MATH 423: Differential Geometry
of Curves and Surfaces
Fall 2007

10:30 - 11:50 a.m., TuTh
445 Altgeld Hall

Instructor:
Stephanie Alexander, Professor
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street (MC-382)
Urbana, Illinois 61801-2975

Office: 322 Illini Hall
Phone: (217) 244-7339
FAX:  (217) 333-9576
e-mail: sba@math.uiuc.edu

This course covers the fundamental theorems that motivate and inspire differential geometry. The techniques are those of vector calculus.

Space curves are studied via the Frenet-Serret moving frames, which give a clear geometric interpretation of curvature and torsion.

Surfaces are defined using patch maps (i.e., as 2-dimensional manifolds). At each point on a surface, the rate of change of the unit normal to the surface may be measured by two principal curvatures. We relate these principal curvatures to the behavior of curves on the surface. The Gaussian curvature function is defined as the product of the principal curvatures. We prove Gauss's famous Theorema Egregium, which says that inhabitants of the surface can calculate this curvature from distance measurements carried out entirely within the surface, even though they cannot calculate the principal curvatures individually. Finally, we introduce covariant derivatives and consider the Gauss-Bonnet Theorem, which relates the integral of the Gaussian curvature over a closed surface to the topological type of the surface.

Prerequisites: Vector Calculus.

Text: Barrett O'Neill, Elementary Differential Geometry.