I'm reading "Finite type submanifolds in pseudo-euclidean spaces and applications", by B.Y. Chen. On Lemma $4$ I have a couple of doubts on the proof, the Lemma says:

Let $\psi\,\colon M\rightarrow\mathbb{E}^{m+1}_s$ be an n-dimensional submanifold of $\mathbb{E}^{m+1}_s$. Then we have: \begin{align} \Delta\psi=-nH \end{align} where $H$ is the mean curvature vector field.

For the proof, an arbitrary vector $a\in\mathbb{E}^{m+1}_s$ is chosen. Then the laplacian of $g(\psi,a)$ is computed, where $\psi$ is understood as a vector valued function. Letting $f=g(\psi,v)\colon p\in M\rightarrow f(p)=g(\psi(p),v)\in\mathbb{R}$ and taking a local orthonormal referential $\{E_i\}_{i=1}^n$ around $p$ with signature $\{\varepsilon_i\}$ verifying $\nabla_{E_i}E_j=0$ (I'm not sure of why we can assure that such referential exists), we have: \begin{align} \Delta(g(\psi,v))_p=-div(\nabla f)=-\sum_{i=1}^n\varepsilon_ig(\nabla_{E_i}(\nabla f),E_i) \end{align} where $\nabla$ is the Levi civita connection of $M$. Using that $\nabla$ is metric, is obvious that: \begin{align} (E_i)_p(g(\nabla f,E_i))=g(\nabla_{E_i}(\nabla f),E_i)+g(\nabla f,\nabla_{E_i}E_i)=g(\nabla_{E_i}(\nabla f),E_i) \end{align} So from this I can deduce the following: \begin{align} \Delta(g(\psi,v))_p=-\sum_{i=1}^n\varepsilon_i(E_i)_p(g(\nabla f,E_i))=-\sum_{i=1}^n\varepsilon_i(E_i)_p(df(E_i))=-\sum_{i=1}^n\varepsilon_i(E_i)_p(E_i(g(\psi,v))) \end{align}

In the proof of the article the following equation is written with no explanation: \begin{align} \Delta(g(\psi,v))_p=-\sum_{i=1}^n\varepsilon_i(E_i)_p(g(E_i,v)) \end{align} Can someone help me to get to this equation?

Also, I don't undertand the meaning of $\Delta\psi$ in this article. Is it the vector field $(\Delta(\psi_1),\dots,\Delta(\psi_{m+1}))\in\mathfrak{X}(\mathbb{E}_s^{m+1})$?. Also, how is $\Delta(g(\phi,v))$ related to $g(\Delta\psi,v)$?. I know that these are a lot of questions, but thank you in advance.



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