On the space of metrics with non-positive curvature

2025/06/24 by Savelyev, Yasha
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.19983

Abstract

Let 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 M (ℝ × S ¹) and M (ℝ × 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 M (X).

Citations

• On the topology of the space of negatively curved metrics
• The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible

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