Spray-Invariant Sets in Infinite-Dimensional Manifolds
Kaveh Eftekharinasab
TL;DR
The paper develops a general theory of spray-invariant sets on infinite-dimensional manifolds, extending geodesic preservation beyond totally geodesic submanifolds by using the admissible set $A_{\EuScript{S},S}$ and the second-order tangent framework. It establishes, in Fréchet and MC$^{k}$-Fréchet contexts, that spray-invariance is characterized by first- and second-order adjacencies to $A_{\EuScript{S},S}$, and provides Nagumo-Brezis and transversality criteria to test invariance. Automorphisms of the ambient manifold and group actions preserving sprays are shown to carry spray-invariant sets into spray-invariant families, including orbit-type stratifications. The Banach and Hilbert sections adapt the framework to more regular settings, with concrete examples in flat sprays, loop spaces, and stratified subspaces to illustrate the subtleties of invariance across regularity classes. Overall, the work offers a robust toolkit for understanding geodesic preservation in singular and infinite-dimensional geometries, with implications for dynamics on complex manifolds and symmetry-reduced systems.
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.