The authors study Sobolev classes of weakly differentiable mappings $f:{\mathbb X}\rightarrow {\mathbb Y}$ between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in ${\mathcal W}{1,n}({\mathbb X}\, ,\, {\mathbb Y})\,$, $n=\mbox{dim}\, {\mathbb X}$.
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