Math 433, Introduction to Differential Geometry, Spring 2019

Instructor: Jun Li MWF 11:00am -12:00, 4088 East Hall

Office Hour: MWF 12:00am -1pm, or by appointment, 4839 East Hall.

Syllabus: W19_433_syllabus. More information on Canvas webpage.

Main reference: [Do Carmo] Differential geometry of curves and surfaces.
[Hitchin] Hitchin_geometry_of_surfaces

Date Topic Reading assignment and HW/Exam

Jan 9-19 Overview, vector valued functions and their differentiations.
3 Methods of describing a curve, Fixed coordinates, Moving frames, Intrinsic way. HW 1

Jan 20-31, Curves in R^n: Arclength Parametrization, curvature, Frenet-Serret equations.
Taylor expansion of a curve, Fundamental Theorem of the local theory of curves

Feb 1-8, Plane curves, Global theory: Isoperimetric Inequality, The Four Vertex Theorem HW2

Feb 9-15, Surfaces in R^3,Definitions and Examples, Compact surfaces, Level sets

Feb 16-22 The First Fundamental Form. Length, Angle, Area
maps preserve Length: Isometry; Angle: conformal; Area: equiareal. HW 3,
Review of Exam 1, Exam 1 in class
Feb 23-28 The Second Fundamental Form, Normal vector fields, Gauss map
Curvatures, Definitions and first properties.
Hitchin's notes introduction to Riemannian curvature tensor

Mar 2-10 Winter break.

Mar 11-15 Calculation of Gaussian, mean curvatures and principal curvatures HW 4,

Mar 16-22 Gauss's Theorema Egregium, Surfaces of constant Gaussian curvature. HW 4 Solution.

Mar 23-29 Parallel transport and covariant derivative. HW 5,
HW 5 Solution.

Apr 1-5 Geodesics, shortest paths, first variation, Geodesics on surfaces of revolutions Exam 2 and solution.

Apr 6-12 Half plane model of hyperbolic plane , Gauss-Bonnet Theorem
Geodesic polygons, Global Gauss-Bonnet, intro to Riemann surfaces. HW 6,
HW 6 Solution.

Apr 13-19 Vector fields and Euler number, Euler-Poincare theorem. Intro to differential form and tensor(non-examinable) Sample final, Final.

Date	Topic	Reading assignment and HW/Exam
Jan 9-19	Overview, vector valued functions and their differentiations. 3 Methods of describing a curve, Fixed coordinates, Moving frames, Intrinsic way.	HW 1
Jan 20-31,	Curves in R^n: Arclength Parametrization, curvature, Frenet-Serret equations. Taylor expansion of a curve, Fundamental Theorem of the local theory of curves
Feb 1-8,	Plane curves, Global theory: Isoperimetric Inequality, The Four Vertex Theorem	HW2
Feb 9-15,	Surfaces in R^3,Definitions and Examples, Compact surfaces, Level sets
Feb 16-22	The First Fundamental Form. Length, Angle, Area maps preserve Length: Isometry; Angle: conformal; Area: equiareal.	HW 3, Review of Exam 1, Exam 1 in class
Feb 23-28	The Second Fundamental Form, Normal vector fields, Gauss map Curvatures, Definitions and first properties. Hitchin's notes introduction to Riemannian curvature tensor
Mar 2-10	Winter break.
Mar 11-15	Calculation of Gaussian, mean curvatures and principal curvatures	HW 4,
Mar 16-22	Gauss's Theorema Egregium, Surfaces of constant Gaussian curvature.	HW 4 Solution.
Mar 23-29	Parallel transport and covariant derivative.	HW 5, HW 5 Solution.
Apr 1-5	Geodesics, shortest paths, first variation, Geodesics on surfaces of revolutions	Exam 2 and solution.
Apr 6-12	Half plane model of hyperbolic plane , Gauss-Bonnet Theorem Geodesic polygons, Global Gauss-Bonnet, intro to Riemann surfaces.	HW 6, HW 6 Solution.
Apr 13-19	Vector fields and Euler number, Euler-Poincare theorem. Intro to differential form and tensor(non-examinable)	Sample final, Final.