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arXiv · 2505.10980

Spray-Invariant Sets in Infinite-Dimensional Manifolds

Kaveh Eftekharinasab
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Abstract

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.

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Kaveh Eftekharinasab. 2025-05-16. Spray-Invariant Sets in Infinite-Dimensional Manifolds. https://doi.org/10.5269/bspm.77114

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