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.