Definition:Induced Metric on Submanifold

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

Let $M : M \subseteq \tilde M$ be a submanifold immersed in $\tilde M$.

Let $i : M \to \tilde M$ be the inclusion mapping.

Let $g$ be a Riemannian metric such that $g = i^* \tilde g$.

Then $g$ is called the induced metric on $M$.