18.950 Differential Geometry (fall 2018)
Instructor: Nick Edelen
Class: Monday and Wednesday, 9:30-11, in 2-146.
Office hours (tentative): Monday 13-14, Tuesday 10-12.
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.
Textbook: Differential Geometry of Curves and Surfaces, do Carmo.
Grading: Midterm (40%), Final (40%), Homework (20%).
Midterm: October 17th, in class.
Final: December 20th, 9am-12pm, 2-146. Both midterm and final are closed book.
Homework: Weekly homework assignments, due Wednesdays in class. No lates accepted, but lowest two scores dropped. Collaboration is allowed, but you must write up your own solutions, and cite any collaborators.
- Homework 1 (due 9/12) solutions
- Homework 2 (due 9/19) solutions
- Homework 3 (due 9/26) solutions
- Homework 4 (due 10/3) solutions
- Homework 5 (due 10/10) solutions
- midterm solutions
- Homework 6 (due 10/24) solutions
- Homework 7 (due 10/31) solutions
- Homework 8 (due 11/7) solutions
- Homework 9 (due 11/21) solutions
- Homework 10 (due 11/28) solutions
- Homework 11 (due 12/10) solutions
Syllabus: A summary of what we've covered will appear here.
- Wed 9/5: curves, arc-length and arc-length parameter, curvature
- Mon 9/10: frenet frame, fundamental theorem of curves, isoperimetric inequality
- Wed 9/12: winding number, regular surface, inverse function theorem
- Mon 9/17: coordinate change, tangent space, differentials
- Wed 9/19: first fundamental form, instrinsic length, distance, isometry, area
- Mon 9/24: area, as depending only on the metric, examples, well-definition
- Wed 9/26: orientation, normal vectors
- Mon 10/1: second fundamental form, principle/Gaussian/mean curvature
- Wed 10/3: umbilic surfaces, computing curvatures in coordinates
- Wed 10/10: vector fields, Lie bracket, Frobenius's theorem
- Mon 10/15: minimal surfaces, conjugate surfaces
- Wed 10/17: midterm!
- Mon 10/22: isometries, conformal maps, Mercator and Riemann projections
- Wed 10/24: connections, Christoffel symbols
- Mon 10/29: Gauss and Codazzi equations, Gauss's theorema egregium, Bonnet's theorem
- Wed 10/31: rigidity of the sphere, parallel transport
- Mon 11/5: more parallel transport, K = 0 implies flat, geodesics
- Wed 11/7: geodesics on rotationally symmetric surfaces, Clairaut's relation
- Wed 11/14: geodesic curvature, parallel transport and Gauss curvature
- Mon 11/19: theorem of turning tangents, local Gauss-Bonnet
- Wed 11/21: global Gauss-Bonnet, classification of orientable surfaces
- Mon 11/26: exponential map, geodesic normal/polar coordinates, Minding's theorem
- Wed 11/28: geodesic balls, intrinsic distance, first variation of length
- Mon 12/3: geodesics locally minimize length, metric completeness
- Wed 12/5: geodesic completeness, Hopf-Rinow theorem
- Mon 12/10: abstract surfaces, hyperbolic plane (half-plane model)
- Wed 12/12: other models for hyperbolic plane, flat torus, classification of spaceforms