# The index theorem by Atiyah and Singer

Prof. Bernd Ammann, Zimmer 119

## Content of the lecture

Let M be a compact connected smooth manifold without boundary. One of the guiding questions of the lecture is whether there is a Riemannian metric g on M, such that g has everywhere positive scalar curvature. We will shortly call such a metric a psc (positive scalar curvature) metric.

The question actually splitts into two parts:
1.) Obstructions against psc metrics, in other words: reasons why such metrics cannot exist
2.) Constructions of psc metrics on large classes of manifolds

A large part of the lecture will study the first part: The Atiyah-Singer index theorem is the most important obstruction against psc metrics. The theorem has attracted a lot of interest within mathematics, because it has connected many fields in mathematics: geometry, topology, and partial differential equations. It also established many links to applications in mathematical physics. For example it provides important tools for a better understanding of scalar curvature in general relativity, leading e.g. to Witten's proof of the positive mass theorem of an asymptotically Euclidean spacetime (e.g. a star or a black hole). The index theorem has attracted many important prizes, e.g. the Fields Medal for Atiyah in 1966 and the Abel prize for Atiyah and Singer in 2004. The Atiyah-Singer index theorem states that the Fredholm index of an elliptic partial differential operator D on M is equal to a characteristic class of the tangent bundle of M, integrated over M. In the classical case, D is the Dirac operator and if M carries a psc metric, then this Fredholm index is 0. On the other hand, characteristic classes are easy to calculate, they do not depend on the choice of a Riemannian metric, and often we see that they are not zero. As a consequence we get manifolds that do not carry a psc metric. The Atiyah-Singer theorem also applies to other types of elliptic operators. One special case is the Gaus-Bonnet-Chern operator which yields a higher-dimensional version of the Gauss-Bonnet formula, and in another version we obtain as an index the signature that some people in the audience might have seen in a topology course. We want to follow the heat-kernel method to prove the index theorem. This approach is considerably simpler that the original approach by Atiyah and Singer, and allows us to understand the proof in many details. If time permits we will then study the second part of the question and we will use surgery methods to construct many metrics of positive scalar curvature.

A good a impression about the course can be obtained from the book(s) by Roe or the lecture notes cited below.

## Recommeded previous knowledge

The most important knowledge is to have a profound understanding of the curvature of riemannian manifolds as e.g. taught in my lecture "Differentialgeometrie I" in the last winter term. We also need several concepts from a course such as "Differentialgeometrie II" such as Lie groups, vector bundles together with connections, but this can be recaptured easily if not present.

## Literature

### Books

• John Roe: Elliptic operators, topology and asymptotic methods, first edition, Pitman Research Notes in Mathematics Series 179, Longman Scientific and Technical, MathSciNet link.
• John Roe: Elliptic operators, topology and asymptotic methods, second edition, CRC Research Notes in Mathematics 395, Chapman and Hall, MathSciNet link.
• Lawson, Michelsohn: Spin Geometry; Princeton Math. Series, 38. Princeton University Press, MathSciNet link
• Berline, Getzler, Vergne: Heat kernels and Dirac operators, Springer Verlag
• O. Hijazi: Spectral properties of the Dirac operator and Geometrical structures, Geometric methods for quantum field theory (Villa de Leyva, 1999), 116–169, World Scientific, 2001, MathSciNet link.
• T. Friedrich: Dirac-Operatoren in der Riemannschen Geometrie, Vieweg, MathSciNet link.
• J.-P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, S.Moroianu, A spinorial approach to Riemannian and conformal geometry, EMS Monographs in Mathematics, 2015, MathSciNet-Link. Available as E-Book of our library.
• P. Gilkey: Invariance Theory, the heat equation and the Atiyah-Singer index theorem MathSciNet link to the second edition and MathSciNet link to the first edition

## Place and Time

Tuesday and Thursday 8-10, M104

## Exercises

Monday, 16-18, M101, Olaf Müller

### Exercise Sheets

(some links are not active yet)
All Exercise sheets in one file

## Kriterien für benotete Leistungsnachweise

Um die üblichen Leistungsnachweise zu erhalten, sind folgende Kriterien zu erfüllen:
1. Regelmäßige Abgabe von Lösungen der Hausaufgaben. Man muss mindestens 50 Prozent der Punkte erhalten, die man bei korrekter Bearbeitung aller Aufgaben (ohne Zusatz-Aufgaben) erhalten kann. Jeder Student muss jede abgegebene Hausaufgabe persönlich an der Tafel vorrechnen können, um zu gewährleisten, dass er die Aufgaben selbst verfasst hat.
2. Die Bearbeitung der Hausaufgaben muss regelmäßig erfolgen. Ein hinreichendes Kriterium ist hierbei: mindestens 25 Prozent der Punkte der letzten 3 Hausaufgabenblätter sollten erreicht werden.
3. Regelmäßige und aktive Teilnahme in den Übungsgruppen. Hierzu gehört das erfolgreiche Vorrechnen von Übungsaufgaben (mind. einmmal).
4. Grundlage der Note ist die mündliche Abschlussprüfung (30 Minuten).

## Unbenotete Leistungsnachweise

Mündliche Prüfung (15 Minuten) nach Zulassung wie oben.

## Related web sites

Bernd Ammann, 29.11.2016 oder später