Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

Abstract

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 ≥ 1 {p\geq 1} and k ≥ 2 {k\geq 2} , this allows us to define the intrinsic higher-order homogeneous Sobolev space W ˙ k , p ⁢ ( M , N ) {\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 W ˙ k , p ⁢ ( M , N ) {\dot{W}^{k,p}(M,N)} is that π ⌊ k ⁢ p ⌋ ⁢ ( N ) ≃ { 0 } {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}} . We investigate the chain rule for higher-order differentiability in this setting.