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# MAI0143 Differential Geometry (8hp)

## Course description

This course gives an introduction to the differential geometry of manifolds. In particular, we shall consider Riemannian manifolds (i.e. manifolds with a metric structure) and their curvatures. Important examples will be provided by surfaces embedded in Euclidean space.

Content: Manifolds, tangent spaces, differential forms, tensors, Riemannian metrics and curvature, connections on vector bundles, surfaces in Euclidean space.

## Lectures

Lectures will be held on Tuesdays 13.15 - 15.00 in Hopningspunkten (B-huset, entrance 23).

1. Tuesday 2018-02-13 (13.15 - 15.00) (w 7)
Content: Chart, atlas, differentiable structure, manifolds, differentiable maps.
Sections: (1.2), 1.3, 1.4.
Homework 1 (due 2018-02-20)
2. Tuesday 2018-02-20 (13.15 - 15.00) (w 8)
Content: Tangent space, cotangent space, tangent bundle, cotangent bundle.
Sections: 2.1-2.3, 2.7.
Homework 2 (due 2018-03-06)
3. Tuesday 2018-03-06 (13.15 - 15.00)
Content: Product manifold, cut-off function, vector fields.
Sections: 1.5, 2.8.
Homework 3 (due 2018-03-13)
4. Tuesday 2018-03-13 (13.15 - 15.00)
Content: Vector fields (continued), differential forms, tensors.
Sections: 2.8, 2.9, 7.1-7.5, 8.1-8.1.1.
Homework 4 (due 2018-03-27)
5. Tuesday 2018-03-27 (13.15 - 15.00)
Content: Differential forms (continued), exterior derivative, de Rahm cohomology.
Sections: 8.1, 8.3, p.441 - p.446.
Homework 5 (due 2018-04-03)
6. Tuesday 2018-04-03 (13.15 - 15.00)
Content: Partition of unity, integration of differential forms.
Sections: 1.5, p. 391 - p. 394 (in Chapter 9).
Homework 6 (due 2018-04-17)
7. Tuesday 2018-04-17 (13.15 - 15.00)
Content: Fibre bundles, vector bundles.
Sections: 6.1, 6.2.
Homework 7 (due 2018-04-24)
8. Tuesday 2018-04-24 (13.15 - 15.00)
Content: Riemannian manifolds: metric, distance, volume form
Sections: 7.6.1, p. 547-550.
Note that in the course litterature, the emphasis lies on Riemannian metrics for vector bundles. For a treatment that is more similar to the lecture, see e.g. Jürgen Jost's book starting from p.13 (at Google Books).
Homework 8 (due 2018-05-08)
9. Tuesday 2018-05-08 (13.15 - 15.00)
Content: Riemannian metrics (continued), affine connections, curvature, parallell transport.
Sections: p.501 - p.503, p.523 - p.527, 12.6.
For a presentation of connections and curvature that do not involve vector bundles, see e.g. Geometry, topology and physics (by M. Nakahara) p. 244-260.
Homework 9 (due 2018-05-15)
10. Tuesday 2018-05-15 (13.15 - 15.00)
Content: Metric connection, torsionfree connection, Levi-Civita connection, Riemannian curvature tensor, geodesics
Sections: 13.1, 13.1.1, 13.2, 13.5
Homework 10 (due 2018-05-22)
11. Tuesday 2018-05-22 (13.15 - 15.00)
Content: Ricci and scalar curvature, submanifolds, induced metrics, surfaces in R3, mean curvature, Gaussian curvature.
Sections: 3.1, 13.3, 4.2, p.174 - p.177.
Homework 11 (due 2018-05-29)
12. Tuesday 2018-05-29 (13.15 - 15.00)
Content: Levi-Civita connection of submanifolds, Gauss equation, Theorema Egregium.
Sections: 13.3, Thm 4.55 (p. 176)
Homework 12 (due 2018-06-05)
13. Tuesday 2018-06-05 (13.15 - 15.00)
Content: A closer look at parallel transport for connections with or without torsion.

## Seminars

Friday 2018-06-08 (10.15 - 12.00 in Kompakta rummet)
The hairy ball theorem and its applications (Christoffer Holm)

Tuesday 2018-06-12 (13.15 - 15.00 in Hopningspunkten)
Characteristic classes and Chern classes: definitions and examples (Kwalombota Ilwale)
An introduction to minimal surfaces (Axel Tiger Norkvist)

Wednesday 2018-06-13 (13.15 - 15.00 in Kompakta rummet)
The Laplace eigenvalue equation on a compact manifold (Abubakar Mwasa)
A curvature bound on eigenvalues of the Laplace operator (Pauline Achieng)

Friday 2018-06-15 (13.15 - 15.00 in Kompakta rummet)
Lie Groups: definitions, examples and Lie algebras (Jennifer Chepkorir)
Graph embeddings on Riemannian manifolds (Olle Abrahamsson)

Monday 2018-06-18 (13.15 - 15.00 in Kompakta rummet)
Mayer-Vietoris Exact Sequences and the Cohomology of Spheres (Vincent Umutabazi)
An introduction to Kähler manifolds (Ahmed Al-Shujary)

Wednesday 2018-10-03 (13.15 - 15.00 in Kompakta rummet)
Felix Järemo

Thursday 2018-10-04 (13.15 - 15.00 in Kompakta rummet)
Andreas Christensen
Jonas Granholm

## Course litterature

Manifolds and Differential Geometry. Jefferey M. Lee. Graduate Studies in Mathematics vol. 107, 2009.

## Examination

Every week, a set of homework exercises will be handed out, and they are to be handed in one week later. Each homework contains a set of exercises. To pass the course, you need to complete at least 70% of the total number of exercises and at least 50% of each set of exercises. Moreover, at the end of the course, each participant is required to give a seminar on a selected topic.

## Instructions for the seminar

You may choose the topic for the seminar quite freely, but please make sure to discuss it with me first (just to see that it is not too difficult or too far from the course). If you can not think of a topic yourself, I have a few suggestions.

The seminar is supposed to be around 45 minutes, and present the topic to the audience in a way that is accessible to those who have taken the course.

## Course coordinator and examiner

Joakim Arnlind. Tel: 013 - 28 14 22. Epost: joakim.arnlind@liu.se.
Room 3A:637, entrance 23 (B-huset, 3rd floor).

Sidansvarig: Joakim Arnlind