Curvature
Central Question
Which aspects of bending can be detected by someone who never leaves the surface?
Why This Matters
Curvature distinguishes geometric structure from topological shape and separates intrinsic measurements from the appearance of an embedding.
Learning Prompt
Motivate curvature by comparing a plane, a cylinder and a sphere. Ask what an inhabitant restricted to measuring distances on each surface could detect. Introduce a Riemannian metric as additional structure on a smooth manifold, distinguish Gaussian curvature from extrinsic bending, and explain the significance of Gauss's Theorema Egregium. Use small circles, geodesic triangles or parallel transport to build intuition without claiming they replace a proof. Make me predict which features survive bending without stretching, then connect the surface story to the idea of curvature in higher dimensions.