Differential-Geometry

Nov 23, 2021 Differential Geometry 72 min read

Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics

Vector bundles generalize the tangent bundle, connections on bundles generalize Levi-Civita, and characteristic classes are topological invariants — this is the geometry underlying gauge theory and general relativity.

Nov 21, 2021 Differential Geometry 70 min read

Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar

The Riemann curvature tensor captures all intrinsic curvature information — its contractions (Ricci and scalar curvature) control volume growth, geodesic deviation, and Einstein's equations.

Nov 19, 2021 Differential Geometry 72 min read

Differential 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 provides the canonical notion of parallel transport and geodesics.

Nov 17, 2021 Differential Geometry 62 min read

Differential 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' theorems into one elegant statement.

Nov 15, 2021 Differential Geometry 64 min read

Differential 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 product turn calculus coordinate-free.

Nov 13, 2021 Differential Geometry 64 min read

Differential 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 to commute, leading to Frobenius integrability.

Nov 11, 2021 Differential Geometry 62 min read

Differential 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 don't live in R^n.

Nov 9, 2021 Differential Geometry 62 min read

Differential 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 differential geometry and global topology.

Nov 7, 2021 Differential Geometry 68 min read

Differential 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 lines' of curved surfaces, minimizing arc length locally.

Nov 5, 2021 Differential Geometry 64 min read

Differential 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 classify every point as elliptic, hyperbolic, or parabolic.

Nov 3, 2021 Differential Geometry 66 min read

Differential 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, angles, and areas on a surface without leaving it.

Nov 1, 2021 Differential Geometry 64 min read

Differential 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 space curves.