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If $M$ is a compact smooth manifold and $X$ is a compact metric space, theSobolev space $W^{1,p}(M,X)$ is defined through an isometric embedding of $X$into a Banach space. We prove that the answer to the question whether Lipschitzmappings ${\rm Lip}\,(M,X)$ are dense in $W^{1,p}(M,X)$ may depend on theisometric embedding of the target.

Sobolev mappings: Lipschitz density is not an isometric invariant of the target