Link to first term: Differential Geometry III (first term by Pavel Tumarkin)
MATH3021
Lecturer: Olaf Post
Term: Epiphany 2014
Lectures:Literature
The following is a list of books on which the lecture is based. They are available in the library. Although we will not follow a book strictly, the material can be found in them and they may sometimes offer a different approach to the material.Content of Lectures
Date | Content (all lectures in one file, as far as they are already typed ...) |
Tuesday, 21 January 2014 (Week 11) | The derivative of a smooth map (from a surface into a vector space) |
Thursday, 23 January 2014 (Week 11) | Lectures 1--2: (local and global) isometries, conformal maps; example: cylinder and plane |
Tuesday, 28 January 2014 (Week 12) | Lecture 3: local isometries in coordinates; example: cylinder and plane; motivation of curvature on surfaces |
Thursday, 30 January 2014 (Week 12) | Lecture 4: The Geometry of the Gauss map: the Weingarten map and its symmetry, the second fundamental form, Gaussian curvature, mean curvature, principal curvatures and principal directions example: sphere |
Tuesday, 4 February 2014 (Week 13) | Lecture 5: local parametrisation: coefficients of the second fundamental form; calculation of Gaussian, mean and principal curvatures in a local parametrisation, example: hyperbolic parabolid; (linear algebra revision sheet) |
Thursday, 6 February 2014 (Week 13) | Lecture 6: (now hopefully without mistakes ...), principal coordinates: calculation of the curvatures in such a parametrisation; example: surface of revolution |
Tuesday, 11 February 2014 (Week 14) | Lecture 7: Gauss curvature: elliptic, hyperbolic and flat points and regions; umbilic and planar points; examples; some (global) theorems on curvature |
Thursday, 13 February 2014 (Week 14) | Lecture 8: Theorema Egregium of Gauss, Christophel symbols |
Tuesday, 18 February 2014 (Week 15) | Lecture 9: Proof of the theorema Egregium of Gauss, example of hyperbolic plane (example part of problem class) |
Thursday, 20 February 2014 (Week 15) | Lecture 10: More examples on how to calculate the Gaussian curvature from the first fundamental form (part of problem class) (there was a factor 2 missing in the formula of the Gauss curvature in Prop. 10.9, corrected in the lecture notes) |
Tuesday, 25 February 2014 (Week 16) | Lecture 11: Curves on a surface; coordinate curves geodesic and normal curvature; examples; |
Thursday, 27 February 2014 (Week 16) | Lecture 12: Theorem of Meusnier; local calculation of normal curvature, asymptotic curves |
Tuesday, 4 March 2014 (Week 17) | Lecture 13: Example of asymptotic curves, lines of curvature, its local calculation and examples |
Thursday, 6 March 2014 (Week 17) | Lecture 14: Lines of curvatures (continued), geodesics; constant speed property of geodesics; examples |
Tuesday, 11 March 2014 (Week 18) | Lecture 15: Main theorem about geodesics: intrinsic property of geodesics; local existence and uniqueness, |
Thursday, 13 March 2014 (Week 18) | Lecture 16: Invariance of geodesics under isometries, examples; Gauss-Bonnet theorem: some topology; Euler characteristic, For a proof of the Gauss-Bonnet theorem, see e.g. these notes of Grant Rotskoff from the University of Chigago |
Tuesday, 18 March 2014 (Week 19) | Lecture 17: Jordan's curve theorem; local and global Gauss-Bonnet theorem, examples |
Thursday, 20 March 2014 (Week 19) | Lecture 18: Problem class: Geodesics on surfaces of revolutions, examples for the Gauss-Bonnet theorem, non-existence of certain geodesics |
Homework | Date | Hand in | Solutions |
Problem Sheet 1 | 2014-01-28 | Problem 1.2 and 1.3 to hand in on 2014-02-06 (small typo: in Problem 1.4, def. of S_2 should be pi/3 < v < pi/2) | Solutions to Problem Sheet 1 (almost completed up to Problem 1.6 --- this is a quite long calculation, or one uses techniques from complex analysis ...) |
Problem Sheet 2 | 2014-02-04 | Problem 2.1 to hand in on 2014-02-20 (another one will be given next week) | Solutions to Problem Sheet 2 |
Problem Sheet 3 | 2014-02-13 | Problem 3.1 to hand in on 2014-02-20 | Solutions to Problem Sheet 3 |
Problem Sheet 3 (still) | 2014-02-20 | Problem 3.7 to hand in on 2014-03-06 | Solutions to Problem Sheet 3 |
Problem Sheet 4 | 2014-02-27 | Problem 4.4 about normal curvature to hand in on 2014-03-06 | Solutions to Problem Sheet 4 |
Problem Sheet 4 (still) | 2014-03-06 | Problem 4.9 about lines of curvature to hand in on 2014-03-18 (Tuesday!) | Solutions to Problem Sheet 4 |
Problem Sheet 5 | 2014-03-13 | Problem 5.15 about lines of curvature to hand in on 2014-03-18 (Tuesday!) | Solutions to Problem Sheet 5 (almost all ...the others are not that relevant ...) |
Last modified: 2014-04-07