# Two results on the mean curvature of hypersurfaces

I am a physicist, now I consider a physically meaningful $N-1$ dimensional hypersurface $M^{N-1}$ embedding in the flat Euclidean space $R^{N}$. We have an explicit form of the hypersurface in the following parametric form: $\mathbf{Y}(u)= (x_1(u), x_2(u), …, x_{N}(u))$, with $u$ denotes the $N-1$ local coordinates on the surface.

I verified that two formulas for 2D surfaces as $div \mathbf{N}=-H$ (Eq. (8.24) of P.224 of Ref 1.) and $\nabla^2 \mathbf{Y}=H\mathbf{N}$ (Eq. (11.31) of P.305 of Ref 1.) that in fact holds true for some simple surface such as spherical surfaces in any dimensions. I then guess that it in fact holds true for any hyperfuace rather than those I know their explicit forms of $\mathbf{Y}(u)$. I use symbols $div$, $\nabla^2$, $\mathbf{N}$, and $H$ to denote the surface divergence, the Laplace-Betrami (surface Laplacian as called in Ref 1), and normal vector, the mean curvature, respectively.

Ref 1, T. Frankel, The geometry of Physics, (Cambridge, 2004)

I have two questions:

1, Can the parametric form $\mathbf{Y}(u)$ of the surface be called the standard form in mathematical community? If yes, references wanted.

2, Can these two formulas $div \mathbf{N}=-H$, $\nabla^2 \mathbf{Y}=H\mathbf{N}$ really hold true for an arbitrary $N-1$ dimensional hypersurface $M^{N-1}$ embedding in the flat Euclidean space $R^{N}$? References wanted.

Thanks you for your help!

In the nutshell, 1) yes, one may call some parametric forms "standard" if needed; 2) these equations hold in any dimension and can be generalized for hypersurfaces in any Riemannian manifold, not necessarily flat.

These facts can be extracted form the Frankel's book that you cite, but I will also give some other references.

Some details follow.

A hypersurface is a submanifold of dimension $n-1$ in a manifold of dimension $n$. In other words, this is a submanifold of codimension 1. This is not a good definition though, because it may allow the hypersurface to be not well behaved, so some additional requirements are included. The complete discussion is rather delicate and is not appropriate here. My favorite reference is Chapter 1 of R.W. Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program (Google Books), but any other standard reference on differential geometry that has a chapter on submanifolds should treat this at a certain level.

The easiest way is to start with regular parametrized hypersurfaces, that is maps of the form $$F \colon U \to \Bbb R^n$$ where $U \subseteq \Bbb R^{n-1}$ is an open subset, and $F$ has the maximal rank: $\mathrm{rank} F = n-1$. This condition applied on the rank is call the regularity. (I use the notation of the book of T.Frankel referred to in the question, see p. 201). Again, this is not very satisfying, since we would like to visualize hypersurfaces as subsets of the space where they live, so it is better to speak about local parametrizations of the hypersurface in question. Indeed, there is an infinite number of parametrizations, and we can choose open subsets of the hypersurface to map onto a subset of $\Bbb R^{n-1}$. An arbitrary parametrization hardly could be termed "standard" for this reason. Sometimes, however, people may pick a particular parametrization and announce it to be "standard", but this is the matter of personal taste, or, maybe, requirements of the problem under consideration. For example, map $(x_1,\dots,x_{n-1}) \mapsto (x_1,\dots,x_n)$ can be thought as the standard parametrization of the plane $x_n = 0$.

I hope, this answers your question 1.

Now, regarding question 2, one needs to have the definitions handy, since there is a lot of various conventions in this area, so depending on a reference you may have the mean curvature being defined as $H = - \mathrm{div} \mathbf{N}$ (and then you can show that this is just the sum of the eigenvalues of the Weingarten operator). On the other hand, the proof on page 225 of Frankel's book can be adapted to any dimension.

The second identity easily follows from equation (8.30) on p.228 of Frankel's book, and the definition of the Laplace-Beltrami operator, eq-n (11.30) on p. 305. It is literally translated to the case of higher dimension.

As for the reference, I suggest to look at these lectures of Greg Galloway, as I find them very useful, even though they follow the tradition of working with surfaces in $\Bbb R^3$. Most of introductory text do that, including W.P.A.Klingenberg, A Course in Differential Geometry, where you can find a more detailed treatment.

Please take a look at the notes of Masato Kimura "Geometry of hypersurfaces and moving hypersurfaces in $\Bbb R^n$", and to these lectures from MIT Open CourseWare. They deal with arbitrary dimensions.