\begin{definition}[Definition:Induced Metric on Submanifold]
Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold with or without a boundary.
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 $M$'''.
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\end{definition}