Spray-Invariant Sets in Infinite-Dimensional Manifolds
Differential Geometry
2026-04-14 v2
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.
Cite
@article{arxiv.2505.10980,
title = {Spray-Invariant Sets in Infinite-Dimensional Manifolds},
author = {Kaveh Eftekharinasab},
journal= {arXiv preprint arXiv:2505.10980},
year = {2026}
}