Diff. Geometry·Course
Differential Geometry
Differential geometry: curves and surfaces, curvature, geodesics, tensor calculus, and manifolds
Part of the Mathematics track — finish it for a verifiable diploma →
§ 01 — Orientation
New here?
What you'll be able to do, who this is for, how long it takes, and where to begin.
By the end, you will be able to
- 01Study curves with curvature and torsion.
- 02Analyze surfaces and their curvature.
- 03Understand geodesics.
- 04Take the first steps into manifolds.
- 05Connect geometry to physics.
Who this is for
How long it takes
- Quick orientationSkim the opening module and the cheatsheet to get the shape of it.~2 h
- Full read-throughRead every article once, in order.~2 h
- Mastery pathRead, take the quizzes, and space out your reviews.~2 weeks
- APA
Stoa. (2026). Differential Geometry [Online course]. Stoa. https://stoa.school/course/diff-geometry
- MLA
Stoa. “Differential Geometry.” Stoa, 2026, https://stoa.school/course/diff-geometry.
- Chicago
Stoa. “Differential Geometry.” Stoa. Accessed September 1, 2026. https://stoa.school/course/diff-geometry.
§ 02 — Curriculum
6 modules.
Each module is a small unit. Most read in sequence — but a determined reader can begin anywhere.
- M ITheory of CurvesFrenet frame, curvature and torsion, natural equations of a curve3 articles
18 minBegin → - M IITheory of SurfacesFirst and second fundamental forms, Gaussian and mean curvature3 articles
18 minBegin → - M IIISmooth ManifoldsCharts, tangent space, vector fields3 articles
18 minBegin → - M IVDifferential Forms and Stokes’ TheoremWedge product, exterior derivative, generalized Stokes’ theorem3 articles
18 minBegin → - M VTopological SpacesTopological spaces, connectedness, compactness3 articles
18 minBegin → - M VIFundamental Group and CoveringsLoops, fundamental group, covering space theory3 articles
18 minBegin →
§ 03 — Learning outcomes
4 outcomes.
Compute curvature and torsion of curves and understand the Frenet–Serret formulas
Study the first and second fundamental forms and Gaussian curvature
Determine geodesics on surfaces and understand parallel transport
Understand the basics of differentiable manifolds and tensor analysis
§ 04 — Practices