The Classification of 7- and 8-dimensional Naturally Reductive Spaces

Abstract

A new method for classifying naturally reductive spaces is presented. This method relies on the structure theory of naturally reductive spaces developed in [15] and the new construction of naturally reductive spaces in [14]. We obtain the classification of all naturally reductive spaces in dimension 7 and 8.1991 Mathematics Subject Classification. Primary 53C30, Secondary 53B20.

Jacobi relations on naturally reductive homogeneous spaces

“…It remains to show that every naturally reductive G 2 -manifold is normal by comparing the list of homogeneous nearly parallel G 2 -manifolds from [19,Ch. 4] with the classification of seven-dimensional naturally reductive structures given in [46]. 44,45,46]) Let (M, g, ∇) be a compact simply connected seven-dimensional naturally reductive space with Ambrose-Singer connection ∇.…”

“…4] with the classification of seven-dimensional naturally reductive structures given in [46]. 44,45,46]) Let (M, g, ∇) be a compact simply connected seven-dimensional naturally reductive space with Ambrose-Singer connection ∇. Suppose that the Lie algebra g of the connected subgroup G ⊂ I 0 (M ) generated by the transvections is not semisimple.…”

“…Proof. We compare the list of seven-dimensional naturally reductive spaces from [46] with the list homogeneous nearly parallel G 2 -manifolds from [19,Ch. 4].…”

“…4]. For this we can restrict us to the naturally reductive structures of semisimple type from [46,Ch. 3.1].…”

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

“…The most important subclass of g.o. spaces are the naturally reductive spaces, whose complete description is also open (see the recent low-dimensional classifications [1], [26]). Another related topic of recent interest is the study of Einstein metrics that are not g.o.…”