Section and question numbers labelled AP-EDG are from the textbook Elementary Differential Geometry by Andrew Presley. Section numbers labelled MN-GTP are from the textbook Geometry, Topology and Physics by Mikio Nakahara. You do not need to submit any solutions! These are just some suggested problems related to our course topics.
Suggested Exercises 1 from AP-EDG (covering Lecture 1-2-3):
Sec 1.1: 2, 3, 4, 5, 9
Sec 1.2: 1, 2, 3
Sec 1.3: 1, 2, 4
Sec 2.1: 1, 2
Sec 2.3: 1, 2, 3, 4, 5
Suggested Exercises 2 from AP-EDG (covering Lecture 4-5):
Sec 4.1: 1, 2
Sec 4.2: 1, 2, 3, 5
Sec 4.4: 1, 2
Sec 6.1: 1, 2, 3
Sec 7.1: 1, 2, 4
Sec 7.2: 1, 2
Sec 8.1: 2, 3, 8
Sec 8.2: 1
Suggested Exercises 3 from MN-GTPP (covering Lecture 6-7):
Read Sec 5.1 and 5.2.2
Sec 5.1: Example 5.2, Example 5.3, Exercise 5.1
Sec 5.2: Exercise 5.2, Example 5.7
Sec 5.3: Exercise 5.9, Exercise 5.10, Exercise 5.11a
Suggested Exercises 4 from MN-GTPP (covering Lecture 8):
Read Sec 5.2.3, 5.2.4, 5.2.5, 7.1.1
Sec 7.1: Exercise 7.1
Prove that tensors of type (p, q) form a module over the ring of smooth functions. Prove that over a point they form a p + q dimensional vector space.
Prove that the following is a metric, and find its components in (u, v) coordinates: g = (1 + 4 u^2) du^2 + 4 u v du dv + 4 u v dv du + (1 + 4 v^2) dv^2.
Write down the standard 3-dimensional Euclidean metric in cylindrical coordinates.