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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.


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up vote 1 down vote accepted

Since this question has some vote i post a solution that i found to my question.

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.

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