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.”

Manifolds on Chen Kai Blog

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