Reduction of Homogeneous Riemannian Structures

The goal of this paper is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group H of isometries. In a first result, H is a normal subgroup of the group of symmetries associated with the reducing tensor . The situation when H is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fibrings of almost contact manifolds over almost Hermitian manifolds. If the structure is, moreover, Sasakian, the obtained reduced tensor is homogeneous Kähler.