The Killing operator on locally homogeneous spaces

Abstract

In this thesis we study a compatibility complex, derived form the Calabi complex, providing conditions for a symmetric 2-tensor on pseudo-Riemannian locally homogeneous space to be in the image of the Killing operator. In the rst chapter, we describe general machinery to study the rst cohomology group of the twisted de Rham complex of an arbitrary vector bundle with connection. This machinery will be applied in the last two chapters to the Killing bundle and the Killing connection, a vector bundle with connection that arises from a prolongation of the Killing equation. In the second chapter we introduce the Killing bundle and the Killing connection, that provides an overdetermined system of linear partial di erential equations for the Killing equation. We prove a theorem analogous to Hano's theorem on the splitting of the Lie algebra of Killing vector elds of a product Riemannian manifold [26], to arbitrary signature. Moreover, we study the structure of special subbundles of the Killing bundle and apply these results in Chapter 3 to provide a characterisation of pseudo-Riemannian locally homogeneous spaces in terms of the maximal parallel at subbundle of the Killing bundle and to give new proof of the Ambrose-Singer theorem regarding homogeneous structures [3]. In the fourth chapter we construct a compatibility complex for the Killing operator, that arises from a modi cation of the Calabi complex, and establish its equivalence to the short twisted de Rham complex of the Killing connection. We make use of this equivalence to provide a characterisation of the image of the Killing operator on pseudo- Hermitian spaces of constant holomorphic sectional curvature by showing that the rst twisted de Rham cohomology group are locally trivial. Even more, we provide several tools to study the rst twisted de Rham cohomology group on product spaces. The last chapter is dedicated to Lorentzian locally symmetric spaces and locally homogeneous plane waves. We prove results on the Singer index of locally homogeneous plane waves and determine exactly which ones have Singer index equal to 0. We make use of this fact to show that the rst twisted de Rham cohomology group of the Killing connections of locally homogeneous plane waves with Singer index 0 is locally trivial. Lastly, we provide a complete characterisation of the image of the Killing operator on Lorentzian locally symmetric spaces, showing in which cases rst twisted de Rham cohomology group of the Killing connection is locally trivial and in which ones it is not.