Homeomorphisms of surfaces that preserve continuously differentiable curves

In this paper, we study Homeo$^1(S)$, the group of homeomorphisms of a surface that preserve the set of one-dimensional $C^1$ submanifolds of that surface. The group Homeo$^1(S)$ belongs to a family of similarly defined groups Homeo$^k(S)$ that were recently introduced by the author. In a separate paper, we have shown that for most closed surfaces, Homeo$^k(S)$ is naturally isomorphic to the automorphisms of a smooth fine curve graph. By contrast, the work in this paper gives local conditions that characterize Homeo$^1(S)$. We show that there exists a collection of conditions that are both necessary and sufficient for a homeomorphism of the surface to be an element of this group. These conditions primarily depend upon the structure of the induced map on the projective tangent bundle. Additionally, we provide examples of several types of elements of Homeo$^1(S)$ that are not diffeomorphisms. These include inducing discontinuous maps on the projective tangent bundle and having infinitely many non-differentiable points.