Recently, Hon Leung and Zorn Leung are teaching differential geometry to some bright secondary school students. So I am affected by them to make my revision on geometry of curves and surfaces in the Euclidean space. In this post, we will study some integral formula for closed hypersurface in Euclidean space. Perhaps the most famous example is the Gauss-Bonnet theorem, which state that for a closed (compact without boundary) surface in ,
where is the Gaussian curvature and is the Euler’s characteristic. We will see that by applying divergence theorem in different settings, we can derive many integral formulas (e.g. Gauss-Bonnet and Wongting’s first and second theorem).
In this post is always an orientable hypersurface in (e.g. Mobius band is not allowed), is a unit normal vector field of , chosen to be “outward” whenever this makes sense (e.g. closed, i.e. compact without boundary). is the position vector of . is the second fundamental form. always denote the mean curvature of w.r.t. , i.e. . Alternatively for an othonormal frame . In our convention, the mean curvature of sphere of radius in w.r.t. outward normal is .
- We will write as the covariant derivative on ; or as partial derivatives. Then for function , but
- A very simple but important observation which we will always use is that for any tangent vector of ,
when we regard as a vector rather than functions. (You may say: no big deal! What’s the difference? This problem becomes subtle when we come to the second derivative of . See the next point. )
- In general, when regarding as a vector-valued function, and regarding as a tangent vector field, then all the following three terms are different:
In particular, given the fact that (the tangential part), it follows that is normal to , whereas is tangential. (1) can be seen as follows. For tangent vector fields ,
Put , the result follows. Using , we also have
Proposition 1 For a closed hypersurface , if is the (intrinsic) Laplacian on , when regarding the position vector as a -valued function (i.e. “independent” functions), then
Proof: Let be a coordinates on , then using as in the first remark, we have
By divergence theorem,
Theorem 3 (Hon Leung’s theorem, 2011) For a closed hypersurface ,
where is the -dimensional area (measure) of .
By noting that if we change the coordinates for any fixed , and and are unchanged, i.e. independent of translation (naturally), thus we can slightly generalize the above theorem to
Theorem 4 For a closed hypersurface , for any ,
From this we can also see that for any , so this in turn implies corollary 2:
Proposition 5 (Codazzi equations) Under any local coordinates , we have
Note that by symmetry of , this means that the 3 indices of can indeed be interchanged in any order. Note: .
Proof: (There is a more intrinsic way of proving it in the more general Riemannian setting, but here I choose to do it in a more clumsy but explicit way. ) By (1), we have
where are the Christoffel symbols:
As , by (4)
Thus with a computation similar to (1), we have
Thus by interchanging and in above, noting that and ,
Note that is independent of our choice of , so it is a well-defined notion regardless of whether is orientable or not.
Remark 3 For , where is the Gaussian curvature.
where is the second fundamental form and (!) For a local coordiantes which is orthonormal at , is defined by
As a corollary, we have
Theorem 8 (Hon Leung’s theorem, 2011) For a closed hypersurface ,
Theorem 9 (Hon Leung’s theorem, 2011) For a closed hypersurface , ,
Example 1 For example when , this becomes
Let us verify this for the sphere of radius . We have , , so . .
First by applying divergence theorem two times, and using Proposition 7: (note that , so it is valid (though a bit strange in notations) to talk about , also we omit for notational convenience)
Consider the first term above, by divergence theorem, (here the always denote the Euclidean dot product, rather than the intrinsic inner product on )
Put this back in the first computation, we have
Noting that , we conclude that
By noting that and are invariant under the change of coordinates , we have the slight generalization
Theorem 10 For a closed hypersurface , , for any ,
In particular, as is arbitrary, we have
Corollary 11 (Hon Leung’s theorem, 2011) For a closed hypersurface , ,