Spray-Invariant Sets in Infinite-Dimensional Manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910120000094208 |
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| author | Eftekharinasab, Kaveh |
| author_facet | Eftekharinasab, Kaveh |
| contents | We introduce the concept of spray-invariant sets on infinite-dimensional manifolds, where any geodesic of a spray starting in the set stays within it for its entire domain. These sets, possibly including singular spaces such as stratified spaces, exhibit different geometric properties depending on their regularity: sets that are not differentiable submanifolds may show sensitive dependence, for example, on parametrization, whereas for differentiable submanifolds invariance is preserved under reparametrization.
This framework offers a broader perspective on geodesic preservation than the rigid notion of totally geodesic submanifolds, with examples arising naturally even in simple settings, such as linear spaces equipped with flat sprays. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spray-Invariant Sets in Infinite-Dimensional Manifolds Eftekharinasab, Kaveh Differential Geometry 58B20, 53C22, 37B35 We introduce the concept of spray-invariant sets on infinite-dimensional manifolds, where any geodesic of a spray starting in the set stays within it for its entire domain. These sets, possibly including singular spaces such as stratified spaces, exhibit different geometric properties depending on their regularity: sets that are not differentiable submanifolds may show sensitive dependence, for example, on parametrization, whereas for differentiable submanifolds invariance is preserved under reparametrization. This framework offers a broader perspective on geodesic preservation than the rigid notion of totally geodesic submanifolds, with examples arising naturally even in simple settings, such as linear spaces equipped with flat sprays. |
| title | Spray-Invariant Sets in Infinite-Dimensional Manifolds |
| topic | Differential Geometry 58B20, 53C22, 37B35 |
| url | https://arxiv.org/abs/2505.10980 |