Flag manifolds, symmetric $\fr{t}$-triples and Einstein metrics

Details Flag manifolds, symmetric $\fr{t}$-triples and Einstein metrics Flag manifolds, symmetric $\fr{t}$-triples and Einstein metrics

2010-10-19

arxiv.org/abs/1010.3992v2

Mathematics

Differential Geometry

24 pages

Scientific paper

We study the structure of a generalized flag manifold $M=G/K$ of a compact connected simple Lie group $G$ by using its $\fr{t}$-root system $R_{\fr{t}}$. We show that an important invariant associated to the isotropy decomposition of $M$, namely the de Siebenthal's invariant of $M=G/K$, can be expressed in terms of $\fr{t}$-roots. Motivated by this result we introduce the notion of symmetric $\fr{t}$-triples and we study their properties. We describe an application of these triples on the structure constants of $M=G/K$, quantities which are straightforward related to the construction of the homogeneous Einstein metric on $M=G/K$. Next we give the general form of symmetric $\fr{t}$-triples for two important classes of generalized flag manifolds $M=G/K$, those with second Betti number $b_{2}(M)=1$, and those with $b_{2}(M)=\ell=\rnk G$, i.e. the full flag manifolds $M=G/T$, where $T$ is a maximal torus of $G$. In the last section, we use the theory of symmetric $\fr{t}$-triples to construct the homogeneous Einstein equation on flag manifolds $G/K$ with five isotropy summands, determined by the simple Lie group $G=SO(7)$. By solving the corresponding polynomial system we get all SO(7)-invariant Einstein metrics, and these are the very first results towards the classification of homogeneous Einstein metrics on flag manifolds with five isotropy summands. We examine also the isometry problem for these metrics.

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