Math 561 — Differential Geometry

(with special relativity)

Instructor:	Sigurd B. Angenent Office: Van Vleck 617.
Course description:	Our main subject is the differential geometry of curves and surfaces in 3-dimensional Euclidean space. We will also consider the geometry of Minkowski space instead of Euclidean space, and its relevance to special relativity.
Lecture Times and Location:	9:30-10:45am, Tuesdays and Thursdays, in B235 Van Vleck Hall
Textbook:	Elementary Differential Geometry, by Andrew Pressley. Springer Undergraduate Mathematics Series. This will be our main textbook. In addition I will also treat material from The Geometry of Spacetime, by James Callahan, Springer Undergraduate Texts in Mathematics. While I will discuss the relevance of Differential Geometry to Special (and perhaps General) Relativity, I do not intend to follow this second book very closely, or assign problems from it.
Midterm, Final Exams and Grading Policy:	There will be two midterm exams and one final exam. Homeworks will count for 15% of the grade, each midterm will count for 25%, and the final will count for 35%. The midterm exams will be held during a regular lecture. See the schedule below.
Homework assignments:	Homework will be assigned on each thursday, and will be due on the following tuesday. Late homework will not be graded. Your two lowest homework scores will be dropped at the end of the semester.

List of topics

Different kinds of geometry: Euclidean space, Galilean spacetime, and Minkowski spacetime.
Curves in the plane and in R³: Curvature, torsion, Frenet formulas, global properties
Curves in Minkowski space, “proper time,” interpretation of curvature.
Surfaces in R³: examples (quadrics, ruled surfaces)
The first fundamental form aka “the metric of the surface.”
Isometries, the notion of intrinsic Differential Geometry.
The second fundamental form aka “the extrinsic curvature of the surface”
Geodesics: geodesic curvature, length minimizing property.
an aside: Calculus of Variations (Euler–Lagrange equations)
Geodesics in General Relativity
Gauss’ Theorema Egregium
The Gauss-Bonnet Theorem

Reading & homework assignments

Date	Topic, Reading and Homework assignment
Sept 2	What is geometry? Review of matrix representation of Euclidean motions. The idea of Invariants.
Sept 4	Invariants in Euclidean geometry. Invariants of pairs of points, of triangles. The osculating circle.
Sept 9	Parametrized curves, arclength, 1st & 2nd chapters of Pressley
Sept 11	Derivative wrt arclength, curvature vector & radius of curvature Homework: 1.10, 1.14, 1.15, and 2.1 ii&iv (problem continues on next page!)
Sept 16	Normal, Binormal, Frenet formulas.
Sept 18	The geometry of Galilean space-time
Sept 23	Invariants of world lines in the Galilean plane. Failure of Maxwell's equations to be Galilean invariant.
Sept 25	Derivation of Lorentz tranformations. Homework due next thursday was handed out in lecture: Click here for the pdf file.
Sept 30
Oct 2
Oct 7
Oct 9
Oct 14
Oct 16
Oct 21	FIRST MIDTERM EXAM Pictures of the curve in problem 2 and and the surface in problem 3. The solution to problem 2 in PDF format.
Oct 23	The Inverse Function Theorem; Surfaces are locally graphs of functions.
Oct 28	Consequences of the Inverse Function Theorem. Definition of the metric, aka, first fundamental form.
Oct 30	Examples of the 1st fundamental form; Definition of Isometry Homework due thursday November 6th: 4.19, 4.21, 5.1 and 5.3; In addition, show that the subset of R³ defined by the equation x⁴+x+y⁴+z⁴=1 is a surface (use theorem 4.1, read example 4.7)
Nov 4	Example of an isometry, from the catenoid to the helicoid. A formula for the angle between two curves on a surface. Definition & property of Conformal mapping. For isometries read §5.2. The isometry from catenoid to helicoid is exercise 5.8.
Nov 6	Area of a portion of a surface. Read: §5.4 for area. For conformal mappings read §5.3
Nov 11	The Second Fundamental Form. Curvature of curves on a surface. Normal and geodesic curvature of a curve on a surface. Normal curvature and the 2nd fundamental form. The second fundamental form is defined in §6.1. Its relation with the Curvature of curves on a surface is in §6.2.
Nov 13	The second fundamental form of the graph of a function, in particular near points with a horizontal tangent. The Principal Curvatures of a surface–EULER's theorem. Homework due thursday November 20: [click here] Note: I will explain how to compute pricipal curvatures,etc in the next lecture. So until then do the first problem.
Nov 18	Examples of computation: sphere, cylinder, surface of rotation. The Gaussian and Mean Curvatures. The transformation formula is exercise 5.4.
Nov 20	The GAUSS map. The WEINGARTEN map (or matrix). Geometric interpretation of the principal curvatures.
Nov 25	SECOND MIDTERM EXAM A list of practice problems for the midterm in PDF
Nov 27	Thanksgiving
Dec 2	The first variation formula for the length of a curve; Shortest Curves have vanishing Geodesic Curvature.
Dec 4	No class. However, to prepare for next week's lecture you should do these problems. Click here for brief solutions.
Dec 9	Geodesics on a surface. The Gauss-Bonnet theorem.
Dec 11	Last class: 1+1D General Relativity