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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.

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    Nobody can prove what is not true in general. Compare this with the [Frenet-Serret formulas](http://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas).2012-11-23
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    The title of the question is quite misleading. The only Levi-Civita connection here is the standard connection in $\Bbb R^n$ whereas $\nabla^{\perp}$ is not the Levi-Civita connection, but a connection in the normal bundle (which is in this case, of course, a part of the ambient L-C connection)2012-11-23

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