# Differential Geometry

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Course at advanced level (course number KTH: SF2722, SU: MM8022), 7.5 credits, spring 2017.

Teachers: Mattias Dahl and Hans Ringström.

Administration: Contact the student affairs office for registration and other administrative matters.

Literature: John Lee, Introduction to Smooth Manifolds, second edition (publishers web page, authors web page, electronic version of the full text should be available at both the KTH and SU libraries). A list of misprints can be found here. The literature for the part of the course devoted to semi-Riemannian geometry and general relativity will partly be based on lecture notes by Christian Bär, and partly on lecture notes that will be posted on this homepage.

Time and place: see schedule in the left hand column.

Description: The central objects in modern differential geometry are differentiable manifolds. In this course we will study differentiable manifolds and see how they are used to define concepts from analysis in a coordinate-independent way. We will see how to define tensors and differential forms and how to formulate the fundamental theorem of calculus in geometric way as Stokes' theorem. In an introduction to (semi-)Riemannian geometry we will see how curvature is described. Ideas and methods from differential geometry are fundamental in modern physical theories.

Content: The course is given in two parts, the first in period 3 with lectures by Mattias Dahl and the second in period 4 with lectures by Hans Ringström.

• First part, Chapters 1-9: Smooth Manifolds; Smooth maps; Tangent Vectors; Submersions, Immersions, and Embeddings; Submanifolds; Sard's Theorem; Lie Groups; Vector fields; Integral Curves and Flows.
• Second part, Chapters 11-16: The Cotangent  Bundle;  Tensors; Riemannian Metrics; Differential Forms; Orientations; Integrations on Manifolds; Semi-Riemannian geometry; Einstein's equations.

Prerequisites: SF2713, Foundations of Analysis, or corresponding background. Also recommended is SF2729, Groups and Rings, or equivalent. A good knowledge of calculus of several variables including the inverse and implicit function theorems is an important prerequisite.

Lectures:

• Tuesday 17/1
Lecture 1: Smooth manifolds, Lee, Chapter 1. Smooth maps, Lee Chapter 2 (until page 40). See also Bär lecture notes sections 1.1-1.2.
Recommended exercises: 1.18 (p. 14), 1-1, 1-6, 1-8, 1-9 (pp. 29-31), 2-2, 2-3ab, 2-5, 2-10 (pp. 48-49).

• Tuesday 24/1
Lecture 2: Partitions of unity, Lee Chapter 2 (pages 44-47), Tangent vectors, Lee, Chapter 3 (pages 50-73).
Recommended exercises: 3.5 (p. 54), 3.7 (p. 56), 3-4 (p. 75).
• Tuesday 31/1
Lecture 3:
Submersions, Immersions, and Embeddings, Lee Chapter 4.
Recommended exercises:
4.3 (p. 79), 4.24 (p. 87), 4.27 (p. 89), 4-5, 4-6, (pp. 96-97).
• Tuesday 7/2
Lecture 4: Submanifolds, Lee Chapter 5. Also read Chapter 7 on Lie groups, at least the definitions and examples 7.27-7.30.
Recommended exercises: 5.10 (p. 104), 5-4, 5-7 (p. 123), 7-11, 7-13, 7-17 (p. 172).
• Tuesday 14/2
Lecture 5: Sard’s theorem, Lee Chapter 6. For another exposition which also includes a discussion on Morse functions, see Guillemin Pollack Differential Topology (pp. 39-56).
Recommended exercises: 6-2, 6-9, 6-13, 6-16 (pp. 147-149)
• Tuesday 21/2
Lecture 6:
Vector Fields, Lee Chapter 8. Integral Curves and Flows, Lee Chapter 9 (until p. 217). Also read the parking car example here (pp. 33-36) and here illustrating Lie brackets and flows.
Recommended exercises: 8-10, 8-13, 8-16 (p. 201)
• Tuesday 28/2
Lecture 7:
Integral Curves and Flows, Lee Chapter 9. (For more on connected sums, surgery, etc, see Wikipedia Surgery Theory and Chapter VI of Kosinski Differential Manifolds.)
Recommended exercises: 9-12, 9-16, 9-18, 9-19 (pp. 246-247)
• Tuesday 21/3
Lecture 8:
The Cotangent Bundle, Lee Chapter 11.

Recommended exercises: 11.12 (p. 278), 11.30 (p. 287), 11-1, 11-7ab (pp. 299-300)
• Tuesday 28/3
Lecture 9:
Tensors, Lee Chapter 12.

Recommended exercises: 12.3 (p. 306), 12.6 (p. 308), 12.17 (p. 316), 12.18 (p. 317), 12.21 (p. 318), 12.26 (p. 320), 12-3, 12-9 (pp. 324-326)
• Tuesday 4/4
Lecture 10:
Differential forms, Lee Chapter 14.
Recommended exercises: 14.4 (p. 351), 14.6 (p. 352), 14.17 (p. 361), 14.28 (p. 368), 14.31 (p. 369), 14-7 (p. 375)
• Friday 7/4
Lecture 11:
Orientations and integration on manifolds, Lee Chapters 15-16.
Recommended exercises: 15.4 (p. 381), 15.10 (p. 382), 15-2 (p. 397), 15-5 (p. 397), 16-2 (p. 434)
• Tuesday 18/4
Lecture 12:
Introduction to Semi-Riemannian geometry, Lee Chapter 13, and Chapters 1 and 2 of the lecture notes.
Recommended exercises: 9, 25, 33 and 39 in the lecture notes
• Tuesday 25/4
Lecture 13:
Levi-Civita connection and geodesics, Chapter 3 of the lecture notes.
Recommended exercises: 46, 48, 58 in the lecture notes and 13.23, 13.24 and 13.27 in Lee
• Tuesday 2/5
Lecture 14:
Curvature, Chapter 4 of the lecture notes.
Recommended exercises: 67, 69, 76, 77 and 88 in the lecture notes
• Tuesday 9/5
Lecture 15:
Introduction to general relativity, Chapter 5 of the lecture notes.
Recommended exercises: 92 and 102 in the lecture notes
• Friday 12/5
Lecture 16:
Simple solutions to Einstein's equations, Chapter 2 in "Skript zur Vorlesung 'Relativity Theory' (Relativitätstheorie)" (in English), available here

Examination: The examination will be in the form of homework problems followed by an oral examination. For grades C-E you only have to solve the homework problems. For grades A-B you must also do the oral examination.

Homework problems: There will be eight sets of homework problems.

• Homework 1, to be handed in 31/1 (strict deadline 7/2).
• Homework 2, to be handed in 7/2 (strict deadline 14/2).
• Homework 3, to be handed in 21/2 (strict deadline 28/2).
• Homework 4, to be handed in 21/3 (strict deadline 28/3).
• Homework 5, to be handed in 4/4 (strict deadline 11/4).
• Homework 6, to be handed in 18/4 (strict deadline 25/4).
• Homework 7, to be handed in 2/5 (strict deadline 9/5).
• Homework 8, to be handed in 12/5 (strict deadline 14/5).

We will not accept solutions handed in after the strict deadlines. Extra assignments to compensate any missed homework will be given at the end of the course.

Oral examination: The oral examination takes place after all homework problems have been handed in and graded. In the oral examination you will be asked about details and background for your solutions to the homework problems, and you  will be asked to present solutions to some of the weekly "recommended exercises". The oral examination will take place May 16-19.

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