Homogeneous spin Riemannian manifolds with the simplest Dirac operator

We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them

Cyclic homogeneous Riemannian manifolds

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and we obtain the classification of simply-connected cyclic homogeneous Riemannian manifolds of dimension less than or equal to four. We also present a wide list of examples of non-compact irreducible Riemannian $3$-symmetric spaces admitting cyclic metrics and give the expression of these metrics

Chern-Simons forms associated to homogeneous pseudo-Riemannian structures

10 pages.-- MSC2000 codes: 57R20, 53C30, 53C50.Forms of Chern-Simons type associated to homogeneous pseudo-Riemannian structures are considered. The corresponding secondary classes are a measure of the lack of a homogeneous pseudo-Riemannian space to be locally symmetric. Explicit computations are done for some pseudo-Riemannian Lie groups and their compact quotients.Partially supported by DGICYT, Spain, under grant BFM2002-00141 and by Xunta de Galicia under grant PGIDT01PXI20704PR.Peer reviewe

Reductive decompositions and Einstein-Yang-Mills equations associated to the oscillator group

9 pages.-- PACS nrs.: 04.40.Nr, 11.15.-q, 02.40.Ma.-- MSC1991 codes: 53C30, 53C50, 53C80.-- Issue title: "Relativity and Gravitation".All of the homogeneous Lorentzian structures on the oscillator group equipped with a biinvariant Lorentzian metric, and then the associated reductive pairs, are obtained. Some of them are solutions of the Einstein-Yang-Mills equations.Partially supported by DGES (Spain) under Project PB95-0124, and by Xunta de Galiza, under Project XUGA 20703B98.Peer reviewe