13.2 Curvature and convexity

Curvature measures how surfaces bend, while convexity describes their shape. These concepts are closely linked in hypersurfaces. Positive Gaussian curvature often indicates local convexity, while negative curvature forms saddle points.

Strictly convex hypersurfaces have positive Gaussian curvature, but the reverse isn't always true. Mean curvature also relates to convexity, with positive values suggesting local convexity. These ideas have applications in surface classification, optimization, and real-world design.

Curvature and Convexity of Hypersurfaces

Curvature and convexity connection

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• Curvature measures quantify surface bending
  • Gaussian curvature product of principal curvatures
  • Mean curvature average of principal curvatures
• Convexity properties describe surface shape
  • Strict convexity no line intersects surface more than twice
  • Weak convexity allows flat regions
• Relationship between curvature and convexity links geometric properties
  • Positive Gaussian curvature indicates local convexity (spheres)
  • Negative Gaussian curvature forms saddle points (hyperbolic paraboloid)
• Second fundamental form determines local surface shape through quadratic approximation
• Principal curvatures maximum and minimum bending directions influence overall curvature

Positive curvature of convex hypersurfaces

• Strictly convex hypersurface lies entirely on one side of any tangent hyperplane
• Gaussian curvature formula $K = \kappa_1 \kappa_2$ product of principal curvatures
• Proof outline demonstrates positive Gaussian curvature:
  1. Assume strict convexity
  2. Show principal curvatures have same sign
  3. Prove both principal curvatures positive
  4. Conclude Gaussian curvature positive
• Counterexamples non-strictly convex surfaces (cylinders)
• Implications for global properties shape restrictions and topological constraints

Mean curvature vs convexity

• Mean curvature formula $H = \frac{1}{2}(\kappa_1 + \kappa_2)$ average of principal curvatures
• Convexity conditions based on mean curvature relate to surface shape
  • Positive mean curvature suggests local convexity (ellipsoid)
  • Zero mean curvature indicates minimal surfaces (catenoid)
• Mean curvature vs Gaussian curvature different geometric information
• Hypersurfaces with varying mean curvature (torus)
• Mean curvature and shape operator eigenvalues of second fundamental form

Applications of curvature and convexity

• Hypersurface classification based on curvature (elliptic, parabolic, hyperbolic)
• Local shape determination using curvature information (surface normals)
• Global hypersurface property analysis
  • Compact hypersurfaces closed and bounded (sphere)
  • Complete hypersurfaces no boundary points (plane)
• Curvature flow problems surface evolution over time
  • Mean curvature flow surface area minimization
  • Gaussian curvature flow shape preservation
• Hypersurface optimization problems
  • Extremal point finding (critical points)
  • Constrained optimization using curvature (Lagrange multipliers)
• Differential geometry applications
  • Geodesics shortest paths on curved surfaces
  • Gauss-Bonnet theorem relates curvature to topology
• Real-world applications of curvature and convexity
  • Computer graphics 3D modeling and rendering
  • Architectural design curved structures and tensile surfaces