# Higher order weak differentiability and Sobolev spaces between manifolds

## TL;DR

This paper introduces a new intrinsic notion of higher-order weak differentiability for maps between manifolds, defining corresponding Sobolev spaces and exploring their properties and differences from classical embedding-based definitions.

## Contribution

It defines higher-order colocally weakly differentiable maps and intrinsic Sobolev spaces between manifolds, highlighting differences from embedding-based approaches and analyzing their properties.

## Key findings

- Intrinsic Sobolev spaces can be larger than embedding-based ones for compact manifolds.
- Necessary condition for density of smooth maps involves trivial homotopy groups.
- Chain rule for higher-order differentiability is examined.

## 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 $\pi_{\lfloor k p \rfloor} (N) \simeq \{0\}$. We investigate the chain rule for higher-order differentiability in this setting.