# A question about laplacian of the second fundamental form

Let $f:M \rightarrow N$ be an immersed oriented hypersurface, $e_{1}, \ldots e_{n},e_{n+1}$ be an orthonormal frame of $N$ such that $e_{1} \ldots e_{n}$ is an orthonormal frame of $M$. Let $h_{ij}$, $i,j=1 \ldots n$ be the coefficients of its second fundamental form. In detail $h_{ij}=h(e_{i},e_{j})=\langle B(e_{i},e_{j}),e_{n+1} \rangle$.

In the article 'Estimates for minimal hypersurfaces' of Schoen Simon and Yau, at (1.16), it asserts that $\Delta h_{ij} = \sum_k h_{ijkk}$ where $h_{ijkk}= \nabla_{e_{k}}(\nabla h)(e_{i},e_{j},e_{k})$ where $\nabla h$ is the covariant derivative in $M$ of the symmetric tensor $h$. Why does this equality hold?

Added: in the same article there is another assertion of the same type:

$|\nabla (h_{ij})|^2=\sum_{k}h_{ijk}^2$ where $h_{ijk}=(\nabla_{e_{k}}h)(e_{i},e_{j})$

In both of this assertions it seems that the authors do not consider some terms: for example we note that

$h_{ijk}= e_{k}(h_{ij})-h(\nabla_{e_{k}}e_{i},e_{j})-h(\nabla_{e_{k}}e_{j},e_{i})$ while

$|\nabla (h_{ij})|^2=\sum_{k}(e_{k}(h_{ij}))^2$

So it seems that they assert: $h_{ijk}=e_{k}(h_{ij})$ . But this fact is false in a general context. An analougous assertion it seems to hold for the first equality in this post.

Thanks

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If we choose a geodesic frame centered at $p$ and we see all of these equality at $p$, the problem is solved. Moreover since we need to work with an orthonormal frame $e_1, \ldots e_n$ such that at $p$ is a base of eigenvectors of the second fundamental form (see (1.23)), we construct a geodesic frame centered at p, starting from a base of eigenvectors $E_1, \ldots E_n \in T_p(M)$ of the second fundamental form.