\begin{definition}[Definition:Normal Space/Manifold]
Let $\struct {\tilde M, \tilde g}$ be a Riemannian manifold.
Let $M \subseteq \tilde M$ be a smooth submanifold with or without boundary in $\tilde M$.
Let $p \in M$ be a point in $M$.
Suppose $v$ is normal to $M$ at $p$.
The set of all such $v$ at $p$ is called the '''normal space (of $M$ at $p$)''' and is denoted by $N_p M = \paren {T_p M}^\perp$
\end{definition}