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 $\mathcal{ M} (X)$.
Categories: math.DG