Certain optimal correspondences between plane curves, I: Manifolds of shapes and bimorphisms

In previous joint work, a theory introduced earlier by Tagare was developed for establishing certain kinds of correspondences, termed bimorphisms, between simple closed regular plane curves of differentiability class at least $C^2$. A class of objective functionals was introduced on the space of bimorphisms between two fixed curves $C_1$ and $C_2$, and it was proposed that one define a ``best non-rigid match'' between $C_1$ and $C_2$ by minimizing such a functional. In this paper we prove several theorems concerning the nature of the shape-space of plane curves and of spaces of bimorphisms as infinite-dimensional manifolds. In particular, for $2\leq j<\infty$, the space of parametrized bimorphisms is a differentiable Banach manifold, but the space of unparametrized bimorphisms is not. Only for $C^\infty$ curves is the space of bimorphisms an infinite-dimensional manifold, and then only a Fréchet manifold, not a Banach manifold. This paper lays the groundwork for a companion paper in which we use the Nash Inverse Function Theorem and our results on $C^\infty$ curves and bimorphisms to show that if $\Gamma$ is strongly convex, if $C_1$ and $C_2$ are $C^\infty$ curves whose shapes are not too dissimilar ($C^j$-close for a certain finite $j$) and if neither curve is a perfect circle, then the minimum of a regularized objective functional exists and is locally unique.