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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.19983 |
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| _version_ | 1866912449027899392 |
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| author | Savelyev, Yasha |
| author_facet | Savelyev, Yasha |
| contents | 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 $\mathcal{M} (X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19983 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the space of metrics with non-positive curvature Savelyev, Yasha Differential Geometry 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 $\mathcal{M} (X)$. |
| title | On the space of metrics with non-positive curvature |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.19983 |