Definition:Normal Space/Manifold

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

Let $M \subseteq \tilde M$ be a smooth submanifold.

Let $p \in M$ be a point in $M$.

Let $T_p \tilde M$ and $T_p M$ be tangent spaces of $\tilde M$ and $M$ at $p$.

Let $v \in T_p \tilde M$, $w \in T_p M$ be tangent vectors.

Let $v$ be orthogonal to all $w$:

$\forall w \in T_p \tilde M : \innerprod v w_g = 0$

The set of all such $v$ at $p$ is called the normal space (of $M$ at $p$) and is denoted by $N_p M = \paren {T_p M}^\perp$