Invariant f-structures on the flag manifolds SO(n)/SO(2)xSO(n-3)

We consider manifolds of oriented flags SO(n)/SO(2)xSO(n-3) (n>=4) as 4- and 6-symmetric spaces and indicate characteristic conditions for invariant Riemannian metrics under which the canonical f-structures on these homogeneous $\Phi$-spaces belong to the classes Kill f, NKf and G_1f of generalized Hermitian geometry.


Introduction
An important place among homogeneous manifolds is occupied by homogeneous Φ-spaces [8,7] of order k (which are also referred to as k-symmetric spaces [16]), i.e. the homogeneous spaces generated by Lie group automorphisms Φ such that Φ k = id. Each k-symmetric space has an associated object, the commutative algebra A(θ) of canonical affinor structures [6,7]. In its turn, A(θ) contains well-known classical structures, in particular, fstructures in the sense of K.Yano [18]. It should be mentioned that an f -structure compatible with a (pseudo-)Riemannian metric is known to be one of the central objects in the concept of generalized Hermitian geometry [13].
From this point of view it is interesting to consider manifolds of oriented flags of the form (1) SO(n)/SO(2) × SO(n − 3) (n ≥ 4) as they can be generated by automorphisms of any even finite order k ≥ 4. At the same time, it can be proved that an arbitrary invariant Riemannian metric on these manifolds is (up to a positive coefficient) completely determined by the pair of positive numbers (s, t). Therefore, it is natural to try to find characteristic conditions imposed on s and t under which canonical f -structures on homogeneous manifolds (1) belong to the main classes of f -structures in the generalized Hermitian geometry. This question is partly considered in the paper. The paper is organized as follows.
In Section 2, basic notions and results related to homogeneous regular Φspaces and canonical affinor structures on them are collected. In particular, this section includes a precise description of all canonical f -structures on homogeneous k-symmetric spaces.
In Section 3, we dwell on the main concepts of generalized Hermitian geometry and consider the special classes of metric f -structures such as Kill f , NKf , and G 1 f .
In Section 4, we describe manifolds of oriented flags of the form SO(n)/ SO(2) × · · · × SO(2) m ×SO(n − 2m − 1) and construct inner automorphisms by which they can be generated. In Section 5, we describe the action of the canonical f -structures on the flag manifolds of the form (1) considered as homogeneous Φ-spaces of orders 4 and 6.
Finally, in Section 6, we indicate characteristic conditions for invariant Riemannian metrics on the flag manifolds (1) under which the canonical f -structures on these homogeneous Φ-spaces belong to the classes Kill f , NKf, and G 1 f.

Canonical structures on regular Φ-spaces
We start with some basic definitions and results related to homogeneous regular Φ-spaces and canonical affinor structures. More detailed information can be found in [17], [8], [16], [7], [5] and some others.
Let G be a connected Lie group, Φ its automorphism. Denote by G Φ the subgroup consisting of all fixed points of Φ and by G Φ 0 the identity component of G Φ . Suppose a closed subgroup H of G satisfies the condition [8,7].
Among homogeneous Φ-spaces a fundamental role is played by homogeneous Φ-spaces of order k (Φ k = id) or, in the other terminology, homogeneous k-symmetric spaces (see [16]).
Note that there exist homogeneous Φ-spaces that are not reductive. That is why so-called regular Φ-spaces first introduced by N.A.Stepanov [17] are of fundamental importance.
Let G/H be a homogeneous Φ-space, g and h the corresponding Lie algebras for G and H, ϕ = dΦ e the automorphism of g. Consider the linear operator A = ϕ−id and the Fitting decomposition g = g 0 ⊕g 1 with respect to A, where g 0 and g 1 denote 0-and 1-component of the decomposition respectively. Further, let ϕ = ϕ s ϕ u be the Jordan decomposition, where ϕ s and ϕ u is a semisimple and unipotent component of ϕ respectively, ϕ s ϕ u = ϕ u ϕ s . Denote by g γ a subspace of all fixed points for a linear endomorphism γ in g. It is clear that h = g ϕ = Ker A, h ⊂ g 0 , h ⊂ g ϕs . Definition 1 [8,17,7,5]. A homogeneous Φ-space G/H is called a regular Φ-space if one of the following equivalent conditions is satisfied: (3) The restriction of the operator A to Ag is non-singular. (4) A 2 X = 0 =⇒ AX = 0 for all X ∈ g. (5) The matrix of the automorphism ϕ can be represented in the form E 0 0 B , where the matrix B does not admit the eigenvalue 1.
A distinguishing feature of a regular Φ-space G/H is that each such space is reductive, its reductive decomposition being g = h ⊕ Ag (see [17]). g = h ⊕ Ag is commonly referred to as the canonical reductive decomposition corresponding to a regular Φ-space G/H and m = Ag is the canonical reductive complement.
It should be mentioned that any homogeneous Φ-space G/H of order k is regular (see [17]), and, in particular, any k-symmetric space is reductive.
Let us now turn to canonical f -structures on regular Φ-spaces. An affinor structure on a smooth manifold is a tensor field of type (1, 1) realized as a field of endomorphisms acting on its tangent bundle. It is known that any invariant affinor structure F on a homogeneous manifold G/H is completely determined by its value F o at the point o = H, where F o is invariant with respect to Ad(H). For simplicity, further we will not distinguish an invariant structure on G/H and its value at o = H throughout the rest of the paper.
Let us denote by θ the restriction of ϕ to m.
Definition 2 [6,7]. An invariant affinor structure F on a regular Φ-space G/H is called canonical if its value at the point o = H is a polynomial in θ.
Remark that the set A(θ) of all canonical structures on a regular Φ-space G/H is a commutative subalgebra of the algebra A of all invariant affinor structures on G/H. This subalgebra contains well-known classical structures such as almost product structures (P 2 = id), almost complex structures The sets of all canonical structures of the above types were completely described in [6] and [7]. In particular, for homogeneous k-symmetric spaces the precise computational formulae were indicated. For future reference we cite here the result pertinent to f -structures and almost product structures only. Put Theorem 1 [6,7]. Let G/H be a homogeneous Φ-space of order k (k ≥ 3). 1) All non-trivial canonical f -structures on G/H can be given by the operators where ζ j ∈ {1, 0, −1}, j = 1, 2, . . . , u, and not all ζ j are equal to zero. 2) All canonical almost product structures P on G/H can be given by Here the numbers ξ j , j = 1, 2, . . . , u, take their values from the set {−1, 1}.
The results mentioned above were particularized for homogeneous Φspaces of smaller orders 3, 4, and 5 (see [6,7]). Note that there are no fundamental obstructions to considering of higher orders k. Specifically, for future consideration we need the description of canonical f -structures and almost product structures on homogeneous Φ-spaces of orders 4 and 6 only.
Corollary 1 [6,7]. Any homogeneous Φ-space of order 4 admits (up to sign) the only canonical f -structure and the only almost product structure Corollary 2. On any homogeneous Φ-space of order 6 there exist (up to sign) only the following canonical f -structures: and only the following almost product structures:

Some important classes in generalized Hermitian geometry
The concept of generalized Hermitian geometry created in the 1980s (see [13]) is a natural consequence of the development of Hermitian geometry. One of its central objects is a metric f -structure, i.e. an f -structure compatible with a (pseudo-)Riemannian metric g = ·, · in the following sense: Evidently, this concept is a generalization of one of the fundamental notions in Hermitian geometry, namely, almost Hermitian structure J. It is also worth noticing that the main classes of generalized Hermitian geometry (see [13,11,12,5,4]) in the special case f = J coincide with those of Hermitian geometry (see [10]).
In what follows, we will mainly concentrate on the classes Kill f , NKf, and G 1 f of metric f -structures defined below.
A fundamental role in generalized Hermitian geometry is played by a tensor T of type (2,1) which is called a composition tensor [13]. In [13] it was also shown that such a tensor exists on any metric f -manifold and it is possible to evaluate it explicitly: where ∇ is the Levi-Civita connection of a (pseudo-)Riemannian manifold (M, g), X, Y ∈ X(M ). The structure of a so-called adjoint Q-algebra (see [13]) on X(M ) can be defined by the formula X * Y = T (X, Y ). It gives the opportunity to introduce some classes of metric f -structures in terms of natural properties of the adjoint Q-algebra.
f is a Killing tensor) (see [11,12]). The class of Killing f -structures is denoted by Kill f . The defining property of nearly Kähler f -structures (or This class of metric f -structures, which is denoted by NKf , was determined in [4] (see also [1,2]). It is easy to see that for f = J the classes Kill f and NKf coincide with the well-known class NK of nearly Kähler structures [9].
The following relations between the classes mentioned are evident: A special attention should be paid to the particular case of naturally reductive spaces. Recall that a homogeneous Riemannian manifold (G/H, g) is known to be a naturally reductive space [14] with respect to the reductive It should be mentioned that if G/H is a regular Φ-space, G a semisimple Lie group then G/H is a naturally reductive space with respect to the (pseudo-)Riemannian metric g induced by the Killing form of the Lie algebra g (see [17]). In [1], [2], [3] and [4] a number of results helpful in checking whether the particular f -structure on a naturally reductive space belongs to the main classes of generalized Hermitian geometry was obtained.
We call a flag L i 1 ⊂ L i 2 ⊂ · · · ⊂ L in (here and below the subscript denotes the dimension of the subspace) oriented if for any L i j and its two basises {e 1 , . . . , e i j } and {e ′ 1 , . . . , e ′ i j } det A > 0, where e ′ t = Ae t for any t = 1, . . . , i j . Moreover, for any two subspaces L i k ⊂ L i j their orientations should be set in accordance.
We need to prove that the group of all fixed points G Φ satisfies the condition According to Corollary 1 the only canonical f -structure on this homogeneous Φ-space is determined by the formula Its action can be written in the form: Taking Corollary 2 into account we can represent the action of the canonical f -structures on this homogeneous Φ-space as follows:   Proof. The explicit form of the reductive complement of (1) was indicated in Section 5. Put Since SO ( In what follows we are going to represent any H ∈ h in the form  H 1 (a 1 a 2 This contradicts our assumption. Continuing the same line of reasoning, we see that neither m 1 nor m 2 can be decomposed into the sum of Ad(H)-invariant summands.
It is not difficult to check that the space in question possesses the following property.
Denote by g 0 the naturally reductive metric generated by the Killing form B: g 0 = −B| m×m . In our case B = −(n − 1) Tr X T Y , X, Y ∈ so(n).
Proof. For the explicit form of m and h see Section 5 and Section 6. It can easily be seen that for any X ∈ m, Y ∈ h Tr X T Y = 0. It should also be noted that it was proved in [17] that h is orthogonal to m with respect to B.
For any almost product structure P put Suppose that P is compatible with g 0 , i.e. g 0 (X, Y ) = g 0 (P X, P Y ) (for example, this is true for any canonical almost product structure P [5]). Clearly, m − and m + are orthogonal with respect to g 0 , since for any Let us consider the action of the canonical almost product structures on the 6-symmetric space (1). Here we use notations of Corollary 2.
It can be deduced from Proposition 3 and Proposition 5 that any invariant Riemannian metric g on (1) is (up to a positive coefficient) uniquely defined by the two positive numbers (s, t). It means that Definition 3. (s, t) are called the characteristic numbers of the metric (5).
It should be pointed out that the canonical f -structures on the homogeneous Φ-space (1) of the orders 4 and 6 are metric f -structures with respect to all invariant Riemannian metrics, which is proved by direct calculations.
Recall that in case of an arbitrary Riemannian metric g the Levi-Civita connection has its Nomizu function defined by the formula (see [14]) where X, Y ∈ m, the symmetric bilinear mapping U is determined by means of the formula Suppose g is an invariant Riemannian metric on the homogeneous Φ-space (1) with the characteristic numbers (s, t) (s, t > 0). The following statement is true.
Outline of the proof. First we apply (5) and the definition of g 0 to (7). We take four matrices X = {x i j }, Y = {y i j }, Z = {z i j } and U = {u i j } and calculate the right-hand and left-hand side of the equality obtained. After that we can represent it in the form where c 1 2 , c 1 3 , c 1 i , c 2 i , c 3 i (i = 1, . . . , n) depend on elements of the matrices X, Y and U . As (9) holds for any Z ∈ m, it follows in the standard way that (10) c 1 2 = c 1 3 = c 1 i = c 2 i = c 3 i = 0, (i = 1, . . . , n).
Using (10), we calculate u i j = u i j (X, Y ). To conclude the proof, it remains to transform the formula for U (X, Y ) into (8), which is quite simple.
In the notations of Section 2 we have the following statement. Theorem 2. Consider SO(n)/SO(2) × SO(n − 3) as a 4-symmetric Φspace. Then the only canonical f -structure f 0 on this space is 1) a Killing f -structure iff the characteristic numbers of a Riemannian metric are (1, 4 3 ); 2) a nearly Kähler f -structure iff the characteristic numbers of a Riemannian metric are (1, t), t > 0; 3) a G 1 f -structure with respect to any invariant Riemannian metric.
Proof. Application of (6) to the definitions of the classes Kill f , NKf and G 1 f yields that The proof is straightforward. For example, it is known that f 0 is a nearly Kähler f -structure in the naturally reductive case, which means that 1 2 [f 0 X, f 2 0 X] m = 0 for any X ∈ m (see [4]). Making use of Proposition 4 and Proposition 6, we obtain U (f 0 X, f 0 X) ∈ Ker f 0 for any X ∈ m, U (f 0 X, f 2 0 X) = 0 for any X ∈ m iff s = 1. Thus we have 2). Other statements are proved in the same manner.
The similar technique is used to prove Theorem 3. Consider SO(n)/SO(2)×SO(n−3) as a 6-symmetric space. Let (s, t) be the characteristic numbers of an invariant Riemannian metric. Then 1) f 1 is a Killing f -structure iff s = 1, t = 4 3 ; f 2 , f 3 , f 4 do not belong to Kill f for any s and t.
2) f 1 is an N Kf -structure iff s=1; f 2 and f 3 are N Kf -structures for any s and t; f 4 is not an N Kf -structure for any s and t. 3) f 1 , f 2 , f 3 , f 4 are G 1 f -structures for any s and t.