Math 473: Introduction to Differential Geometry

Theory of curves in space R^3. Regular curves and reparametrization, Serret-Fernet apparatus, existence and uniqueness theorem for space curves. Local theory of surfaces: Simple surfaces, coordinate transformations, tangent vectors and tangent spaces, first and second fundamental forms, normal and geodesic curvatures, Weingarten map, principal, Gaussian and mean curvatures. Geodesics, equations of Gauss and Codazzi-Mainardi.