We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups and in particular, on cotangent bundles of simple Lie groups. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T*G of a Lie group G, will always be endowed with their Lie group structures induced by the right trivialization. We show that TG and T*G are isomorphic if G itself possesses a biinvariant Riemannian or pseudo-Riemannian metric. We also show that, if on a perfect Lie group, there exists a Cartan-Schouten metric, then it must be biinvariant. We compute all such metrics on the cotangent bundles of simple Lie groups. We further show the following. Endowed with their canonical Lie group structures, the set of unit dual quaternions is isomorphic to T*SU(2), the set of unit dual split quaternions is isomorphic to the cotangent bundle of the group of unit split quaternions. The group SE(3) of special rigid displacements of the Euclidean 3-space is isomorphic to T*SO(3). The group SE(2,1) of special rigid displacements of the Minkowski 3-space is isomorphic to T*SO(2,1). So some results on SE(3) by N. Miolane and X. Pennec, and M. Zefran, V. Kumar and C. Croke, are generalized to SE(2,1) and to T*G, for any simple Lie group G.

翻译：我们讨论了完美李群上的Cartan-Schouten度量（即关于Cartan-Schouten典范联络平行的黎曼或伪黎曼度量），特别关注单李群余切丛上的这类度量。该研究在信息几何中具有潜在应用。本文始终假设李群G的切丛TG和余切丛T*G均赋有由右平凡化诱导的李群结构。我们证明：当G本身具有双不变黎曼或伪黎曼度量时，TG与T*G同构；若完美李群上存在Cartan-Schouten度量，则该度量必为双不变度量。我们计算了单李群余切丛上的所有此类度量，并进一步证明：在典范李群结构下，单位对偶四元数集同构于T*SU(2)，单位对偶分裂四元数集同构于单位分裂四元数群的余切丛；欧氏三维空间中的特殊刚体位移群SE(3)同构于T*SO(3)，闵可夫斯基三维空间中的特殊刚体位移群SE(2,1)同构于T*SO(2,1)。由此，N. Miolane、X. Pennec以及M. Zefran、V. Kumar、C. Croke关于SE(3)的若干结论被推广至SE(2,1)及任意单李群G的余切丛T*G。