Surface Curvature Summary

Differential Geometry · Axiom Academy

Reviewing the fundamental concepts of how surfaces bend and curve in space. Shape Operator & Second Fundamental Form Shape Operator (Weingarten Map): A linear map S: T p M → T p M that measures how the normal vector changes as we move along the surface Formula: S(v) = -dN p (v) where N is the unit normal field Second Fundamental Form: II(v,w) = ⟨S(v), w⟩ quantifies how the surface deviates from its tangent plane Why It Matters: These encode all extrinsic curvature information about how a surface sits in 3D space Principal Curvatures & Directions Principal Curvatures: The eigenvalues κ 1 and κ 2 of the shape operator, representing maximum and minimum normal curvatures Principal Directions: The corresponding eigenvectors, which are always orthogonal on a surface Normal Curvature Formula: κ n (v) = II(v,v) / I(v,v) for any direction v Physical Meaning: Principal directions show where the surface bends most and least Definition: K = κ 1 κ 2 (product of principal curvatures) Intrinsic Property: Can be computed using only distances measured on the surface (Theorema Egregium) K > 0: Elliptic points (bowl-like, e.g., sphere) K = 0: Parabolic points (flat in at least one direction, e.g., cylinder) K Hyperbolic points (saddle-like, e.g., hyperboloid) Definition: H = (κ 1 + κ 2 )/2 (average of principal curvatures) Extrinsic Property: Depends on how the surface is embedded in space H = 0: Minimal surfaces (soap films, catenoids)

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