  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 a
Riemannian metric, this allows us to define intrinsically the homogeneous
Sobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent with
the definition by embedding in the Euclidean space and with the definition of
Sobolev maps into a metric space. The colocal weak derivative is an approximate
derivative. The colocal weak differentiability is stable under the suitable
weak convergence. The Sobolev spaces can be endowed with various intrinsinc
distances that induce the same topology and for which the space is complete.
