The tangential gradient of a function $f$ defined on a manifold

Question

I'm self study Riemannian Geometry to be able to understand this lecture notes about Mean Curvature Flow. The first chapter is a review of Riemannian Geometry and I'm stuck in the following part:

"Using the isomorphism induced by the metric $g$ we can regard $\nabla f$ also as element of the tangent space, in this case it is called the gradient of $f$. The gradient of $f$ can be identified with a vector in $R^{n+1}$ via the differential $dF$; such a vector is called the tangential gradient of $f$ and is denoted by $\nabla^M f$, given in coordinates by"

$$\nabla^M f = \nabla^i f \frac{\partial F}{\partial x_i} = g^{ij} \frac{\partial f}{\partial x_j} \frac{\partial F}{\partial x_i}$$

The word ”tangential” comes from the equivalent definition of $\nabla^M f$ in case $f$ is a function defined on the ambient space $R^{n+1}$. It can be checked that $\nabla^M f$ is the projection of the standard Euclidean gradient $DF$ onto the tangent space of $M$, that is,

$$\nabla^M f = Df - \langle Df, \nu \rangle_e \nu,$$

where $\nu$ is a local choice of unit normal to $M$.

A relevant consideration done in the beginning of the chapter is

We restrict ourselves to manifolds of codimension 1 in an Euclidean ambient space, i.e. we consider a n-dimensional smooth manifold $M$, without boundary, either closed or complete and non-compact and an immersion (or embedding)

$$F: M \longrightarrow \mathbb{R}^{n+1}$$

[...]We denote by $g_{ij} := \langle \frac{\partial F}{\partial x_i}. \frac{\partial F}{\partial x_j} \rangle_e$ the induced metric on $M$, where $\langle \cdot, \cdot \rangle_e$ is the Euclidean scalar product on $\mathbb{R}^{n+1}$.

My doubt is what I need to check exactly? I need to show that $g^{ij} \frac{\partial f}{\partial x_j} \frac{\partial F}{\partial x_i} = Df - \langle Df, \nu \rangle_e$ or I need prove that the $\textit{gradient}$ of $f$ (which is different of the definition of $\textit{tangential gradient}$ of $f$) is equal to $Df - \langle Df, \nu \rangle_e$ identifying $\textit{gradient}$ of $f$ with the $\textit{tangential gradient}$ of $f$? In the first case, I think the argument is just as it is in this topic, but I don't know how to prooced if the case is the second.

Thanks in advance!

$\textbf{P.S.:}$

Other consideration that maybe be relevant: my background are Linear Algebra, Metric Spaces, Analysis on $\mathbb{R^n}$ and Differential Geometry.

Answer 1

The only thing you need to "check" (essentially) is that $$D \widetilde{f} \circ DF=df, $$ which is trivial by the chain rule, where I am denoting by $\widetilde{f}$ the function as defined on $\mathbb{R}^{n+1}$ (so that $\widetilde{f} \circ F=f$). $^{(1)}$

Now the rest is linear algebra (which may also be considered something to "check"), since given $X_p \in T_pM$, \begin{align*} df(X_p)=D\widetilde{f}( DF(X_p))&=\langle \nabla \widetilde{f}, DF(X_p\rangle) \\ &=\langle (\nabla \widetilde{f})^{\perp}+(\nabla \widetilde{f})^{\top}, DF(X_p)\rangle \\ &=\langle (\nabla \widetilde{f})^{\top}, DF( X_p) \rangle \\ &=\langle DF^{-1}(\nabla \widetilde{f}^{\top}), X_p \rangle, \end{align*} so the gradient of $f$ at some $p \in M$ is just the tangential gradient vector of $\widetilde{f}$ (under the identification of $DF$ $^{(2)}$). I am assuming you are giving $M$ the pull-back metric, of course.

$^{(1)}$Just an observation: There may be some identifications here, depending on definitions. If you take $Df$ to mean the usual definition as given in a standard course in analysis, then you must identify $T_p\mathbb{R}^n$ and $\mathbb{R}^n$ in the usual way. This is also the difference between $df: T_pM \to \mathbb{R}$ or $df: T_pM \to T_p\mathbb{R}$, which you allude to in the comments.

$^{(2)}$This is the identification he mentions at

The gradient of $f$ can be identified with a vector in $R^{n+1}$ via the differential $dF$.