Differential Geometry

Standard differential geometry topics from the canonical curriculum.

Core Topics

Arc Length & Reparametrization — Measure curves by arc length so geometry is intrinsic and unit-speed.
Curvature of a Plane Curve — Signed curvature: turning rate of the unit tangent.
Torsion & the Frenet-Serret Formulas — The TNB frame, curvature, torsion, and the Frenet equations.
Regular Surfaces & Tangent Planes — Parametrizations, the regularity condition, and the tangent plane.
First Fundamental Form — The metric E, F, G measuring length, angle, and area on a surface.
The Gauss Map & Second Fundamental Form — The Gauss map, the shape operator dN, and the form e, f, g.
Principal Curvatures & Normal Curvature — Euler's formula, shape-operator eigenvalues, and point classification.
Gaussian & Mean Curvature — Product and average of the principal curvatures, and the sign of K.
Geodesics — Zero geodesic curvature: the surface's straight lines and shortest paths.
Covariant Derivative & Parallel Transport — Differentiate vector fields on a surface; transport them along curves.
Theorema Egregium — Gaussian curvature is intrinsic, so isometries preserve it.
Gauss-Bonnet Theorem — Total curvature equals 2π times the Euler characteristic.
Minimal Surfaces — Zero mean curvature: soap films, the catenoid, and the helicoid.