Abstract

Learn Differential Geometry

Calculus on curved spaces: manifolds, curvature, geodesics, and the bridge between local geometry and global topology.

Free to start · adaptive placement finds your level · reviews timed so it stays learned.

What you'll learn

28 lessons in Differential Geometry

Smooth manifolds & chartsTangent & cotangent spacesVector fields & Lie bracketsDifferential forms on manifoldsRiemannian metricsConnections & covariant derivativesCurvatureGeodesicsGauss–Bonnet theoremLie groups & Lie algebrasThe hyperbolic planeGeometric flows (Ricci flow preview)First & second fundamental formsGaussian & mean curvature; Theorema EgregiumParallel transport & holonomyThe exponential map & normal coordinatesSectional, Ricci & scalar curvatureStokes' theorem on manifoldsThe Riemann curvature tensor & Bianchi identitiesThe Hopf–Rinow theoremJacobi fields & conjugate pointsComparison theoremsPrincipal bundles & connectionsChern–Weil theoryde Rham cohomology & Hodge theorySymplectic geometryComplex & Kähler manifoldsThe Atiyah–Singer index theorem
How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the moment just before you'd forget it. A short placement check finds what you already know, so you start Differential Geometry exactly where it's useful.

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