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Invariant structures on homogeneous Φ-spaces of order 5

Let Φ be an analytic automorphism of order k (Φ* = id) of a connected Lie group G, and Η a closed subgroup of G such that Go C Η C G®, where Go is the connected component of the identity in the subgroup G* of points fixed under Φ. Homogeneous spaces G/H of this sort containing symmetric spaces were first introduced by Stepanov in [1], and they are called homogeneous Φ-spaces of order k. Such spaces occur [1] in the wider class of homogeneous regular Φ-spaces, which are reductive [1]. For odd k, it is well known (see [2], for example) that there is an invariant almost complex structure / on G/H; its geometry has been studied in greatest detail for k = 3 (see [3], [4]). We note that this is stipulated by the explicit description of J when k = 3 (see [3]) in terms of the automorphism φ = άΦ, of the Lie algebra a, of G. For k = 5, we shall construct here an invariant almost product structure C on G/H; integrability conditions for it are given, and we investigate its consonance with other invariant structures on this space. We note that the structure expounded here enables us to look at the constructions contained in [5] —[7] from a unified point of view. Let the canonical reductive decomposition [1] the regular Φ-space G/H. the of automorphism of G commuting it [8]