From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: newbie question on differential geometry
Date: 02 Oct 1999 21:30:40 GMT
Newsgroups: sci.math
Keywords: sectional curvature

In article <7svq9o$ca$1@nnrp1.deja.com>, rge11@my-deja.com writes:
|Now I can see that a sphere, whose is K =
|1/R^2 is constant everywhere, cannot be bent at all, but what happens
|if I cut a slice out of it. Then it seems that the spherical cap should
|be bendable, am I right?. But then how can K stay constant?

Well, in any case we can consider a surface which can be changed in
shape isometrically. Gauss found it remarkable that the value for his
notion of curvature remained the same, so it is reasonable for you to
find it a surprise.

The Gaussian curvature is the product of the two sectional curvatures
of the surface. If you take a surface which initially has both of the
sectional curvatures 1/R and isometrically bend it so that one
sectional curvature increases, then the other sectional curvature
decreases in such a way as to keep the product of the two the same.

For comparison, think about what happens when you roll up a flat
piece of paper. The Gaussian curvature remains 0, because although
there is now a nonzero sectional curvature (which you can make
whatever you like, in principle) the other sectional curvature is
still 0.

Keith Ramsay