# Proof on inmersions involving the Laplacian

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.