How curves bend and surfaces curve, measured from the inside: geodesics, parallel transport and Gauss–Bonnet, on the way to general relativity.
Same Curve, Different Speeds Draw a Curve from Its Curvature Osculating Circle and Evolute Huygens' Cycloidal Pendulum Frenet Frame on a Helix Build a Space Curve from κ and τ Turning Tangents and the Rotation Index The Four-Vertex Theorem The Isoperimetric Inequality Principal Curvatures and the Sign of K The Gauss Map The Metric and Map Distortion Möbius Strip and Orientability Bend a Flat Sheet Soap Film Between Two Rings Catenoid to Helicoid Ruled Surfaces Geodesics and Clairaut's Relation Shortest Path by Unrolling Conjugate Points on the Sphere Parallel Transport and Holonomy Holonomy on a Flat Cone Circles in Three Geometries Triangle Angle Excess Gauss–Bonnet on a Torus Euler Characteristic and Genus Flat 2-Manifolds: the Wrap-Around World The Poincaré Disc An Atlas of the Sphere Integral Curves of a Vector Field When Two Flows Fail to Commute Parallel Parking Is a Lie Bracket Lines on a Torus Index of a Vector Field and the Hairy Ball The Hopf Fibration Line Integral of a 1-Form The Angle Form and Winding Christoffel Symbols in Polar Coordinates Geodesic Deviation