On the space of metrics with non-positive curvature

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 $\mathcal{M}(X)$.