Consider a parametrization of a hypersurface $M \subset \mathbb{R}^n$ given by $x: U \longrightarrow \mathbb{R}^n$. It is known that the Second Fundamental Form at a point $p=x(u)$ is given by $$ II_p(w)= w^T(h_{ij})w,$$
$$ h_{ij}=\langle N(u),x_{u_iu_j}(u)\rangle.$$
When we have a manifold $M \in \mathbb{R}^{n+r}$ of codimention $r$, the Second Fundamental Form is defined by $$ II_p: T_pM \longrightarrow N_pM$$ $$II_p(w(p))=B(w,w)(p),$$ for $w \in \mathcal{X}(U)$, that is $w $ is a vector field on $M$. Considering $\mathcal{N}(U)$ the set of normal vector fields, the map $B$ is given by $$B: \mathcal{X}(U)\times\mathcal{X}(U) \longrightarrow \mathcal{N}(U) $$ $$B(w_1,w_2)= {\nabla}_{\overline{w}_1}\overline{w}_2 - ({\nabla}_{\overline{w}_1}\overline{w}_2)^T $$
Here, the bars indicates that we are taking extensions of $w_1,w_2$ to $\mathbb{R}^{n+r}$, and the superscrip T indicates the tangent component of the vector field. My questions are: 1. How this extension mentioned is done ? 2. From the general definition, how can I get the definition for the case of codimention 1 ?