CHAPTER 1

Cartan geometries

The concept of a “generalized space” was introduced by

´

Elie Cartan in order to

build a bridge between geometry in the sense of Felix Klein’s Erlangen program and

differential geometry. In the Erlangen program, a geometry is given by a manifold

endowed with a transitive action of a Lie group, and thus (up to the choice of a

base point) by a homogeneous space G/H of a Lie group G. Cartan’s idea was

to associate to such a homogeneous space a differential geometric structure, whose

objects may be thought of as curved analogs of the homogeneous space G/H, just

like n–dimensional Riemannian manifolds may be thought of as curved analogs of

Euclidean space, viewed as a homogeneous space of the group of Euclidean motions.

In modern terminology, such structures are called Cartan geometries, and they are

defined as principal bundles endowed with Cartan connections.

This chapter provides a quick introduction to Cartan geometries as well as a

sketch of the necessary background from differential geometry. A more detailed

explanation of the relation to the Erlangen program and a comprehensive study of

several basic examples of Cartan geometries can be found in the book [Sh97].

There are several results on general Cartan geometries, which are highly non-

trivial even for special cases like Riemannian or conformal structures. Among these

are, for example, the fact that the automorphisms of a Cartan geometry always form

a Lie group, or that local automorphisms of the homogeneous model always uniquely

extend to global automorphisms. Apart from these aspects, in which the Cartan

geometry is viewed as an input, there is also a second important point of view:

The most interesting examples of Cartan geometries are those, in which the Cartan

geometry is actually determined by some underlying geometric structure, so that

results on the Cartan geometry directly give results on the underlying structure.

Of course, one may also turn this around and ask about the structures underly-

ing a Cartan geometry of given type and whether (assuming a suitable normaliza-

tion condition) the Cartan geometry is determined by these underlying structures.

This point of view will also be important in several parts of this chapter. The

underlying structures can already be seen on the homogeneous model G/H of a

Cartan geometry, so this question boils down to understanding G–invariant geo-

metric structures on G/H.

Let us briefly describe the contents of the individual sections: Section 1.1 should

be viewed as a prologue. We consider several simple actions of Lie groups on spheres

and projective spaces and show how to find geometric structures which are invariant

under these actions. On the one hand, this shows the diversity of structures showing

up already in simple situations. On the other hand, except for the Riemannian

round sphere, all the examples we consider are actually the homogeneous models

of some parabolic geometry, so this provides the first overview of the structures

available in that way.

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http://dx.doi.org/10.1090/surv/154/01