MATH 423: Differential Geometry

Abstract The course is the study of curves using multi-variable calculus and the study of surfaces using exterior calculus. The course is essentially the study of the concept of curvature. Students are also expected to improve ability in writing mathematical arguments.

Topics to be covered by the course:

THE GEOMETRY OF CURVES

• Basic notions of the theory of curves: regular curves, tangent lines, arc length, parameterization by arc length • Plane curves: signed curvature, Frenet frame • Space curves: curvature and torsion, Frenet frames, local approximation, spherical image, canonical form of a curve up to isometry, parallel curves, congruence of curves • (Hopf's Umlaufsatz) • (The Four-vertex theorem) • (Total curvature)

EXTERIOR CALCULUS

• 1-from and differential of a function • Tensors in a vector space: alternating tensors • Wedge products • Differential k-forms • Exterior derivatives • Covariant derivatives • Regular maps

CLASSICAL SURFACE THEORY

• Regular surfaces • The tangent plane • Differential forms on a surface and exterior derivative • Inverse Function Theorem for a surface • Pullback of differential forms • Stokes’ Theorem • The Gauss map • The first fundamental form • Normal fields and orientation of surfaces • The second fundamental form • The third fundamental form • Asymptotic directions • Curvature: principal curvature, Gaussian and mean curvatures • Examples: ruled surfaces, surfaces of revolution, minimal surfaces • Geodesic • (Surface area and integration on surfaces)

INTRINSIC SURFACE THEORY

• Connection forms • The first and second structural equations • The structural equations of a surface • The symmetry equations of a surface • The Gauss equations of a surface • The Godazzi equations of a surface • Isometries • Local isometries • Conformal mapping • Covariant derivatives of adapted frame field • Gauss's Theorema Egregium • (The fundamental theorem of surfaces (Bonnet's theorem)) • (Parallel transport) • (Geodesics and the exponential map) • (The Euler-Lagrange equation) • (The Gauss-Bonet theorem and applications (Poincare index theorem))