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. Applications are foreseen in Information Geometry. Throughout this work, the tangent bundle TG and the cotangent bundle T*G of a Lie group G, are always endowed with their Lie group structures induced by the right trivialization. We show that TG and T*G are isomorphic if G 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 TSU(2), the set of unit dual split quaternions is isomorphic to T*SL(2,R). 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). 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：摘要：本文讨论了完美李群上的Cartan-Schouten度量（相对于Cartan-Schouten标准联络平行的黎曼或伪黎曼度量）。其应用前景出现在信息几何领域。全文始终假定李群G的切丛TG和余切丛T*G均赋予由右平凡化诱导的李群结构。我们证明：若G具有双不变黎曼或伪黎曼度量，则TG与T*G同构。同时表明：若完美李群上存在Cartan-Schouten度量，则该度量必为双不变的。我们计算了单李群余切丛上的所有此类度量。进一步证明：在标准李群结构下，单位对偶四元数集合同构于TSU(2)，单位对偶分裂四元数集合同构于T*SL(2,R)；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)。我们将N. Miolane与X. Pennec、以及M. Zefran、V. Kumar和C. Croke关于SE(3)的部分结论推广至SE(2,1)及任意单李群G的T*G情形。