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Homogeneous Geodesies in Generalized Wallach Spaces.

Introduction

Let (M, g) be a homogeneous Riemannian manifold, i.e. a connected Riemannian manifold on which the largest connected group G of isometries acts transitively. Then M can be expressed as a homogeneous space ( G/K, g) where K is the isotropy group at a fixed point o of M, and g is a G-invariant metric. In this case the Lie algebra g of G has an Ad(K)-invariant decomposition g = [??] [direct sum] m, where m [subset] g is a linear subspace of g and [??] is the Lie algebra of K. In general such decomposition is not unique. The Ad(K)-invariant subspace m can be naturally identified with the tangent space [T.sub.o]M via the projection [pi] : G [right arrow] G/K.

A geodesic [gamma](t) through the origin o of M = G/K is called homogeneous if it is an orbit of a one-parameter subgroup of G, that is

[gamma](t)= exp(tX)(o), t [member of] R, (1)

where X is a non zero vector of g.

A homogeneous Riemannian manifold is called a g.o. space, if all geodesies are homogeneous with respect to the largest connected group of isometries. All naturally reductive spaces are g.o. spaces ([15]), but the converse is not true in general. In [13] A. Kaplan proved the existence of g.o. spaces that are in no way naturally reductive. These are generalized Heisenberg groups with two-dimensional center. In [16] O. Kowalski, F. Prufer and L. Vanhecke made an explicit classification of all naturally reductive spaces up to dimension five. In [18] O. Kowalski and L. Vanhecke gave a classification of all g.o. spaces, which are in no way naturally reductive, up to dimension six. In [11] C. Gordon described g.o. spaces which are nilmanifolds and in [29] H. Tamaru classified homogeneous g.o. spaces which are fibered over irreducible symmetric spaces. In [6] and [7] O. Kowalski and Z. Dusek investigated homogeneous geodesics in Heisenberh groups and some H-type groups. Examples of g.o. spaces in dimension seven were obtained by Dusek, O. Kowalski and S. Nikcevic in ([9]). Also, in [1] the first author and D.V. Alekseevsky classified generalized flag manifolds which are g.o. spaces.

Concerning the existence of homogeneous geodesics in homogeneous Riemannian manifold, we recall the following. V.V. Kajzer proved that a Lie group endowed with a left-invariant metric admits at least one homogeneous geodesic ([12]). O. Kowalski and J. Szenthe extended this result to all homogeneous Riemannian manifolds ([17]). An extension of the result of [17] to reductive homogeneous pseudo-Riemannian manifolds has been also obtained ([7], [24]). Also, O. Kowalski, S. Nikcevic and Z. Vlasek studied homogeneous geodesics in homogeneous Riemannian manifolds ([14]), and G. Calvaruso and R. Marinosci studied homogeneous geodesics in three-dimension Lie groups ([21], [5]). Homogeneous geodesics were studied by J. Szenthe ([25], [26], [27], [28]). In addition, D. Latifi studied homogeneous geodesics in homogeneous Finsler spaces ([19]), and the first author investigated homogeneous geodesics in the flag manifold SO(2l + 1)/U(l - m) x SO(2m + 1) ([3]).

Homogeneous geodesics in the affine setting were studied in [10]. Finally, D.V. Alekseevsky and Yu. G. Nikonorov in [2] studied the structure of compact g.o. spaces and gave some sufficient conditions for existence and non existence of an invariant metric with homogeneous geodesics on a homogeneous space of a compact Lie group G. They also gave a classification of compact simply connected g.o. spaces of positive Euler characteristic.

Because of these results, it is natural to study g.o. spaces as well as to describe homogeneous geodesics for other large classes of homogeneous spaces. In this paper, we study this problem for generalized Wallach spaces. These spaces were well known before as three-locally-symmetric spaces ([20]), however they were recently classified by Z. Chen, Y. Kang and K. Liang ([4]) and Yu.G. Nikonorov ([23]). We search for homogeneous geodesics in these spaces.

One of the main results in the present paper is Theorem 2, which classifies generalized Wallach spaces which are g.o. spaces. For those which are not, we show how to obtain homogeneous geodesics. We make explicit computations for the three dimensional Lie group SU(2) (thus recovering a result of R.A. Marinosci), and for the Stiefel manifold SO(4)/SO(2).

The paper is organized as follows: In Section 1 we recall the basic definitions and properties of homogeneous geodesics in a Riemannian manifold. In Section 2 we recall the definition of generalized Wallach spaces as well as their classification from [23]. In Section 3 we classify g.o. spaces among generalized Wallach spaces. For those which are not g.o. spaces, in Section 4 we discuss how to find all homogeneous geodesics for a given G-invariant Riemannian metric.

Acknowledgements. The authors express their thanks to the referee for his suggestion to improve the proof of Theorem 2. The first author was supported by Grant # E.037 from the research committee of the University of Patras.

The second author is supported by NSFC 11501390, by Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing 2014QZJ03, by funding of Sichuan University of Science and Engineering grant 2014PY06, 2015RC10, and by funding of opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (Grant No.2016WYJ04).

The second author also acknowledges the hospitality of the University of Patras, during her visit 2015-2016.

1 Homogeneous geodesies in homogeneous Riemannian manifolds

Let (M = G/K, g) be a homogeneous Riemannian manifold, where G is compact and semisimple. Let g and [??] be the Lie algebras of G and K respectively and let

g = [??] [direct sum] m (2)

be a reductive decomposition. The canonical projection [pi] : G [right arrow] G/K induces an isomorphism between the subspace m and the tangent space [T.sub.o]M at the identity o = eK. The G-invariant metric g induces a scalar product <*, *> on m which is Ad(K)-invariant. Let B(*, *) = --Killing form on g. Then any Ad(K)-invariant scalar product <*, *> on m can be expressed as <x,y> = B([LAMBDA]x, y) (x,y [member of] m), where [LAMBDA] is an Ad(K)-equivariant positive definite symmetric operator on m. Conversely, any such operator [LAMBDA] determines an Ad(K)-invariant scalar product <x, y> = B([LAMBDA]x, y) on m, which in turn determines a G-invariant Riemannian metric g on m. We say that [LAMBDA] is the operator associated to the metric g, or simply the associated operator. Also, a Riemannian metric generated by inner product B(*, *) is called standard metric.

Definition 1. A nonzero vector X [member of] g is called a geodesic vector if the curve (1) is a geodesic.

Lemma 1 ([18]). A nonzero vector X [member of] g is a geodesic vector if and only if

<[[X, Y].sub.m], [X.sub.m]> = 0, (3)

for all Y [member of] m. Here the subscript m denotes the projection into m.

A useful description of homogeneous geodesics (1) is provided by the following :

Proposition 1 ([1]). Let ( M = G/K, g) be a homogeneous Riemannian manifold and [LAMBDA] be the associated operator. Let a [member of] [??] and x [member of] m. Then the following are equivalent:

(1) The orbit [gamma](t) = expt(a + x) * o of the one-parameter subgroup expt(a + x) through the point o = eK is a geodesic of M.

(2) [a + x, [LAMBDA]x] [member of] [??].

(3) <[a, x], y> = <x, [[x, y].sub.m]> for ally [subset] m.

(4) <[[a + x,y].sub.m], x> = 0 for all y [subset] m.

An important corollary of Proposition 1 is the following:

Corollary 1 ([1]). Let (M = G/K, g) be a homogeneous Riemannian manifold. Then (M = G/K, g) is a g.o. space if and only if for every x [subset] m there exists an a(x) [subset] [??] such that

[a(x) + x, [LAMBDA]x] [subset] [??].

The following Proposition is very important for us to investigate g.o. spaces in generalized Wallach spaces.

Proposition 2. ([2, Proposition 5]) Let (M = G/K, g) be a compact g.o. space with associated operator [LAMBDA]. Let X, Y [subset] m be eigenvectors [LAMBDA] with different eigenvalues [lambda], [mu]. Then

[X, Y]=-[[lambda]/[lambda] - [mu]][h,X] + [[mu]/[lambda] - [mu]][h,Y] (4)

for some h [member of] [??].

If we want to decide whether a homogeneous Riemannian manifold (M = G/K,g) is a g.o. space or not, we need to find a decomposition of the form (2) and look for geodesic vectors of the form

X = [s.summation over (i=1)][a.sub.i][e.sub.i] + [l.summation over (j=1)] [x.sub.j][A.sub.j]. (5)

Here {[e.sub.i] : i = 1,2, ..., s} is a basis of m and {[A.sub.j] : j = 1,2, ..., l} is a basis of [??]. By substituting X into equation (3) we obtain a system of linear algebraic equations for the parameters xj. If for any X [member of] m \ {0} this system for the variables [x.sub.j] has real solutions, then it follows that the homogeneous Riemannian manifold (M = G/K, g) is a g.o. space.

If, on the other hand, we need to find all homogeneous geodesics in the homogeneous Riemannian manifold (M = G/K, g), then we have to calculate all geodesic vectors in the Lie algebra g. Condition (3) reduces to a system of s quadratic equations for the variables [x.sub.j] and [a.sub.i]. Then the geodesic vectors correspond to those solutions of this system for the variables [x.sub.1], ..., [x.sub.l], [a.sub.1], ..., [a.sub.s], which are not all equal to zero.

2 Generalized Wallach spaces

Let G/K be a compact homogeneous space with connected compact semisimple Lie group G and a compact subgroup K. Denoted by g and [??] Lie algebras of G and K respectively. We assume that G/K is almost effective, i.e., there are no non-trivial ideals of the Lie algebra g in [??] C g. Let <*, *> = B([LAMBDA]*, *) be an Ad(K)-invariant scalar product on m, where [LAMBDA] is the associated operator.

We assume that the homogeneous space G/K has the following property. The module m decomposes into a direct sum of three Ad(K)-invariant irreducible modules pairwise orthogonal with respect to B, i.e.

m = [m.sub.1] [direct sum] [m.sub.2] [direct sum] [m.sub.3], (6)

such that

[[m.sub.i], [m.sub.i]] [subset] [??] i = 1,2,3. (7)

A homogeneous space with this property is called generalized Wallach space.

Some examples of these spaces are the manifolds of complete flags in the complex, quaternionic, and Cayley projective planes, that is SU(3)/[T.sub.max], Sp(3)/Sp(1) x Sp(1) x Sp(1), [F.sub.4]/Spin(8) (known as Wallach spaces), and the generalized flag manifolds SU([n.sub.1] + [n.sub.2] + [n.sub.3])/S(U([n.sub.1]) x U([n.sub.2]) x U([n.sub.3])), SO(2n)/U(1) x U(n - 1) and [E.sub.6]/U(1) x U(1) x SO(8).

Every generalized Wallach space admits a three parameter family of invariant Riemannian metrics determined by Ad( K) -invariant inner products

<*, *> = [[lambda].sub.1] B(-, *) |[m.sub.1] +[[lambda].sub.2]B(*, *) |[m.sub.2] +[[lambda].sub.3]B(*, *) |[m.sub.3], (8)

where [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3] are positive real numbers.

Let [d.sub.i] be the dimension of [m.sub.i]. Let {[e.sup.j.sub.i]} be an orthogonal basis of [m.sub.j] with respect to B, where j = 1,2,3 and 1 [less than or equal to] i [less than or equal to] [d.sub.j]. Consider the expression [ijk] defined by the equality

[ijk] = [summation over ([alpha],[beta],[gamma])]B[([[e.sup.i.sub.[alpha]], [e.sup.j.sub.[beta]]], [e.sup.k.sub.[gamma]]).sup.2], (9)

where [alpha], [beta], [gamma] range from 1 to [d.sub.i], [d.sub.j] and [d.sub.k] respectively (cf. [30]). The symbols [ijk] are symmetric in all three indices due to the bi-invariance of the metric B. Moreover, for the spaces under consideration we have [ijk] = 0, if two induces coincide.

We recall the classification of generalized Wallach spaces that was recently obtained by Yu.G. Nikoronov ([23]) and Z. Chen, Y. Kang, K. Liang ([4]):

Theorem 1 ([23], [4]). Let G/K be a connected and simply connected compact homogeneous space. Then G/K is a generalized Wallach space if and only if one of the following types:

1) G/K is a direct product of three irreducible symmetric spaces of compact type ([ijk] = 0 in this case).

2) The group is simple and the pair (g, k) is one of the pairs in Table 1.

3) G = F x F x F x F and H = diag(F) C Gfor some connected simple connected compact simple Lie group F, with the following description on the Lie algebra level:

(g,[??]) = (f [direct sum] f [direct sum] f [direct sum]f, diag(f) = {(X,X,X,X) | X [member of] f},

where f is the Lie algebra of F, and (up to permutation) [m.sub.1] = {(X,X, -X, -X) | X [member of] f}, [m.sub.2] = {(X, -X, X, -X) | X [member of] f}, [m.sub.3] = {(X, -X, -X, X) | X [member of] f}.

3 g.o. generalized Wallach spaces

Let ( G/K, g) be a generalized Wallach space with reductive decomposition g = [??] [direct sum] m, equipped with a G-invariant metric corresponding to a scalar product of the form (8). Let l = dim[??] and [d.sub.i] = dim([m.sub.i]) (i = 1,2,3), and let {[e.sup.0.sub.i]} and {[e.sup.j.sub.s]} be orthogonal bases of [??] and [m.sub.j] respectively with respect to B, where 1 [less than or equal to] i [less than or equal to] l, j = 1,2,3 and 1 [less than or equal to] s [less than or equal to] [d.sub.j]. For any X [member of] g \ {0} we write

[mathematical expression not reproducible] (10)

Then by Lemma 1 it follows that X is a geodesic vector if and only if

[mathematical expression not reproducible] (11)

for all Y [member of] m. Hence we obtain the following system of [d.sub.1] + [d.sub.2] + [d.sub.3] equations

[mathematical expression not reproducible] (12)

If we set

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

then System (12) is equivalent to AX = B. Then we have the following:

Proposition 3. Let (G/K, g) be a generalized Wallach space, where the metric g is determined by the scalar product (8). Then for any [a.sub.j] (j = 1,..., [d.sub.1]), [b.sub.k] (k = 1, ..., [d.sub.2]) and [c.sub.s] (s = 1, ..., [d.sub.3]) not all equal to zero, (G/K,g) is a g.o. space if and only if rank( A) = rank( A, B).

Proof. By Proposition 1 and Corollary 1 it follows that (G/K,g) is a g.o. space if and only if for any X [member of] m \ {0} there exists an a(X) such that the curve expt(a(X) + X) * o (t [member of] R) is a geodesic. This means that system (12) of [d.sub.1] + [d.sub.2] + [d.sub.3] equations for the variables [x.sub.i (i = 1, ..., l) has real solutions for any [a.sub.j], [b.sub.k], [c.sub.s] (j = 1, ..., [d.sub.1], k = 1, ..., [d.sub.2], s = 1, ..., [d.sub.3]) not all equal to zero. Then system AX = B has real solutions if and only if rank (A) = rank (A, B).

Next, we shall investigate which of the families of spaces listed in Theorem 1 are g.o. spaces.

Theorem 2. Let (G/K, g) be a generalized Wallach space as listed in Theorem 1. Then

1) If (G/K, g) is a space of type 1) then this is a g.o. space for any Ad(K) -invariant Riemannian metric.

2) If (G/K, g) is a space of type 2) or 3) then this is a g.o. space if and only if g is the standard metric.

Proof. Assume that (G/K,g) is a g.o. space, and the corresponding metric ([[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3]) is geodesic orbit with [[lambda].sub.i] [not equal to] [[lambda].sub.j]. Proposition 2 implies that [[m.sub.i], [m.sub.j]] [subset] [m.sub.i] [direct sum] [m.sub.j]. Since (G/K,g) is a generalized Wallach space, it follows that [[m.sub.i], [m.sub.j]] [subset] [m.sub.k], k [??] {i, j}, hence we get [[m.sub.i], [m.sub.j]] = 0.

For case 1) it is [ijk] = 0 for all i, j, k [member of] {1,2,3}. Indeed, formula (9) implies that B([[e.sup.i.sub.[alpha]], [e.sup.j.sub.[beta]]],[e.sup.k.sub.[gamma]]) = 0 for any i, j,k [member of] {1,2,3} and for any [alpha],[beta],[gamma] ranging from 1 to [d.sub.i], [d.sub.j] and [d.sub.k] respectively. Hence [[e.sup.i.sub.[alpha]], [e.sup.j.sub.[beta]]] = 0 for i [not equal to] j and for any [alpha], [beta] ranging from 1 to [d.sub.i] and [d.sub.j]. Therefore, a generalized Wallach space (G/K, g) of type 1) is naturally reductive for any Ad(K)-invariant Riemannian metric ([[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3]), so it is a g.o. space.

For case 2) the only non zero [ijk] is [123] (cf. [22]). Then formula (9) implies that there exist [alpha], [beta], [gamma] such that B ([[e.sup.i.sub.[alpha]], [e.sup.j.sub.[beta]]],[e.sup.k.sub.[gamma]]) [not equal to] 0, where i [not equal to] j [not equal to] k [not equal to] i. It follows that there exist [alpha], [beta] such that [[e.sup.i.sub.[alpha]], [e.sup.j.sub.[beta]]] [not equal to] 0 for i [not equal to] j. This implies that [[m.sub.i], [m.sub.j]] [not equal to] 0, which is a contradiction. Hence we obtain that [[lambda].sub.i] = [[lambda].sub.j]. Therefore, a generalized Wallach space (G/K,g) of type 2) or 3) is a g.o. space if and only if g is the standard metric.

4 Homogeneous geodesics in generalized Wallach spaces

Let (G/K,g) be a generalized Wallach space with B-orthogonal decomposition g = [??] [direct sum] m, where m = [m.sub.1] [direct sum] [m.sub.2] [direct sum] [m.sub.3] and B = -Killing form on g. Let {[e.sup.j.sub.i]} and {[e.sup.j.sub.s]} be the orthogonal bases of [??] and [m.sub.j] respectively with respect to B (1 [less than or equal to] i [less than or equal to] l, j = 1,2,3,1 [less than or equal to] s [less than or equal to] [d.sub.j]). For any X [member of] g \ {0} we write

[mathematical expression not reproducible]

In order to find all homogeneous geodesics in G/K, it suffices to find all the real solutions of the system (12), of [d.sub.1] + [d.sub.2] + [d.sub.3] quadratic equations for the variables [x.sub.i], [a.sub.j], [b.sub.k], [c.sub.s], which are not all equal to zero.

By Theorem 2 we only consider homogeneous geodesics in generalized Wallach spaces types 2) and 3) given in Theorem 1 for the metric ([[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3]), where at least two of [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3] are different. Then the geodesic vectors correspond to those solutions of the system (12) for the variables [x.sub.i], [a.sub.j], [b.sub.k], [c.sub.s], which are not all equal to zero. However, for many generalized Wallach spaces it is difficult to find all the real solutions of the system (12).

Next, we will give two examples of generalized Wallach spaces and give all the homogeneous geodesics for any given metric.

Example 1. We consider the generalized Wallach space SU(2)/{e}.

Let {[square root of 1][h.sub.[alpha]], [[A.sub.[alpha]]/[square root of 2]], [[B.sub.[alpha]]/[square root of 2]]} be an orthogonal basis of su(2) with respect to B, where [alpha] denotes a simple root of the Lie algebra su(2), and [A.sub.[alpha]] = [E.sub.[alpha]] - [E.sub.-[alpha]], [B.sub.[alpha]] = [square root of 1]([E.sub.[alpha]] + [E.sub.-[alpha]]). Here {[E.sub.[alpha]]} denotes the Weyl basis of su(2). We set [X.sub.[alpha]] = [A.sub.[alpha]]/[square root of 2], [Y.sub.[alpha]] = [B.sub.[alpha]]/[square root of 2]. Then we have that

[mathematical expression not reproducible]

The Lie algebra su(2) has an orthogonal decomposition su(2) = [m.sub.1] [direct sum] [m.sub.2] [direct sum] [m.sub.3], where [m.sub.1], [m.sub.2], [m.sub.3] are spanned by [square root of -1][h.sub.[alpha]], [X.sub.[alpha]] and [Y.sub.[alpha]] respectively. For any X [member of] su(2) we write

X = a[square root of -1][h.sub.[alpha]] + b[X.sub.[alpha]] + c[Y.sub.[alpha]], a,b,c [member of] R.

We will find all geodesic vectors of SU(2)/{e} for a given metric ([[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3]). The system (12) for SU(2)/{e} is

[mathematical expression not reproducible] (13)

Case 1. [[lambda].sub.i] = [[lambda].sub.i] [not equal to] [[lambda].sub.3]. The solutions of the system (13) are c = 0 or c [not equal to] a = b = 0, so the geodesic vectors are X = a[square root of -1][h.sub.[alpha]] + b[X.sub.[alpha]] and X = c[Y.sub.[alpha]].

Case 2. [[lambda].sub.1] = [[lambda].sub.3] [not equal to] [[lambda].sub.2]. The solutions of the system (13) are b = 0 or b [not equal to] 0, a = c = 0, so the geodesic vectors are X = a[square root of -1][h.sub.[alpha]] + c[Y.sub.[alpha]] and X = b[X.sub.[alpha]].

Case 3. [[lambda].sub.2] = [[lambda].sub.3] [not equal to] [[lambda].sub.1]. The solutions of the system (13) are a = 0 or a [not equal to] 0, b = c = 0, so the geodesic vectors are X = b[X.sub.[alpha]] + c[Y.sub.[alpha]] and X = a[square root of -1][h.sub.[alpha]]

Case 4. [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3] are distinct. Then the system (13) reduces to ab = ac = bc = 0, whose solutions are a = b = 0 or a = c = 0 or b = c = 0. In this case any vector X [member of] [m.sub.i]\{0} (i = 1,2,3) is a geodesic vector.

Therefore we obtain the following theorem, which recovers a result on R.A. Marinosci [21, p. 266]

Proposition 4. For the generalized Wallach space SU(2)/{e} the only geodesic vectors for a given metric ([[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3]) are the following:

1) If [[lambda].sub.i] = [[lambda].sub.j] = [[lambda].sub.k] (i,j,k [member of] {1,2,3}), then any vector X [member of] [m.sub.k]\{0} or X [member of] ([m.sub.i] [direct sum] [m.sub.j])\{0}.

2) If [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3] are distinct, then any vector X [direct sum] [m.sub.1] [union] [m.sub.2] [union] [m.sub.3].

Example 2. We consider the generalized Wallach space SO(n)/SO(n - 2), (n [greater than or equal to] 4).

This is the Stiefel manifold of orthogonal 2-frames in [R.sup.n]. Let so (n) and so (n - 2) be the Lie algebras of SO(n) and SO(n - 2) respectively. Let [E.sub.ab] denote the n x n matrix with 1 in the (ab)-entry and 0 elsewhere. If [e.sub.ab] = [E.sub.ab] - [E.sub.ba], then the set B = {[e.sub.ij] = [E.sub.ij] - [E.sub.ji] : 1 [less than or equal to] i < j [less than or equal to] n} is a B-orthogonal basis of so(n). The multiplication table of the elements in B is given as follows:

Lemma 2. If all four indices are distinct, then the Lie brackets in B are zero. Otherwise, it is [[e.sub.ij], [e.sub.jl]] = [e.sub.ik], where i, j, k are distinct.

Let so(n) = [??] [direct sum] [m.sub.1] [direct sum] [m.sub.2] [direct sum] [m.sub.3] be an orthogonal decomposition of so(n) with respect to B, where [??] = [span.sub.R]{[e.sub.ij] : 3 [less than or equal to] i < j [less than or equal to] n}, [m.sub.1] = [span.sub.R]{[e.sub.12]}, [m.sub.2] = [span.sub.R]{[e.sub.1j] : 3 [less than or equal to] j [less than or equal to] n}, and [m.sub.3] = [span.sub.R]{[e.sub.2j] : 3 [less than or equal to] j [less than or equal to] n}. For any X [member of] so(n) we write

[mathematical expression not reproducible]

Then the system (12) for SO(n)/SO(n - 2) takes the form

[mathematical expression not reproducible] (14)

As the above system is difficult to handle, we restrict to the Stiefel manifold SO(4)/SO(2) and look for geodesics X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.12][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24]. Then the above system simplifies to

[mathematical expression not reproducible] (15)

In order to find all geodesic vectors in the generalized Wallach space SO(4)/SO(2) for a given metric, we should find all the non zero real solutions of the system (15).

Case 1. [[lambda].sub.1] = [[lambda].sub.2] = [[lambda].sub.3]. Then the system (15) reduces to

[mathematical expression not reproducible] (16)

If [a.sub.34] = 0 and [a.sub.12] = 0 then the geodesic vectors are X = [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24] [e.sub.24] with [a.sub.13] [a.sub.23] + [a.sub.14] [a.sub.24] = 0.

If [a.sub.34] = 0 and [a.sub.12] = 0 we get [a.sub.23] = [a.sub.24] = 0, so the geodesic vectors are X = [a.sub.12] [e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14].

If [a.sub.34] [not equal to] 0 and [a.sub.12] = 0 we get [a.sub.13] = [a.sub.14] = 0, so the geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] = 0 we have [a.sub.13] = [a.sub.14], geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] - [a.sub.23][e.sub.24].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] [not equal to] 0 we have [a.sub.13] [not equal to] [a.sub.14]. If [a.sub.13] = -[a.sub.14] we have [a.sub.23] = [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] - [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.23][e.sub.24]. If [a.sub.13] [not equal to] -[a.sub.14] we have [a.sub.23] [not equal to] [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

Case 2. [[lambda].sub.1] = [[lambda].sub.3] = [[lambda].sub.2]. Then the system (15) reduces to

[mathematical expression not reproducible] (17)

If [a.sub.34] = 0 and [a.sub.12] = 0, geodesic vectors are X = [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][a.sub.24] with [a.sub.13][a.sub.23] + [a.sub.14] [a.sub.24] = 0.

If [a.sub.34] = 0 and [a.sub.12] [not equal to] 0 we get [a.sub.13] = [a.sub.14] = 0, geodesic vectors are X = [a.sub.12][e.sub.12] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

If [a.sub.34] [not equal to] 0 and [a.sub.12] = 0 we get [a.sub.23] = [a.sub.24] = 0, geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] = 0 we have [a.sub.13] = [a.sub.14], geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] - [a.sub.23][e.sub.24].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] [not equal to] 0 we have [a.sub.13] [not equal to] [a.sub.14]. If [a.sub.13] = -[a.sub.14] we have [a.sub.23] = [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] - [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.23][e.sub.24]. If [a.sub.13] [not equal to] -[a.sub.14] we have [a.sub.23] [not equal to] [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

Case 3. [[lambda].sub.3] = [[lambda].sub.2] = [[lambda].sub.1]. Then the system (12) reduces to

[mathematical expression not reproducible] (18)

If [a.sub.34] = 0 and [a.sub.12] = 0, geodesic vectors are X = [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

If [a.sub.34] = 0 and [a.sub.12] [not equal to] 0 we have [a.sub.13] = [a.sub.14] = [a.sub.23] = [a.sub.24] = 0, geodesic vectors are X = [a.sub.12][e.sub.12].

If [a.sub.34] [not equal to] 0 and [a.sub.12] = 0 we have [a.sub.13] = [a.sub.14] = [a.sub.23] = [a.sub.24] = 0, geodesic vectors are X = [a.sub.34][e.sub.34].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] = 0 we have [a.sub.13] = [a.sub.14], geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] - [a.sub.23][e.sub.24].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] [not equal to] 0 we have [a.sub.13] [not equal to] [a.sub.14]. If [a.sub.13] = -[a.sub.14] we have [a.sub.23] = [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] - [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.23][e.sub.24]. If [a.sub.13] [not equal to] -[a.sub.14] we have [a.sub.23] [not equal to] [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.32] + [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

Case 4. [[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3] are all different.

If [a.sub.34] = 0 and [a.sub.12] = 0, geodesic vectors are X = [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24] with [a.sub.13][a.sub.23] + [a.sub.14][a.sub.24] = 0.

If [a.sub.34] = 0 and [a.sub.12] [not equal to] 0 we have [a.sub.13] = [a.sub.14] = [a.sub.23] = [a.sub.2] = 0, geodesic vectors are X = [a.sub.12][e.sub.12].

If [a.sub.34] [not equal to] 0 and [a.sub.12] = 0 we have [a.sub.13] = [a.sub.14] = [a.sub.23] = [a.sub.24] = 0, geodesic vectors are X = [a.sub.34][e.sub.34].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] = 0 we have [a.sub.13] = [a.sub.14], geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] - [a.sub.23][e.sub.24].

If [a.sub.34] [not equal to] 0, [a.sub.12] [not equal to] 0 and [a.sub.23] + [a.sub.24] [not equal to] 0 we have [a.sub.13] [not equal to] [a.sub.14]. If [a.sub.13] = -[a.sub.14] we have [a.sub.23] = [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] - [a.sub.13][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.23][e.sub.24]. If [a.sub.13] [not equal to] -[a.sub.14] we have [a.sub.23] = [a.sub.24], so geodesic vectors are X = [a.sub.34][e.sub.34] + [a.sub.12][e.sub.12] + [a.sub.13][e.sub.13] + [a.sub.14][e.sub.14] + [a.sub.23][e.sub.23] + [a.sub.24][e.sub.24].

References

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[2] D.V. Alekseevsky and Yu. G. Nikonorov: Compact Riemannian manifolds with homogeneous geodesies, SIGMA: Symmetry Integrability Geom. Methods Appl. 5(93) (2009) 16 pages.

[3] A. Arvanitoyeorgos: Homogeneous geodesies in the flag manifold SO(2l + 1)/U(l - m) x SO(2m + 1), Linear Alg. Appl. 428 (2008) 1117-1126.

[4] Z. Chen, Y. Kang and K. Liang: Invariant Einstein metrics on three-locally-symmetric spaces, Commun. Anal. Geom. (to appear) arXiv:1411.2694.

[5] G. Calvaruso and R.A. Marinosci: Homogeneous geodesies of three-dimensional unimodular Lie groups, Mediterr. J. Math. 3 (2006) 467-481.

[6] Z. Dusek: Explieit geodesie graphs on some H-type groups, Rend. Circ. Mat. Palermo Ser. II, Suppl. 69 (2002) 77-88.

[7] Z. Dusek and O. Kowalski: Geodesic graphs on the 13-dimensional group of Heisenberg type, Math. Nachr. (254-255) (2003) 87-96.

[8] Z. Dusek and O. Kowalski: Light-like homogeneous geodesies and the geodesic lemma for any signature, Math. Debrecen. 71 (2007) 245-252.

[9] Z. Dusek, O. Kowalski and S. Nikcevic New examples of g.o. spaces in dimension 7, Differential Geom. Appl. 21 (2004) 65-78.

[10] Z. Dusek, O. Kowalski and Z. Vlasek : Homogeneous geodesics in homogeneous affine manifolds, Result. Math. 54 (2009) 273-288.

[11] C. S. Gordon: Homogeneous manifolds whose geodesics are orbits, in: Topics in Geometry in Memory of Joseph D'Atri, Birkha user, Basel, 1996,155-174.

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[13] A. Kaplan: On the geometry of groups of Heisenberg type, Bull. London. Math. Soc. 15 (1983) 32-44.

[14] O. Kowalski, S. NikCeviC and Z. Vlasek: Homogeneous geodesics in homogeneous Riemannian manifolds. Examples, Reihe Mathematik, TU Berlin, No. 665/2000 (9 pages).

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[20] A.M. Lomshakov, Yu.G. Nikonorov and E.V. Firsov: Invariant Einstein metrics on three-locally-symmetric spaces, Matem. tr. 6(2) (2003), 80-101 (Russian); English translation in: Siberian Adv. Math. 14(3) (2004) 43-62.

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University of Patras, Department of Mathematics, GR-26500 Rion, Greece

email: arvanito@math.upatras.gr

Sichuan University of Science and Engineering,

Mathematic Department,

Zigong, 643000, China

email: wangyu_813@163.com

Andreas Arvanitoyeorgos Yu Wang (*)

(*) Corresponding author.

Received by the editors in May 2016.

Communicated by J. Fine.

2010 Mathematics Subject Classification : 53C20, 53C22, 53C30.

Key words and phrases : Homogeneous geodesic; g.o. space; invariant metric; geodesic vector; naturally reductive space; generalized Wallach space.
Table 1. The pairs (g,k) corresponding to generalized Wallach spaces
G/K with G simple.

g                            [??]

so(k + l + m)                so(k) [direct sum] so(l) [direct sum] so(ra)
su(k + l + m)                su(k) [direct sum] su(l) [direct sum] su(m)
sp(k + l + m)                sp(k) [direct sum] sp(l) [direct sum] sp(ra)
su(2l),l                     u(l)
[greater than or equal to]
2
so(2l), l                    u(l) [direct sum] u(l - 1)
[greater than or equal to]
4
[e.sup.6]                    su(4) [direct sum] 2sp(1) [direct sum] R
[e.sup.6]                    so (8) [direct sum] [R.sup.2]
[e.sup.6]                    sp(3) [direct sum] sp(1)

g           [??]

[e.sup.7]   so(8) [direct sum] 3sp(1)
[e.sup.7]   su(6) [direct sum] sp(1) [direct sum] R
[e.sup.7]   so(8)
[e.sup.8]   so(12) [direct sum] 2sp(1)
[e.sup.8]   so(8) [direct sum] so(8)
[f.sup.4]   so(5) [direct sum] 2sp(1)
[fsup.4     so(8)
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Author:Arvanitoyeorgos, Andreas; Wang, Yu
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Date:Apr 1, 2017
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