Cartan Geometries and Differential Equations

Cartan geometries offer a general concept for describing certain geometric structures in the sense of differential geometry. The starting point for the notions of a Cartan geometry is a so-called homogeneous model, which form the most symmetric instance of the geometric structure. An important subclass of Cartan geometries is formed by the so-called parabolic geometries. For these, the homogeneous model is a generalized flag variety, i.e. the quotient of a semisimple Lie group by a parabolic subgroup. On the one hand, this subclass contains several of the most important examples of Cartan geometries, in particular the ones which are equivalent to conformal structures, projective structures, hypersurface-type CR structures, and quaternionic contact structures. These examples lead to relations to several other parts of mathematics and, in particular via conformal geometry and the AdS/CFT- correspondence also to theoretical physics. On the other hand, in the case of parabolic geometries one can apply methods of representation theory of semisimple Lie algebras, which lead to strong results. Parabolic geometries have been intensively studied during the last year, which also lead to new results for general Cartan geometries. The topic of the project is the study of Cartan geometries and, in particular, parabolic geometries and of their relations to partial differential equations. There are two basic sources of such relations. On the one hand, any geometric structure leads to the concept of invariant differential operators, i.e. differential operators, which are intrinsically associated to the geometry and correspondingly to geometric differential equations. For some of these equations, existence of solutions (or of solutions with special properties) leads to very interesting conditions on the underlying geometric structures. Therefore, the study of invariant differential equations forms an important part of the theory of parabolic geometries and of other Cartan geometries. On the other hand, there is the classical geometric approach to the theory of differential equations. Here on describes general differential equations by associating a geometric structure to the equation. Geometric properties of this structure then give information on the equation, which by construction does not depend on choices of coordinates or anything like that. In several cases of interest, these geometric structures can be equivalently described by a Cartan geometry or even by a parabolic geometry. For parabolic geometries, the machinery of Bernstein-Gelfand-Galfand sequences provides a construction for a large class of invariant differential operators. In particular, this provides examples leading to overdetermined geometric partial differential equations, for which existence of solutions leads to interesting conditions on the geometry. During the last years an equivalent description of the solutions of these equations as parallel sections for a certain natural linear connection was found. Of course, these natural connections for a perfect starting point for the study of solutions of the equations. An important special case of this leads to the study of infinitesimal automorphisms of parabolic geometries. In this case, also ideas from dynamics should be involved in the study during the project. A second major topic of the project is the application of methods for Cartan geometries and parabolic geometries to the geometric theory of differential equations. In this area, some important progress was made during the last years (for example in the study of systems of ordinary differential equations and of a single partial differential equation in two independent variables), which indicate applicability of such methods. Finally, subgeometries of parabolic geometries will be an important topic in the project, where preliminary research shows that a generalization of the Bernstein-Gelfand-Galfand machinery will be applicable. Results on such subgeometries can on the one hand be applied to problems in the geometric theory of differential equations. On the other hand, there are very interesting connections to results for certain special types of parabolic geometries, for example to rigidity theorems on embeddings between CR manifolds of hypersurface type.

The project contributed to an area of pure mathematics, which was internationally very active during the last years. The general topic was the relation between geometric structures in the sense of differential geometry and differential equations. The structures studied in the project were Cartan-geometries and, more specifically, parabolic geometries. These form a rather large class of geometric structures, which are very diverse when viewed in elementary terms, but admit a uniform description based on a certain concept of symmetry. Via the so-called homogeneous model, any Cartan geometry has a connection to a geometry in the classical sense and thus to the theory of (Lie-)groups. On the other hand, in many situations, there are differential equations, which are naturally associated to a geometric structure. In the case of the homogeneous model, these equations and their solutions are well understood. One basic problem in this area of research is relating general instances of a geometric structure to the homogeneous model. Second, there is the fundamental question of relating geometric properties (for example the existence of additional symmetries) to the differential equations associated to the geometry. The main result of the project was the development of holonomy theory for Cartan geometries, which provides solutions to both these basic problems in many interesting situations. In addition, this theory leads to conceptual conditions for compatibility of various geometric structures on spaces of different dimensions. Such conditions could become important for several areas of mathematics (like scattering theory) and of theoretical physics (like general relativity). Finally, holonomy reductions of parabolic geometries lead to several interesting relations to simpler geometric structures, which are intensively studied from other points of view. For example, several kinds of holonomy reductions lead to Einstein-metrics, which provide an interesting connection to Riemannian geometry.