Let $M$ be a compact manifold with boundary $\partial M$. Suppose that we are given a hyper surface $A$ which is transversal to $\partial M$.
Is there a way to construct a metric $g$ on $M$ such that for any $x \in A \cap \partial M$, we have $$ T_x \partial M^{\perp_g} \subset T_x A ? $$ In other words, can we force the orthogonal of the boundary to lie in a prescribed hypersurface ?
Thanks a lot in advance for your help.