Let $f \colon M^{n} \rightarrow \overline{M}^{n+m}$ be an immersion. Identifying $TpM$ with $Im(df_{p})$, we have the following decomposition
$$T_{p}\overline{M} = T_{p}M \oplus (T_{p}M)^{\perp}$$
Consider $X, Y$ two tangent vector fields in $M$ and $\overline{X}, \overline{Y}$ some extensions to $\overline{M}$, define
$$B(X,Y) = \overline{\nabla}_{\overline{X}} \overline{Y} - (\overline{\nabla}_{\overline{X}} \overline{Y})^{T}.$$
Here's my doubt: Do Carmo's book affirms that $B(X,Y)p$ only depends on $X_p$ and $Y_p$. Apparently this comes immediately from the bilinearity and symmetry of $B$. I can't understand this.
My thoughts: In general, we know that $(\nabla_X Y)p$ only depends on the value of $X$ and $Y$ in $p$ and $Y$ along a curve which passes through $p$ with velocity $X_p$. It seems the fact that $B(X,Y)$ lies on $(T_pM)^{\perp}$ "kills" the dependence of $Y$ along a curve passing through $p$ with velocity $X_p$ and it only remains the dependent with $X_p$ and $Y_p$.
Can anyone help me?