# TMA4190 Manifolds

### Messages

• Here are suggestions for solutions to the exam problems: pdf
• Final exercise class on Tuesday, May 5, 16:15-18:00 in Seminar room 734 in Sentralbygg 2.
• Here are some exercises you may want to think about: pdf
• New Office Hours until the exam: Mondays and Wednesdays 15:00-16:00, room 1246 Sentralbygg 2.

## Lectures

• Tuesday 12:15-14:00, room R73.
• Friday 14:15-16:00, room R73.

## Exercises

• Thursday 16:15-17:00, room R73

## Textbook

We will mostly follow the book:

Other good textbooks are:

• L. W. Tu, An Introduction to Manifolds, Springer Verlag.
• J.M. Lee, Introduction to Smooth manifolds, Springer-Verlag.
• J. Milnor, Topology from the differentiable viewpoint, The University Press of Virginia, 1969.

### Lecture Plan

Week Topic Reference Exercises
2.1 Warm-up and introduction to manifolds Dundas § 2
2.2 Topological manifolds and first examples Dundas § 3.1 Dundas: 3.1.5, 3.1.6, Lee: 1-1
3.1 Smooth atlas and more examples Dundas § 3.2 Dundas: 3.2.8, 3.2.11, Lee: 1-8
3.2 Maximal atlas and smooth manifolds Dundas § 3.3 + 3.4 Dundas: 3.3.4, 3.3.9, 3.4.4, 3.4.5, 3.4.6, 3.4.11
4.1 More on smooth maps and submanifolds Dundas § 3.4 + 3.5 Dundas: 3.5.4, do 3.4.19 again
4.2 Products and Sums, the tangent space Dundas § 3.6 + 4.0 Dundas: 3.5.12, 3.5.15, 3.6.5, 3.6.10
5.1 The tangent space Dundas § 4.2 Dundas: 4.1.6, 4.2.3, 4.2.6
5.2 Tangent and cotangent spaces Dundas § 4.2 + 4.3 Dundas: 4.2.17, 4.2.18
6.1 Tangent spaces revisited
6.2 Cotangent space Dundas § 4.3 Dundas: 4.3.16
7.1 Rank of smooth maps and regular values Dundas § 5.1
7.2 Ranks and the fundamental theorem of algebra Dundas § 5.2 + 5.3
8.1 Regular values and submanifolds Dundas § 5.4 Dundas: 5.4.4, 5.4.11-13
8.2 Transversality Dundas § 5.5 Dundas: 5.5.2, 5.5.5
9.1 Immersions and imbeddings Dundas § 5.5 Dundas: 5.7.2, 5.7.3, 5.7.6, 5.7.7
9.2 Topological vector bundles Dundas § 6.1 Dundas: 6.1.3
10.1 Sections and transition functions Dundas § 6.1 + 6.2 Dundas: 6.2.4, 6.2.5
10.2 Smooth vector bundles Dundas § 6.3 Dundas: 6.3.10, 6.3.14, 6.3.15. 6.4.4
11.1 The tangent bundle Dundas § 6.5 Dundas: 6.5.9
11.2 Vector fields and parallelizable manifolds Dundas § 6.5 Dundas: 6.5.17, 6.5.20
12.1 Flows and velocity fields Dundas § 8.1 Dundas: 8.1.9, 8.1.11
12.2 Integrability Dundas § 8.2 Dundas: 8.2.6
13.1 Local flows and vector fields Dundas § 8.3 + 8.4 Dundas: 8.4.3
13.2 The hairy ball theorem Milnor § 5
14 NTNU Spring Break in Week 14
15.1 NTNU Spring Break
15.2 Smooth bump functions and partition of unity
16.1 Embeddings in Euclidean space Dundas § 9.1 + 9.2 Dundas: 9.2.3
16.2 Riemannian manifolds and normal bundles Dundas § 9.3 + 9.4 Dundas: 9.4.8, 9.4.10
17.1 Manifolds with boundary and Brouwer's FPT Milnor § 2

## Exam

• Monday 18 May 2015, 9:00.

### Reference Group

• August Peter Brådelen Sonne
• Haakon Holm Gulbrandsrud
• Juraj Palenik
• Marte Lovise Nilsen

### Reference Group Meetings

• First meeting: January 30, 16:00, in R73.
• Second meeting: March 20, 16:00, in R73.
• Third meeting: April 28, 12:15 in 1246.

### Alternative books and additional literature

• M.Spivak, A comprehensive Introduction to Differential Geometry, Vol.1. Publish or Perish.
• A.A. Kosinski, Differential Manifolds. With a New Appendix by J.W. Morgan on the Work of Grigory Pererman, Dover Publications, 2007.
• T.Bröcker, K.Jänich, Introduction to differential topology, Cambridge Univ. Press, 1982.
• F. Brickell,R.S.Clark, Differentiable manifolds, Van Nostrand Reinhold, London 1970
• W.M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press, 1975.
• J. R. Munkres, Analysis on manifolds, Westview Press 1991
• Guillemin and Pollack, Differential Topology, Prentice Hall, 1974 (eller nyare)
• Per Holm, Differensialformer og mangfoldigheter, http://www.math.uio.no/~pholm/mangfold.html

Note. The recent edition of Kosinski's book (see above) is also interesting because of the new appendix on Perelman's work and the Poincare conjecture.