  A contravariant pseudo-Hessian manifold is a manifold $M$ endowed with a pair
$(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric
bivector field satisfying a contravariant Codazzi equation. When $h$ is
invertible we recover the known notion of pseudo-Hessian manifold.
  Contravariant pseudo-Hessian manifolds have properties similar to Poisson
manifolds and, in fact, to any contravariant pseudo-Hessian manifold
$(M,\nabla,h)$ we associate naturally a Poisson tensor on $TM$. We investigate
these properties and we study in details many classes of such structures in
order to highlight the richness of the geometry of these manifolds.
