Big geometric theme: local to global
local-to-global:
- sum exterior angles -> winding number
- Descartes' theorem on total angular defect -> Gauss-Bonnet
- Chern-Weil characterization of characteristic classes
(of which Gauss-Bonnet (curvature -> Euler class) is the simplest EG)
- Atiyah-Singer index theorem
positive v. restrictive; negative not v. restrictive
- sphere theorems
- hyperbolic is generic geometrizable
- positive Ricci restrictive; negative Ricci no restriction at all, in higher dim
EG of Kodaira dimension:
low Kodaira dimension = positive curvature = very restrictive;
high Kodaira dimension = negative curvature = not restrictive
both:
- specific computations of global/topological data from local/geometric
(Gauss-Bonnet)
- restrictions on topology, given info about geometry
(sphere theorems)