Symmetries, Geometric Structures and Holonomy

Beyond Riemannian and Lorentzian geometry, there are various other notions of geometry that arise naturally in mathematics and physics. Classical examples are conformal, projective, and CR geometry, and there are many others. The main focus of this course will be on the study of conformal

structures, which are given by equivalence classes of (pseudo-) Riemannian metrics, where two metrics are equivalent if one is a rescaling of the other by a positive smooth function. But we will meet other, equally important, geometric structures. The main goal of the course is to introduce students to the topic and to provide them with tools to study these various geometries in an invariant manner.

Beyond Riemannian and Lorentzian geometry, there are various other notions of geometry that arise naturally in mathematics and physics. Classical examples are conformal, projective, and CR geometry, and there are many others. The main focus of this course will be on the study of conformal

structures, which are given by equivalence classes of (pseudo-) Riemannian metrics, where two metrics are equivalent if one is a rescaling of the other by a positive smooth function. But we will meet other, equally important, geometric structures. The main goal of the course is to introduce students to the topic and to provide them with tools to study these various geometries in an invariant manner. I plan to cover the following topics:

1. Lie groups and homogeneous spaces

a. Lie groups, Lie algebras and their representations

b. the Frobenius theorem

c. the Maurer-Cartan form

d. Lie group actions on manifolds and homogeneous spaces

2. Bundles, connections, and holonomy

a. principal bundles and associated bundles

b. principal bundle connections and induced connections

c. homogeneous bundles and invariant sections

d. parallel transport, curvature and holonomy

e. G-structures

f. holonomy groups of Riemannian manifolds and Berger's classication

3. Conformal structures

a. decomposition of the Riemannian curvature tensor, the Weyl tensor

b. the normal conformal Cartan connection and tractor connection

c. conformal invariants and invariant differential operators

d. the Killing equation and the conformal Killing equation and their

prolongations

e. conformal holonomy

4. Other geometric structures and Cartan connections:

a. projective structures and projectively invariant differential operators

b. geometries determined by non-integrable vector distributions

c. Cartan geometries

d. parabolic geometries

e. rudiments of Cartan's equivalence method

The course is aimed at both physics and mathematics students. Depending on the knowledge and the interests of the students, we shall keep the discussion of the more standard differential geometric background material shorter or not, and decide which of the later topics to cover in detail.

1. Transformation Groups in Differential Geometry, Shoshichi Kobayashi

2. Notes on projective differential geometry, Michael Eastwood

3. Parabolic geometries I: Background and General Theory, Andreas Cap and

Jan Slovak

4. An introduction to conformal geometry and tractor calculus, with a view to

applications in general relativity, Sean Curry and A. Rod Gover

5. Cartan for Beginners: Differential Geometry via Moving Frames and Exterior

Differential Systems, Thomas A. Ivey and J.M. Landsberg

1. Understanding of a part of (local) differential geometry, its techniques, and the ability to explain important notions and results in the eld.

2. Solving simple problems about differential geometric structures.

1. Homework

2. Exam

not applicable