A local characterization for homeomorphisms that preserve continuously differentiable curves

Abstract

We study the subgroup of the homeomorphism group consisting of homeomorphisms of a manifold that preserve the set of one-dimensional $C^1$-submanifolds. For many surfaces, the similarly-defined family of related groups preserving the set of one-dimensional $C^r$-submanifolds has been shown by the author to be isomorphic to the automorphism groups of smooth fine curve graphs. In this paper, we give a necessary and sufficient local characterization for elements of this group using properties on the projective tangent bundle. In addition, we give several new examples of elements from this group that are not $C^1$-diffeomorphisms.

arXiv comment16 pages, 8 figures. Previous argument has been extended to higher dimensions and significantly simplified