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Ben Niehoff
Ben Niehoff is offline
Nov1-11, 09:27 PM
Sci Advisor
P: 1,562

The Gauss-Bonnet formula has nothing to do with ambient spaces. The Ricci scalar is an intrinsic curvature and the Euler characteristic is an intrinsic topological invariant. Therefore the Gauss-Bonnet formula is intrinsically true regardless of the ambient space.

The relevance to your previous business about embedding spheres in hyperbolic spaces in order to somehow put flat metrics on them was to explain that you are wrong. You cannot put a flat metric on a sphere no matter what the ambient space is, because the Gauss-Bonnet formula is a statement about facts intrinsic to the sphere itself. The best you can hope to do is make the sphere locally flat everywhere except at a finite number of points.

Also, the topology is more fundamental than the metric. This is an important thing for people coming from differential geometry to realize. A manifold is a topological object. It has topological features (such as handles, etc.) that exist independently of any local definition of "distance".

When we say a "sphere", we mean a closed 2-dimensional manifold of Euler characteristic 2. There is no need to make any reference to metrics. Then one can ask, what sorts of metrics can be put on a sphere? The Gauss-Bonnet formula gives the only constraint: any metric whose total Ricci curvature is [itex]8 \pi[/itex] (the Ricci scalar is exactly twice the Gauss curvature). Some parts of the manifold might have R = 0 or even R < 0, so long as the integral over the whole manifold is [itex]8 \pi[/itex].