Math 423. Differential Geometry
Instructor Syllabus

Topics to be covered:

THE GEOMETRY OF CURVES

  • Basic notions of the theory of curves: regular curves, tangent lines, arc length, parameterization by arc length
  • Plane curves: curvature, Frenet frame
  • Space curves: curvature and torsion, Frenet frames, canonical from of a curve up to rigid motion
  • Hopf's Umlaufsatz
  • The Four-vertex theorem
  • Total curvature

CLASSICAL SURFACE THEORY

  • Regular surfaces
  • The tangent plane
  • The first fundamental form
  • Normal fields and orientation of surfaces
  • The Gauss map
  • The second fundamental form
  • Curvature: principal curvature, Gaussian and mean curvatures
  • Surface area and integration on surfaces
  • Examples: ruled surfaces, surfaces of revolution, minimal surfaces

INTRINSIC SURFACE THEORY

  • Isometries
  • The fundamental theorem of surfaces (Bonnet's theorem)
  • Covariant derivatives
  • Gauss's Theorema Egregium
  • Parallel transport
  • Geodesics and the exponential map
  • The Euler-Lagrange equation
  • The Gauss-Bonet theorem and applications (Poincare index theorem)

Texts used previously:

  • Christian Bär, Elementary Differential Geometry
  • Bartlett O'Neill, Elementary Differential Geometry, Second Edition
  • Wolfgang Kühnel, Differential Geometry: Curves - Surfaces - Manifolds, Second Edition

 

Approved by UAC 4/17/13.