2025/05/16 by Eftekharinasab, Kaveh
#37B35 #53C22 #58B20 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.10980
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.