A Cartan geometry from a correspondence space in Riemannian geometry

Abstract

Cartan geometries are curved analogues of homogeneous spaces, and the correspondence space construction produces, from a Cartan geometry of one type, another of a different type over a larger base manifold. The unit tangent bundle of a Riemannian manifold is, in this language, a Cartan geometry of type $(\operatorname{Euc}(n),O(n-1))$ obtained in this way; we study Cartan geometries of this type in general, whether or not they arise as correspondence spaces. We show that every such geometry induces on its base manifold an almost contact metric structure, together with an orthogonal splitting of the tangent bundle and a compatible linear connection; conversely, these data determine the geometry, so that the correspondence is a bijection. When $n\ge3$, we single out a distinguished Cartan connection, which we call normal, by a condition on its torsion formulated in terms of the Spencer differential. This normalization is dictated by the correspondence space construction: the Cartan connection induced on the unit tangent bundle of a Riemannian manifold is normal.