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TitleThe classification of naturally reductive homogeneous spaces in dimensions n≤6
Author(s)Ferreira, Ana Cristina
Agricola, Ilka
Friedrich, Thomas
Keywordsnaturally reductive homogeneous space
connection with skew torsion
characteristic connection
holonomy algebra
quasi-Sasakian manifold
almost complex manifold
Issue dateApr-2015
JournalDifferential Geometry and its Applications
Abstract(s)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.
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AccessRestricted access (UMinho)
Appears in Collections:CMAT - Artigos em revistas com arbitragem / Papers in peer review journals

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