Firstly, I would like to say that I didn't do a course in Riemannian geometry, so my doubt is probably something that is not clear for me from the basis of the Riemannian geometry, which I'm self-studying.

I'm reading Interior estimates for hypersurfaces moving by mean curvature and I'm trying understand the statement:

"Now note that $\nabla u = \omega - \langle \nu , \omega \rangle$ and $\nabla (|\textbf{x}|^2 - u^2) = 2(\textbf{x} - \langle \textbf{x} , \nu \rangle \nu - u \nabla u)$",

where $M$ is an hypersurface of dimension $n$ which evolves under Mean Curvature Flow by the immersion $F: M \times [0,T) \longrightarrow \mathbb{R}^{n+1}$, $\textbf{x} = F(p,t)$, $u := \langle \textbf{x} , \omega \rangle$, $\nu$ is the normal unit vector and $\omega$ is a fixed vector on $\mathbb{R}^{n+1}$ such that $\langle \omega, \nu \rangle > 0$.

First, I think the authors wanted to write $\nabla u = \omega - \langle \nu, \omega \rangle \nu$, i.e., $\nabla u$ it's the projection of $\omega$ on tangent space of $M$.

I tried compute $\nabla u$ and I found $\nabla u = \langle \nabla \textbf{x}, \omega \rangle$, which is the length of the projection of $\omega$ on tangent space of $M$. I did this computation by the fact that $u$ is a function defined on $M$, by the compatibility of the tensor metric, but I found some problems because my computation of $\nabla u$ doesn't match with the computation of $\nabla u$ did by the authors and the other problem is that the compatibility of the tensor metric it's relationed with the derivation of a function along to a vector field by relation

$$\nabla_X \langle Y,Z \rangle = X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y, \nabla_X Z \rangle,$$

but I don't have explicitly along what vector field $u$ is being differentiated. I think this vector field defined on $M$ is arbitrary and this explain why the direction was omitted in the differentiation of $u$. I had these same problems when I tried compute $\nabla |\textbf{x}|^2$ by an analagous argument.

I would like to know how compute $\nabla u$ and $\nabla |\textbf{x}|^2$. Thanks in advance!


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