# 18.950 - Differential Geometry (Fall 2016)

Instructor: Heather Macbeth

Lectures: MWF 1 - 2 pm, 2-131

Office Hours: W 2 - 4 pm, 2-246

## COURSE DESCRIPTION

Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space.

Prerequisites: 18.100; 18.06 or equivalent; 18.02 or equivalent

Textbook: do Carmo, Differential Geometry of Curves and Surfaces (optional)

## RECENT UPDATE

 [12.19.2016] Grades for the final are posted on Stellar. Mean 75, median 81, max 93. Well done all, and thanks for a great semester! [12.18.2016] In response to some questions about the final: Individual ruled surfaces could appear on the final. No question concerning ruled surfaces as a class (directrices, lines of striction, the formulas involving the "distribution parameter" λ, etc) will appear on the final. The Gauss-Bonnet theorem, including the formulation involving the degree of the Gauss map, could appear on the final. [12.14.2016] Special instructions for final: * bring one one-sided page of notes; * first ten mins (9:00-9:10) are reading only (no writing). Extra office hours before the final: 3-4 pm Fri 16 Dec, 2-4 pm Sun 18 Dec. Tangent surfaces of some curves: helix (image, Wolfram CDF Player interactive demonstration), twisted cubic (image, Java plugin interactive demonstration), rational curve of degree four (image, Java plugin interactive demonstration), unspecified curve (video). [12.12.2016] The Wikipedia article on ruled surfaces has some beautiful pictures, including real-life rulings of the helicoid and the one-sheeted hyperboloid. [12.09.2016] Solutions to problem set 11 are now available. Check out this animation of a homotopy between a coffee cup and a doughnut. [12.04.2016] Solutions to problem set 10 are now available. Check out this animation of Enneper's surface. [12.03.2016] Problem set 11 is now available. [11.28.2016] Some notes regarding problem 4 of problem set 10 are now available. You may cite (no need to copy out the calculations) any part of them in your own problem set. [11.23.2016] Problem set 10 is now available. [11.18.2016] Solutions to problem set 9 are now available. [11.14.2016] Solutions to problem set 8 are now available. The two parametrizations of the sphere constructed in problem 3(i) are (a) Mercator's projection and (b) the Lambert cylindrical equal-area projection. The transformation h_1 in problem 4 turned the Poincare half-plane model representation of the hyperbolic plane (our usual representation) into a new representation, the Poincare disk model. [11.11.2016] Problem set 9 is now available. [11.07.2016] Visualization of the surface from problem set 8, problem 2: image and code. [11.06.2016] Problem set 8 and solutions to problem set 7 are now available. [11.04.2016] Visualizations of hyperbolic geometry: animations of the families of isometries (i) fixing the point (0,1), (ii) fixing the y-axis; correspondence with the tractroid. [10.31.2016] Solutions to problem set 6 are now available. [10.28.2016] Problem set 7 is now available. [10.22.2016] Problem set 6 is now available. [10.21.2016] Solutions to the midterm are now available, and grades are posted on Stellar. Mean 52, median 55, max 71. It will be curved. [10.13.2016] Solutions to problem set 5 are now available. [10.12.2016] Developing the helicoid onto the catenoid: animation 1, animation 2, animation 3. [10.10.2016] Try out this interactive demonstration of the geodesics of the Schwarzschild metric. (Your computer must have the Wolfram CDF player installed.) [10.08.2016] Problem set 5 and solutions to problem set 4 are now available. [10.03.2016] Problem set 4 is now available. [09.30.2016] Solutions to problem set 3 are now available. [09.28.2016] Try out these interactive demonstrations of geodesics on surfaces of revolution: one-sheeted hyperboloid, torus. (Your computer must have the Wolfram CDF player installed.) [09.22.2016] Problem set 3 and solutions to problem set 2 are now available. [09.21.2016] The schedule of final exams has been announced. The 18.950 exam will be held on Monday 19 December, 9 am - 12 pm. [09.19.2016] Try out this interactive demonstration of parallel transport on the sphere. (Your browser must have the Java plugin.) [09.16.2016] Problem set 2 is now available. Wikipedia is a reasonable reference for the formula for the Christoffel symbols in terms of the first fundamental form (missing from do Carmo). Note that a sum over the index m (m=1,2,... n) is implied. [09.15.2016] Solutions to problem set 1 are now available. [09.09.2016] Problem set 1 is now available. [09.07.2016] Please vote on office hours. As mentioned in class, please refer to do Carmo, Appendix to Ch 2, "A Brief Review of Continuity and Differentiability'" (pp 118-133) for a review of foundational material in real analysis (18.100, etc). [08.01.2016] Welcome to the fall semester!