A contravariant pseudo-Hessian manifold is a manifold M endowed with a pair $(\mathrm{\nabla },h)$ where ∇ 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 $(M,\mathrm{\nabla },h)$ 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.

中文翻译：

---

逆变伪Hessian流形及其相关的泊松结构

逆伪Hessian流形是赋有一对的流形M$（\mathrm{\nabla }，H）$其中∇是平坦连接，h是满足对变Codazzi方程的对称双矢量场。当h是可逆的时，我们恢复了伪Hessian流形的已知概念。逆伪Hessian流形具有与泊松流形相似的性质，并且实际上与任何逆伪Hessian流形相似$（\mathrm{中号}，\mathrm{\nabla }，H）$我们自然将TM上的泊松张量关联起来。我们调查这些特性，并详细研究此类结构的许多类，以突出这些歧管的几何形状的丰富性。