## Differentiable manifolds 1. Instructor: Roland van der Veen (snellius 234) Assistant: Kevin van Helden. Email us at manifolds17@gmail.com

Lectures: Monday 13:30-15:15, Snellius 402. Problem session: Thursday 15:30-17:15, Snellius 401.

Course description. In this course we sharpen and combine our tools from linear algebra and calculus to address geometric questions such as: What is the shortest path from A to B in a curved space? What do we mean by ‘curved’ anyway? Why does a soap bubble curve like it does? How can we understand differential equations qualitatively? And what does the graph of a holomorphic function look like in R^4?
A natural setting for discussing such questions are spaces that locally look like R^n, these are called manifolds. We will clean up the the vector calculus of R^n and lift it to the context of manifolds using differential forms. This includes a vast generalization of the fundamental theorem of calculus: the general Stokes theorem.
At the end of the course we will be able to understand the famous Gauss-Bonnet theorem:
The integral of the curvature of a surface equals the total index of any vector field on the surface and this in turn equals the Euler characteristic. This statement illustrates the beautiful interaction of geometry, differential equations and topology that manifolds are all about.

Lectures and examination. Weekly lectures and problem sessions. Written exam plus regular homework. Homework should be handed in (or emailed) on Monday before class. The final grade is the weighted average of the written exam and the homework grade. The weight of the homework is the area of the unit sphere in 21-dimensional space. For people that need to retake the exam, it will be an oral exam by appointment and the homework grade will be weighted by the area of the unit sphere in 22-dimensional space.

Literature: Elementary topics in differential geometry by J.A. Thorpe. Pdf version available through the library. A hard copy may be bought for 25 euro through the same site

Tentative schedule:

• Sept 4: Chapters 1-4. Introduction, vector fields and hypersurfaces. Homework due sept 11 before class: Additional exercises 1,2 (See Hints for a hint for exercise 2).
• Sept 7: Problem session: Ch1: 4,7,10; Ch2: 1d,2a,3,4,6,10; Ch3: 2,4; Ch4: 7,9,10,15,16,17.
• Sept 11: Chapters 5-6, Orientation, Gauss map. Homework due Sept 18 before class: Additional exercises 7,8.
• Sept 14: Problem session: Ch4: 3,11,12,13,14; Ch5: 2,4,5,10; Ch6: 1-6, 8,9.
• Sept 18: Chapters 7-8, Geodesics, parallel transport. Homework: due Sept 25: Additional exercise 9.
• Sept 21: Problem session: Ch7: 1a,e, 2, 5, 7, 10; Ch8: 2,3,6 and 7.
• Sept 25: Chapter 8, More parallel transport. Homework: due Oct 9 Additional exercises 10,11.
• Sept 28: Problem session: Ch8 2,3,4,5,7.
• Oct 2: NO CLASS
• Oct 9: Chapter 9,11,12, Curvature of surfaces, 1-forms. Homework: due Oct 16: Additional exercises 12,13.
• Oct 12: Problem session: Ch9: 1a,d, 2, 3c, 8, 10, 11, 15; Ch11: 11,12c,13; Ch12: 3,4,7,8,10,16.
• Oct 16: Chapter 14,15 Surfaces and Parametrized surfaces, Manifolds. Homework due Oct 23: Additional exercises 14,15.
• Oct 20: Problem session: Ch14: 1,5,7,11,12,13,14,19.
• Oct 23: Chapter 17, Volume and integration of forms. No Homework this week.
• Oct 27: Problem session: Ch15: 5,7,9; Ch17: 3,6,8,9,11.
• Oct 30: Chapter 17, More on integration. Homeweork additional exercises 16,17, due Nov 9 before the problem session starts.
• Nov 2: Problem session: exercises CH17: 2,7,10,18.
• Nov 6: Chapter 17,20 Differential forms, boundary.
• Nov 9: Problem session: no session.
• Nov 13: NO CLASS
• Nov 16: Problem session: exercises Ch17: 12-15, 19;.
• Nov 20: Chapter 19, Exponential map
• Nov 23: Ch19 :1-5;
• Nov 27: Chapter 20 surfaces with boundary. Homeweork additional exercises 18,19,20 due Dec 4 before class.
• Nov 30: Problem session: exercises Ch 9: 12, Ch 20 1,2,3,5.
• Dec 4: Chapter 20 Stokes theorem. Homework: additional exercises 21,22,23 due Dec 11 before class. Exercise 21d is bonus.