I have two questions about tensor calculation.
First question : In the book, Lectures on mean curvature flows written by Xi-Ping Zhu, there exists the equaility $g^{mn} \nabla_m \nabla_n h_{ij} = g^{mn} \nabla_m \nabla_i h_{jn}$. I do not understand this.
The situation is as follows : $X(\cdot, t) : M^n \rightarrow {\bf R}^{n+1}$ is a one-parameter family of smooth hypersurface immersions in ${\bf R}^{n+1}$, and $ X_t = H \nu$ where $H$ and $\nu$ is the mean curvature and unit normal to $X$. $g_{ij} = (X_i,X_j)$, $h_{ij} = (\nu, X_{ij})$
The question is found in the proof of Lemma 2.3 in 19 page. Please help me.
Second question : In the same book, there exists the equality $\Delta h_{ij} -\epsilon \Delta H g_{ij} = \Delta( h_{ij} - \epsilon H g_{ij}) $ (See the proof of Proposition 2.6 in 22 page)
I cannot understand the equality. Please help me.