I'd like to have a good reference to understand how the Laplacian operator get generalized over differential manifolds.
More concretely, I want to understand and prove the equation :
$$\Delta Id_{\mathbf{X}}=H.\mathbf{N}$$
Where $\mathbf{X}$ is a smooth surface in $\mathbb{R}^3$, $Id_\mathbf{X}$ the identity function defined on $\mathbf{X}$, $H$ its mean curvature and $N$ the normal vector to the surface $\mathbf{X}$ (ie its Gaussian map).
I have read Andrew Pressley's book Elementary Differential Geometry.
I'm used to the classical Laplacian of differential calculus that operates on scalar fields, but I dont really understand what the Laplacian of a vector field represents, how it relates to the scalar version or how you define a Laplacian over manifolds.
I have not finished Pressley, but it does not seem to cover this topic.
But trying to look into it, I also saw lot of references to things like tensors, connection form, volum form, that I dont know.
If you have also good recommandation regarding those topics, I would be grateful.
Basically, I would like to understand how to do multivariable calculus over surfaces.
Thank you.
Best Answer
This is rather standard notation which admittedly could be a bit confusing if you've never seen it before. In this equation $\Delta X$ has nothing to do with differential forms, all you need to know is the definition of the Laplacian for a submanifold. FYI, I think that my sign convention is different than yours.
Write $X = (x_1,x_2,x_3)$ and the normal vector $N=(n_1,n_2,n_3)$. All this equation claims is that for $\Sigma^2\hookrightarrow \mathbb{R}^3$,
$$\Delta_\Sigma x_i = -H n_i$$
for $i=1,2,3$. Here, we just mean that $x_i$ are the Euclidean coordinates, restricted to the surface. In particular, a often used consequence of this equation is that $\Sigma$ is a minimal surface if and only if $\Delta_\Sigma x_i=0$ for $i=1,2,3$.
To prove this, you need to know about the second fundamental form, and how it relates the Levi-Civita connection on $\Sigma$ with the Euclidean connection. Remember that $$ (D^\Sigma)_X Y = (D^{\mathbb{R}^3})_X Y + II(X,Y) N $$ (I don't know exactly what notation you're used to, so feel free to ask for clarification). Also, everyone has different sign conventions here, so be careful.
Then, if $e_1,e_2$ is an orthonormal basis for $T_p\Sigma$, we can extend it to a neighborhood of $p$ in $\mathbb{R}^3$ so that it is parallel with respect to $D^{\mathbb{R}^3}$. Then, we compute: \begin{align*} \Delta_\Sigma x_i & = tr_{g_\Sigma}(D^\Sigma)^2 x_i\\ & = \sum_{j=1}^2 ((D^\Sigma)_{e_j}d x_i)(e_j) \\ & = \sum_{j=1}^2 ((D^\Sigma)_{e_j}(d x_i(e_j)) - dx_i((D^\Sigma)_{e_j}e_j)\\ & = \sum_{j=1}^2 ((D^{\mathbb{R}^3})_{e_j} dx_i)(e_j) - dx_i((D^{\mathbb{R}^3})_{e_j}e_j) - II(e_j,e_j)dx_i(N)\\ & =- Hn_i \end{align*}