Homogeneous spaces with invariant Koszul-Vinberg structures

Let M be a manifold endowed with a flat torsionless connection ∇. A symmetric bivector field h on (M,∇) is said to be a Koszul-Vinberg (K-V for short) bivector field if it satisfies: (∇α#h)(β,γ)=(∇β#h)(α,γ), where α#:=h#(α) and h#:T⁎M→TM is the contraction of h. The pair (∇,h) is called a K-V structure and the triple (M,∇,h) is called a K-V manifold. When h is non-degenerate, (M,∇,h−1) is a pseudo-Hessian manifold, otherwise, we have a singular foliation defined by Imh# called the associated affine foliation whose leaves are pseudo-Hessian manifolds. In this paper, we study invariant K-V structures on homogeneous spaces. More precisely, we give an algebraic characterization of these structures in the same spirit of Nomizu's theorem on invariant connections on homogeneous spaces, and we give many classes of examples mainly for reductive or symmetric pairs (G,H). We establish many properties of pseudo-Hessian homogeneous manifolds, in particular, we show that when G is a semi-simple Lie group then G/H does not admit any non trivial G-invariant pseudo-Hessian structure. Finally, we show that the leaves of the affine foliation associated to an invariant K-V structure are homogeneous pseudo-Hessian manifolds.