Differential Geometry

The mathematical language of smooth spaces.

Part I · The Smooth World

Smooth Manifolds Topological spaces, charts, atlases, smooth maps, and what it means for a space to have local coordinates

The Tangent Bundle Tangent vectors as derivations, pushforward and pullback, vector fields, and the flows they generate

Differential Forms and Integration The cotangent bundle, exterior algebra, the exterior derivative, and Stokes' theorem

Part II · Riemannian Geometry

The Riemannian Metric Lengths, angles, geodesics, the exponential map, and what it means to measure on a curved space

Connections and Parallel Transport The Levi-Civita connection, covariant derivatives, parallel transport, and holonomy

Curvature The Riemann tensor, sectional and Ricci curvature, and the Gauss-Bonnet theorem

Part III · Structure, Symmetry, and Applications

Lie Groups and Symmetric Spaces Lie groups, Lie algebras, the exponential map revisited, and spaces with maximal symmetry

Fiber Bundles Principal bundles, associated bundles, connections, and the geometric unification of structure

Symplectic Geometry Symplectic manifolds, Hamiltonian vector fields, moment maps, and the geometry of mechanics

Information Geometry Statistical manifolds, the Fisher-Rao metric, natural gradient, and the geometry of generative models