M365G Curves and Surfaces

Prerequisite and degree relevance: Credit with a grade of at least C- or registration for Mathematics 365C. Students should know multivariable calculus and a little linear algebra.

Course Description: Calculus applied to curves and surfaces in three dimensions: graphs and level sets, tangent spaces, vector fields, surfaces, orientation, the Gauss map, geodesics, parallel transport, the second fundamental form and the Weingarten map, length and curvature of plane curves, curvature of surfaces, the exponential map, and the Gauss--Bonnet Theorem.

Textbook:  Elementary Topics in Differential Geometry, John A. Thorpe, Springer--Verlag, New York, 1979. (ISBN 0-387-90357-7)

INTRODUCTION: Differential geometry is a rich and active area of research in pure mathematics. It also provides powerful tools for disciplines like general relativity and other branches of mathematical physics as well as for applications in engineering and computer graphics. This course will introduce the basic language and methods of differential geometry by studying the geometry of n-dimensional hypersurfaces in (n+1)-dimensional Euclidean space. The course would be excellent preparation for graduate courses in Differential Topology or Riemannian Geometry, as well as for further study in applications like those listed above.