An affine connection is said to be flat if its curvature tensor vanishes identically. Koszul-Vinberg (KV for abbreviation) cohomology has been invoked to study the deformation theory of flat and torsion-free affine connections on tangent bundle. In this Note, we compute explicitly the differentials of various specific KV cochains, and study their relation to classical objects in information geometry, including deformations associated with projective and dual-projective transformations of a flat and torsion-free affine connection. As an application, we also give a simple yet non-trivial example of a KV algebra of which second cohomology group does not vanish.

翻译：仿射联络若其曲率张量恒为零，则称为平坦联络。Koszul-Vinberg（简称KV）上同调已被用于研究切丛上平坦无挠仿射联络的形变理论。本注释中，我们显式计算了若干特定KV上链的微分算子，并研究了它们与信息几何中经典对象的关系，包括与平坦无挠仿射联络的射影变换及对偶射影变换相关的形变。作为应用，我们给出了一个简单却非平凡的KV代数实例，其第二上同调群非零。