Spray-Invariant Sets in Infinite-Dimensional Manifolds

TL;DR

This paper introduces spray-invariant sets on infinite-dimensional manifolds, exploring their properties and differences from traditional geodesic-preserving submanifolds, with examples including linear spaces.

Contribution

It broadens the concept of geodesic invariance to include sets with varying regularity, including singular and stratified spaces, extending classical notions.

Findings

Invariance depends on regularity of the set

Differentiable submanifolds preserve invariance under reparametrization

Examples include linear spaces with flat sprays

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.