Differential geometry of curves and surfaces

Builds on

Required prerequisites, all the way down: 37

You will learn to

• Compute curvature and torsion using the Frenet–Serret frame.
• Compute the first and second fundamental forms and the Gaussian curvature of standard surfaces.
• State the Theorema Egregium and the Gauss–Bonnet theorem and check them on spheres and tori.

Definition

• The curvature of a unit-speed curve γ is κ = |γ″|.
• The Gaussian curvature K of a surface at a point is the product of its principal curvatures.
• A geodesic on a surface is a curve whose acceleration is everywhere normal to the surface.

Theorem

• Theorema Egregium: Gaussian curvature depends only on the first fundamental form, so it is intrinsic.
• Gauss–Bonnet: for a compact oriented surface S without boundary, ∫_S K dA = 2πχ(S).
• No map from an open region of a sphere to the plane preserves all distances, because their Gaussian curvatures differ.

Common misconceptions

• Thinking a cylinder is curved in the Gaussian sense; its Gaussian curvature is zero.
• Confusing the curvature of a curve with the Gaussian curvature of a surface.

Optional discoveries

• Smooth manifolds The Theorema Egregium motivates intrinsic geometry, which is studied on abstract manifolds with no surrounding space.
• ← Euclidean geometry and proof

Learning pathways

• Undergraduate mathematics major

Further reading

• Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces (1976)