🚀 go-pugleaf

The Four-Vertex Theorem states that the curvature function on a smooth 
simple closed curve in the plane has at least 4 critical points.

Now consider a closed hypersurface M^n smoothly embedded in R^(n+1). 

Let "Gaussian curvature" K denote the determinant of the Jacobian of the
Gauss map G: M -> S^n (obtained by setting G(x) =  the outward unit normal 
to M at x).

QUESTION:
	Is there some generalization of the Four-Vertex Theorem for the 
	Gaussian curvature of such hypersurfaces in R^(n+1) ?

	Intuitively, if M were a non-degenerate ellipsoid, it would apparently
	have 2n critical points of its Gaussian curvature K: M -> R.  So, is 
	there evidence for or against the possibility that

		The Gaussian curvature K: M -> R of an 
		arbitrary smooth hypersurface 
		M^n in R^(n+1) must have at least 2n critical points.
	?

	Possible cases to consider:	a)	M is convex 
					b)	M is topologically S^n 
					c)	M is arbitrary.


Dan Asimov
Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799

P.S.  To avoid questions of differentiability I have assumed that everything
in sight is smooth, although a finite number of derivatives would almost
certainly suffice.