### Math 439. Introduction To Differential Geometry - Fall 18 - Hans Lindblad

Syllabus Differential Geometry can be seen as continuation of Vector Calculus. Geometry is the part of mathematics concerned with questions of size, shape and position of objects in space. Differential geometry uses the methods of differential calculus to study the geometry. A curve in the plane is determined by its initial point and direction and the curvature at each point along the curve, that measures how fast the curve pulls away from the tangent line. Similarly a surface is determined by its curvatures in different directions at each point on the surface. The curvatures is the main focus of the course, these are essentially the coefficients in the second order terms in the Taylor expansion of a surface, or the second order derivatives of the parameterization. The theory of the curvatures was used in Einstein's theory of General Relativity to study how space-time curves under the influence of gravity.
Text: Do Carmo Differential Geometry of Curves and Surfaces
Other books: Oprea Differential Geometry and Its Applications, O'Neill Elementary Differential Geometry.
Thorpe Elementary Differential Geometry, Spivak A comprehensive introduction to Differential Geometry II-III
Exams/Grades: Exam 1 (15%), Exam 2 (15%), Exam 3 (15%), Final 12/12 at 9-12 am (30%), Homework (25%).