In Riemannian differential geometry, the Levi-Civita connection governs pretty much everything. Anything constructed from it is manifestly invariant and, consequently, there are just heaps of invariant tensors and differential operators. If anything, there are rather too many. But in other differential geometries, such as conformal or projective, it's actually quite difficult to write down anything at all that is invariant. Even basic examples in conformal geometry such as the Weyl curvature, the Bach tensor in dimension four, and the conformal Laplacian, are surprising when first encountered. Well clearly, there is something going on here, crying out for an explanation.
So why not start with the flat model and see what happens there? It's still hard but the paucity of invariant differential operators is actually a good thing: it leads to a classification! Having understood the flat case, it is then remarkable that many of its features persist in the curved setting (once we've come up with a satisfactory notion of invariance)!
Parabolic geometry is modelled on homogeneous spaces of the form G/P where G is a semisimple Lie group and P is a parabolic subgroup. Classical examples include projective, conformal, and CR differential geometry. In parabolic geometry, BGG stands for Bernstein-Gelfand-Gelfand and the BGG complex on projective space is the simplest instance of a classification of invariant differential operators to be explained in these lectures.
There will be some assumed background for this course. A familiarity with elementary differential geometry and Lie groups (more through examples than general theory) will do.
Given a vector bundle with connection, one can define the parallel transport of fibres along curves in the manifold. The holonomy group the is the group of parallel transports along closed curves. It carries geometric information about the vector bundle, and, since it acts linearly on the fibre, it can be treated by powerful algebraic tools such as representation theory.
Holonomy groups gained importance in the case where the vector bundle is the tangent bundle of a Riemannian manifold and the connection is the Levi-Civita connection. In this case Marcel Berger essentially classified all possible holonomy groups in a surprisingly short list. Many fundamental developments in modern Riemannian geometry are related to this classification. In the course I will however focus on holonomy in a non-Riemannian setting. This includes the holonomy for indefinite semi-Riemannian manifolds, in particular Lorentzian manifolds, as well as holonomy groups associated to other geometric structures, such as conformal geometry. In both situations we will introduce the fundamental notions as well as give an overview about what is known.
The course will start with Part 1 in which we give the the basic definitions about holonomy groups and algebras and fundamental results such as the Ambrose-Singer holonomy theorem. In Part 2 we will focus on holonomy groups for semi-Riemannian manifolds, briefly explaining also the Riemannian case and Berger’s result as a guiding principle. More detail will be then be given on holonomy groups of Lorentzian manifolds, their classification, and the geometric structures related to them.
In the Part 3 we will concentrate on holonomy theory in conformal geometry. Here the canonical connection is not defined on the tangent bundle (as in semi-Riemannian geometry) but on a vector bundle that is however closely related to the tangent bundle, the so called conformal tractor bundle (at this point the course will be linked to some of the other courses). We will explain some of the results that are available about conformal holonomy and its relations to other invariant descriptions of conformal geometry, such as the ambient metric construction. If time permits, we might briefly look at holonomy in other parabolic geometries such as projective geometry.
A projective structure on a smooth manifold is an equivalence class of torsion-free connections on its tangent bundle, where two connections are called equivalent if they share the same unparametrised geodesics. Projective structures fall within the class of parabolic geometries and hence admit a description in terms of a Cartan geometry. In particular, a projective structure has a finite dimensional automorphism group, local invariants such as the Weyl projective curvature tensor and associated invariant differential operators.
Despite having a long and fascinating history, global results about projective structures are somewhat lacking, except for two special cases: Projective structures that are flat and projective structures that arise from the Levi-Civita connection of a Riemannian metric.
After discussing the basic definitions, we will construct the Cartan geometry of a projective manifold and proceed to study the metrisability problem in two dimensions, that is, the problem of characterising the 2D projective structures that arise from a Levi-Civita connection. We will see how the metrisability problem naturally connects to interesting global questions and a variety of topics of current interest, including invariant differential operators, projective transformations, twistor geometry, higher Teichmüller theory and Lagrangian minimal surfaces.
A familiarity with the foundations of modern differential geometry on smooth (and ideally complex) manifolds is assumed.
In differential geometry many important geometric structures are geometrically rigid in the sense that their automorphism groups (in some natural topology) are finite-dimensional Lie groups. Prominent examples of such structures are Riemannian manifolds, conformal and projective structures and in general all geometric structures admitting equivalent description as so-called Cartan geometries, which comprise a huge variety of geometric structures. Generically these geometric structures have trivial automorphism groups and so the ones among them with large automorphism groups or special types of automorphisms are typically geometrically and topologically very constrained and hence can often be classified. Recall for instance that a Riemannian manifold (dim>2) whose group of conformal transformations is of largest possible dimension is conformally diffeomorphic to a sphere. Another famous further manifestation of the rigidity of conformal structures is the Ferrand-Obata Theorem, which proves the Lichnerowicz conjecture. It states that a compact conformal manifold whose automorphism group does not preserves any metric in the conformal class is conformally diffeomorphic to a sphere.
In this course we will discuss these and other results along these lines, concerned with (local) automorphism groups of geometric structures and local and global phenomena of geometric rigidity. Since we will see that these phenomena are intimately related to the existence of Cartan connections for the discussed geometric structures, the course can be also seen as providing an introduction to the concept of a Cartan geometry.
Prerequisites: Familiarity with modern differential geometry and basic knowledge about Lie groups.
Cartan's equivalence method gives an algorithm for checking if two geometric structures of the same type are locally diffeomorphically equivalent. Its byproduct is a full set of local differential invariants for a given type of geometric structure. Sometimes this set of invariants is given in terms of the curvature of a Cartan connection. It is this aspect of the Cartan method that is understood (at least in the parabolic geometries case) and mostly used by mathematicians. However, as the authors of this minicurse feel, these were not the applications that were Cartan's motivation for designing his method. A case in point is one of Cartan's most celebrated paper on the subject, the `five variable paper' from 1910, in which he did not bother to built a Cartan connection. Instead, he mainly used his method to built sufficiently many invariants which, when worked out by a subprocedure of Cartan's method called `reduction', produced homogeneous models of the considered G2 geomtery.
It is this aspect of Cartan's equivalence method - namely its use to produce algorithmically all locally nonequivalent homogeneous models for a geometry of a given type - that we believe was the main motovation of Cartan. And this story is not so well known among mathematicians. Our minicourse will have take this aspect of Cartan's equivalence method as the central theme.
The course will be elementary, mainly based on the examples. We will discuss and illustrate with examples: equivalence of rigid coframes, moving coframes, absorption and normalization procedures, Tanaka prolongations, Cartan connections, reduction procedure, and finally the algorithm of constructing homogeneous models.
No other techniques than exterior differentiation will be needed to understand the course.