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## Hypersurfaces and Elliptic Points

I'm reading a paper, in which we have $M^n$ an n-dimensional compact hypersurface embedded in $\mathbb{R}^{n+1}$. We take the scalar cuvature $R$ to be the elementary symmetric polynomial of degree 2 in the principal curvatures of $M$. We know that $R$ is constant.

The author then says "As $M$ has one elliptic point, $R$ is a positive constant and the mean curvature is positive somewhere".

I'm lost here - Why does $M$ have an elliptic point? And how does this affect $R$ and the mean curvature?

Thanks for any help.

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A generic coordinate function on M has a single maximum. My geometric intuition says that such a maximum is an elliptic point, but I don't actually know any Riemannian geometry. – Qiaochu Yuan Jan 19 2011 at 22:58
I believe this is explained, at least for a compact surface in $R^3$, in any basic differential geometry textbook or at least those that talk about surfaces in $R^3$. I suggest you look there first. Or ask on math.stackexchange.com. Or there's probably someone a lot nicer than a grumpy old differential geometer like me who's willing to tell you the answer. – Deane Yang Jan 19 2011 at 23:02
@Qiaochu: it is easier to consider the maximum point(s) on $M$ of the smooth function $f(x) = |x|^2$, I think. Then you don't have to worry about genericity and actually get local convexity. – Willie Wong Jan 20 2011 at 0:11
Willie, that's right. I just noticed that your answer is the same as Igor's below with "far-away" removed. – Deane Yang Jan 20 2011 at 4:21

Elliptic point is, by definition, a point where all the principal curvatures are positive, hence $R$ is positive. A point of maximal distance from some far-away basepoint is elliptic.

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Igor, you're much kinder person than me. – Deane Yang Jan 20 2011 at 3:43
Igor, why does the base point have to be far away? Doesn't the argument work for any base point? – Deane Yang Jan 20 2011 at 3:45
Yes, of course, where there is a sphere, there is a way. For some reason I always think of the basepoint as being somewhere far. An amusing fact, rediscovered several times (the last in a Rutgers PhD thesis about five years ago) is that this does not work for PL surfaces (more precisely, there are compact embedded polyhedral surfaces in $R^3$ whose curvature (in the PL/cone sense) is nowhere positive. – Igor Rivin Jan 20 2011 at 4:15
Igor, that's amazing. Is there a picture or description somewhere? Also, can you name names? – Deane Yang Jan 20 2011 at 4:20
I can name names, but I have to remember them first... Stay tuned... – Igor Rivin Jan 20 2011 at 4:55
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