Well if $\Sigma$ is a submanifold of $R^{n+p}$ and $\{e_i,e_\alpha\}$ is orthonormal frame over $\Sigma$ where the $e_i$'s are tangent and the $e_\alpha$'s are normal to $\Sigma$.
Can anyone prove (with an adequate frame) that $\nabla_{e_i}^{\perp} e_\alpha=0$?
Obs: The result is pretty easy when we have only one normal direction, but in this case there are more.