Pullback of Riemannian Metric by Smooth Mapping is Riemannian Metric iff Mapping is Immersion

Theorem
Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold with or without boundary.

Let $M$ be a smooth manifold with or without boundary.

Let $F : M \to \tilde M$ be a smooth mapping.

Let $F^* \tilde g$ be the pullback of $\tilde g$ by $F$.

Let $g = F^* \tilde g$ be a smooth $2$-tensor field.

Then $g$ is a Riemannian metric $F$ is an immersion.