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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