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Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define

$H^{\alpha}_{ij,k}=e_k\langle e_{\alpha},\nabla _{e_i} e_j\rangle $

Can anyone provide a proof that $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in ${\bf R}^n$?

PS: The $e_k$ vector is deriving the function.

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