The classification of naturally reductive homogeneous spaces in dimensions n≤6

Abstract

We present a new method for classifying naturally reductive homogeneous spaces -i. e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions 3, 4, and 5, and yields as a new result the classification in dimension 6. In each dimension, one obtains a 'hierarchy' of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a 'generic' parallel torsion form defines a quasi-Sasakian structure in dimension 5 and an almost complex structure in dimension 6. within the priority programme 1388 "Representation theory". Ana Ferreira thanks Philipps-Universität Marburg for its hospitality during a research stay in May-July 2013, and she also acknowledges partial financial support by the FCT through the project PTDC/MAT/118682/2010 and the University of Minho through the FCT project PEst-C/MAT/UI0013/2011. We also thank Andrew Swann (Aarhus) for discussions on Section 4 during a research visit to Marburg in June 2013 and Simon G. Chiossi (Marburg) for many valuable comments on a preliminary version of this work. Metric connections with skew torsionConsider a Riemannian manifold (M n , g). The difference between its Levi-Civita connection ∇ g and any linear connection ∇ is a (2, 1)-tensor field A,The curvature of ∇ resp. ∇ g will always be denoted by R resp. R g . Following Cartan, we study the algebraic types of the torsion tensor for a metric connection. Denote by the same symbol the (3, 0)-tensor derived from a (2, 1)-tensor by contraction with the metric. We identify T M n with (T M n ) * using g from now on. Let T be the n 2 (n − 1)/2-dimensional space of all possible torsion tensors,A connection ∇ is metric if and only if A belongs to the spaceThe spaces T and A are isomorphic as O(n) representations, reflecting the fact that metric connections can be uniquely characterized by their torsion. For n ≥ 3, they split under the action of O(n) into the sum of three irreducible representations,

Integrable properties of σ-models with non-symmetric target spaces

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On the structure of geodesic orbit Riemannian spaces

“…Since B is negatively definite on h, m is a complement to h in g. Note also that n(g) ⊂ m due to (2). Advantages of this reductive decomposition follow from the next lemma.…”

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“…The naturally reductive connection is here also used to find new examples of generalized Killing spinors. More examples of this phenomena are presented in [1]. Naturally reductive spaces have also been used to find new homogeneous Einstein metrics.…”