countable power of the Hilbert cube span.

Results also apply to spaces of complete smooth Riemannian metrics on arbitrary manifolds.

Abstract

We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on surfaces of positive Euler characteristic equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is infinite or is not an integer. If $\gamma=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $\gamma$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics on an arbitrary manifold.