\begin{definition}[Definition:Pseudo-Riemannian Submanifold]
Let $\struct {\tilde M, \tilde g}$ be a pseudo-Riemannian manifold.
Let $i : \tilde M \to M$ be a smooth submanifold, where $i$ is the inclusion mapping.
Suppose $i^* \tilde g$ is nondegenerate with constant signature.
Then $\struct {M, i^* \tilde g}$ is called the '''pseudo-Riemannian submanifold (of $\struct {\tilde M, \tilde g}$)'''.
\end{definition}