Introduction to Differential Geometry

MTH

324

Introduction to Differential Geometry

This course is designed to provide students in the Mathematical Sciences Program with an introduction to the classical (local) differential geometry of curves and surfaces in R3 using vector methods. The concepts of arc length, curvature, torsion along with the fundamental systems of basic unit vectors and the associated lines and planes will be discussed. The Serret-Frenet formulas and their application and the moving trihedron will be investigated in detail. The representation problem in terms of the natural parameter (arc lengths) and the general theory of smooth space (twisted or gauche) curves will be emphasized, as will the representation problem and elementary theory of smooth surfaces embedded in Euclidean space. The First and Second Fundamental Forms will be presented and the various curves on embedded surfaces (such as lines of curvature, asymptotic lines, and directions) will be discussed, as will Meusnier's theorem, Euler's theorem and the Dupin indicatrix. Elementary principles and methods of the tensor calculus will be introduced as a means of investigating the Fundamental Theorem of Surface Theory, the Gauss-Weingarten equations, and the

Fall, Spring

Undergraduate

No

024339