Let $M$ be a submanifold of the riemannian manifold $\overline{M}$, with the induced metric. Denote by $\nabla$ and $\overline{\nabla}$ the riemannian connections of $M$ and $\overline{M}$, respectively. Given a point $p \in M$ and a unit normal vector $\eta \in T_p M ^{\perp}$, the second fundamental form of $M$ at $p$ in the direction on $\eta$ is the linear map $S_\eta : T_p M \to T_p M$ given by $$ S_\eta(v) = \left( \overline{\nabla}_v N \right)^T, \quad v \in T_p M, $$ where $N$ is a local extension of $\eta$ normal to $M$ and $x^T$ denotes the orthogonal projection of $x$ onto $T_p M$.
My questions:
$1$. Why can we always extend a unit normal vector to a local normal vector field?
$2$. Can we extend a unit normal vector to a local unit normal vector field?
If $2$. is true and $N$ is a unit normal extension of $\eta$, then
$$ S_\eta(v) = \overline{\nabla}_v N, \quad v \in T_p M $$
right?
Thanks in advance!