Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups

Details Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups

2009-11-20

arxiv.org/abs/0911.4067v2

Mathematics

Differential Geometry

Corrected typos; added references and the corresponding naturally reductive homogeneous structures are found for the Lie group

Scientific paper

This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved here that the simply connected Lie group $N$ can be constructed starting from a real representation $(\pi,\vv)$ of a certain Lie algebra $\ggo$. We study the geometry of $(N, \la \,,\,\ra_N)$ and we find the corresponding naturally reductive homogeneous structure. On the other hand, related to the case of degenerate center we provide another family of naturally reductive spaces, both non compact and also compact examples.

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