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Naturally reductive metrics on homogeneous systems. (English) Zbl 0546.17002
An invariant form on the Lie triple algebra $${\mathcal G}$$ is characterized as a symmetric bilinear form invariant with respect to the left multiplications and inner derivations of $${\mathcal G}$$. Using this fact, the author shows that the naturally reductive metrics on homogeneous systems are determined by nondegenerate invariant forms of their tangent Lie triple algebras and obtains the de Rham decomposition of a simply connected regular homogeneous system G, which does not depend on the choice of the metric. The Riemannian manifold G is irreducible if and only if the tangent Lie triple algebra is simple.
Reviewer: K.Yamaguti

##### MSC:
 17A40 Ternary compositions 53C30 Differential geometry of homogeneous manifolds