MAT 342E - Differantial Geometry

Course Objectives

1. To give the student essentials of curves and surfaces in 3-dimension,
2. To reinforce their advanced calculus and linear algebra knowledge and to give an ability to apply this knowledge on geometry problems

Course Description

Differential geometry of curves; osculating plane, curvature and torsion, Frenet’s trihedral, Frenet’s formulas, osculating sphere. Helices, involutes and evolutes, Bertrand’s curves. Natural equations, the fundamental theorem of the theory of curves. Differential geometry of surfaces; first and second fundamental forms of a surface, normal curvature, the lines of curvature, Dupin indicatrix, Meunier’s theorem, mean curvature, Gaussian curvature. Gauss’s Theorema Egregium, Mainardi-Codazzi and Weingarten equations. Asymptotic lines, geodesic lines. Euler’s, Ossian-Bonnet’s and Liouville’s formulas. The fundamental theorem of the theory of surfaces.