Harmonic Functions on Connected Riemannian Manifold with Matching Restrictions to Boundary are Identical

Theorem
Let $\struct {M, g}$ be a connected Riemannian manifold with the boundary $\partial M \ne \empty$.

Let $u, v \in \map {\CC^\infty} M$ be smooth harmonic functions on $M$.

Suppose restrictions of $u$ and $v$ to $\partial M$ agree.

Then $u = v$ identically.