Higher order weak differentiability and Sobolev spaces between manifolds

Abstract

We define the notion of higher-order colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. When $M$ and $N$ are endowed with Riemannian metrics, $p\ge 1$ and $k\ge 2$, this allows us to define the intrinsic higher-order homogeneous Sobolev space ${\dot{W}}^{k,p}(M,N)$. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of $N$ in a Euclidean space; if the manifolds $M$ and $N$ are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space ${\dot{W}}^{k,p}(M,N)$ is that ${\pi }_{\lfloor kp\rfloor }(N)\simeq \{0\}$. We investigate the chain rule for higher-order differentiability in this setting.

Cited by in corpus (1)

• Uniform boundedness principles for Sobolev maps into manifolds