Differential Geometry
From curves and surfaces to manifolds, connections, and Gauss-Bonnet.
01Differential Geometry (1): Curves in Space — Curvature, Torsion, and the Frenet Frame
Parametrized curves, arc length, curvature, torsion, and the Frenet-Serret apparatus — the complete local theory of …
02Differential Geometry (2): Surfaces and the First Fundamental Form: Intrinsic Measurements
Regular surfaces, coordinate patches, the tangent plane, and the first fundamental form — how to measure lengths, …
03Differential Geometry (3): The Shape Operator — Curvature of Surfaces
The Gauss map and shape operator capture how a surface bends in space — principal, Gaussian, and mean curvatures …
04Differential Geometry (4): Intrinsic Geometry — Theorema Egregium and Geodesics
Gauss's Theorema Egregium reveals that Gaussian curvature depends only on the first form — geodesics are the 'straight …
05Differential Geometry (5): The Gauss-Bonnet Theorem — Where Geometry Meets Topology
The Gauss-Bonnet theorem connects total Gaussian curvature to the Euler characteristic — a stunning bridge between local …
06Differential Geometry (6): Smooth Manifolds — Geometry Beyond Embedded Surfaces
Manifolds free geometry from ambient space — charts, atlases, and smooth structure let us do calculus on spaces that …
07Differential Geometry (7): Vector Fields, Flows, and the Lie Bracket
Vector fields generate flows — one-parameter families of diffeomorphisms. The Lie bracket measures the failure of flows …
08Differential Geometry (8): Differential Forms — The Natural Language of Integration on Manifolds
Differential forms unify gradient, curl, and divergence into a single framework — the exterior derivative d and wedge …
09Differential Geometry (9): Integration on Manifolds and Stokes' Theorem
Stokes' theorem — the fundamental theorem of calculus on manifolds — unifies Green's, Gauss's, and the classical Stokes' …
10Differential Geometry (10): Riemannian Geometry — Metrics, Connections, and Parallel Transport
A Riemannian metric lets us measure lengths, angles, and volumes on any smooth manifold — the Levi-Civita connection …
11Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar
The Riemann curvature tensor captures all intrinsic curvature information — its contractions (Ricci and scalar …
12Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics
Vector bundles generalize the tangent bundle, connections on bundles generalize Levi-Civita, and characteristic classes …