Differential Geometry (6): Smooth Manifolds — Geometry Beyond Embedded Surfaces

Thu, 11 Nov 2021 09:00:00 +0000

The first five chapters of this series lived inside $\mathbb{R}^3$ . We had curves and surfaces, parametrized explicitly, with all the geometric data — first and second fundamental forms, principal curvatures, Christoffel symbols, the Theorema Egregium, Gauss-Bonnet — built up from coordinates we could write down. The Theorema Egregium revealed that the intrinsic story can be told without reference to the embedding. But “without reference to the embedding” still meant “the embedding exists; we just choose not to use it.”

Differential Geometry (5): The Gauss-Bonnet Theorem — Where Geometry Meets Topology

Tue, 09 Nov 2021 09:00:00 +0000

The Theorema Egregium of the previous chapter showed that Gaussian curvature is intrinsic — bend a surface without stretching and $$K$$ does not change. The Gauss-Bonnet theorem, which we develop here, says something equally remarkable in a different direction: integrate $$K$$ over a closed surface and you get a topological invariant. The total curvature of any sphere is $4\pi$ . The total curvature of any torus is $$0$$ . The total curvature of a double torus is $-4\pi$ . These facts are blind to the specific geometry — bend, twist, or smoosh the surface as you please, the total curvature does not change.

Topology on Chen Kai Blog

Differential Geometry (6): Smooth Manifolds — Geometry Beyond Embedded Surfaces

Differential Geometry (5): The Gauss-Bonnet Theorem — Where Geometry Meets Topology