A Note on the Topology of a Generic Subspace of Riem

TL;DR

This paper investigates the topological properties of a dense subspace of Riemannian metrics on a 3-manifold without boundary, focusing on metrics with no symmetries and their implications for geometrodynamics.

Contribution

It extends known topological results of the space of all metrics to a subspace with no Killing vectors under certain conditions, clarifying its homotopy structure.

Findings

Riem'(M) has trivial homotopy groups under specified conditions

Riem'(M)/Diff(M) forms a proper manifold with homotopy groups linked to Diff(M)

The space of metrics with no symmetries is suitable for geometrodynamical analysis

Abstract

For Riem(M) the space of Riemannian metrics over a compact 3-manifold without boundary $M$, we study topological properties of the dense open subspace Riem'(M) of metrics which possess no Killing vectors. Given the stratification of Riem(M), we work under the condition that, in a sense defined in the text, the connected components of each stratum do not accumulate. Given this condition we find that one of the most fundamental results regarding the topology of Riem(M), namely that it has trivial homotopy groups, would still be true for Riem'(M). This would make the topology of Riem'(M) completely understood. Coupled with the fact that for Riem'(M), we have a proper principal fibration with the group of diffeomorphisms, which makes Riem'(M)/Diff(M) a proper manifold (as opposed to Riem(M)/Diff(M)), we would have that the homotopy groups of the quotient are given by the homotopy groups of Diff(M), which reflects the topology of $M$. These results would render the space of metrics with no symmetries subject to the above condition, as an ideal setting for geometrodynamical analysis.