Definition:Metric Induced by Smooth Immersion

Definition
Let $\struct {\tilde M, \tilde g}$, $\struct {M, g}$ be Riemannian manifolds.

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

Suppose, the metric $g$ is such that $g = F^* \tilde g$.

Then $g$ is called the metric induced by $F$.