# 18.950 Differential Geometry (fall 2018)

Instructor: Nick Edelen

Email: edelen@mit.edu

Office: 2-181

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.

Assignments:

1. Homework 1 (due 9/12) solutions
2. Homework 2 (due 9/19) solutions
3. Homework 3 (due 9/26) solutions
4. Homework 4 (due 10/3) solutions
5. Homework 5 (due 10/10) solutions
6. midterm solutions
7. Homework 6 (due 10/24) solutions
8. Homework 7 (due 10/31) solutions
9. Homework 8 (due 11/7) solutions
10. Homework 9 (due 11/21) solutions
11. Homework 10 (due 11/28) solutions
12. Homework 11 (due 12/10) solutions

Syllabus: A summary of what we've covered will appear here.

1. Wed 9/5: curves, arc-length and arc-length parameter, curvature
2. Mon 9/10: frenet frame, fundamental theorem of curves, isoperimetric inequality
3. Wed 9/12: winding number, regular surface, inverse function theorem
4. Mon 9/17: coordinate change, tangent space, differentials
5. Wed 9/19: first fundamental form, instrinsic length, distance, isometry, area
6. Mon 9/24: area, as depending only on the metric, examples, well-definition
7. Wed 9/26: orientation, normal vectors
8. Mon 10/1: second fundamental form, principle/Gaussian/mean curvature
9. Wed 10/3: umbilic surfaces, computing curvatures in coordinates
10. Wed 10/10: vector fields, Lie bracket, Frobenius's theorem
11. Mon 10/15: minimal surfaces, conjugate surfaces
12. Wed 10/17: midterm!
13. Mon 10/22: isometries, conformal maps, Mercator and Riemann projections
14. Wed 10/24: connections, Christoffel symbols
15. Mon 10/29: Gauss and Codazzi equations, Gauss's theorema egregium, Bonnet's theorem
16. Wed 10/31: rigidity of the sphere, parallel transport
17. Mon 11/5: more parallel transport, K = 0 implies flat, geodesics
18. Wed 11/7: geodesics on rotationally symmetric surfaces, Clairaut's relation
19. Wed 11/14: geodesic curvature, parallel transport and Gauss curvature
20. Mon 11/19: theorem of turning tangents, local Gauss-Bonnet
21. Wed 11/21: global Gauss-Bonnet, classification of orientable surfaces
22. Mon 11/26: exponential map, geodesic normal/polar coordinates, Minding's theorem
23. Wed 11/28: geodesic balls, intrinsic distance, first variation of length
24. Mon 12/3: geodesics locally minimize length, metric completeness
25. Wed 12/5: geodesic completeness, Hopf-Rinow theorem
26. Mon 12/10: abstract surfaces, hyperbolic plane (half-plane model)
27. Wed 12/12: other models for hyperbolic plane, flat torus, classification of spaceforms