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We define the notion of colocally weakly differentiable maps from a manifold$M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with aRiemannian metric, this allows us to define intrinsically the homogeneousSobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent withthe definition by embedding in the Euclidean space and with the definition ofSobolev maps into a metric space. The colocal weak derivative is an approximatederivative. The colocal weak differentiability is stable under the suitableweak convergence. The Sobolev spaces can be endowed with various intrinsincdistances that induce the same topology and for which the space is complete.

Intrinsic weak derivatives and Sobolev spaces between manifolds