Structure theory of naturally reductive spaces

Abstract

The main result of this paper is that every naturally reductive space can be explicitly constructed from the construction in [23]. This gives us a general formula for any naturally reductive space and from this we prove reducibility and isomorphism criteria.1991 Mathematics Subject Classification. Primary 53C30, Secondary 53C10.

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

“…Remark 2.6. In [15] it is shown that every naturally reductive pair of type I admits exactly one compact partial dual pair of type I. Moreover, for spaces of type II there exists exactly one partial dual pair for which the semisimple part of the canonical base space is compact.…”

“…Such a parametrization breaks down in higher dimensions. The recent developments in [14] and [15] tell us however that naturally reductive spaces are still very rigid. This gives us the ability to present here a completely new way to classify naturally reductive spaces.…”

“…This will be carried out explicitly for naturally reductive spaces of dimension 7 and 8. An important point is the division of naturally reductive spaces into two types as in [15]:…”

“…Type II: The transvection algebra is not semisimple. Another simplification of the classification comes from the partial duality of naturally reductive spaces defined in [15]. This makes the classification much more transparent.…”

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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.

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