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I posted this question in MSE but got no response (even after giving a bounty), so I am trying here. Let $M,N$ be smooth $d$-dimensional Riemannian manifolds. Suppose $f:M \to N$ is a differentiable ...

Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth). Suppose we have a map $\phi:M \to N$ which satisfies the following properties: $$(1) \, \,...