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Владикавказский математический журнал 2013, Том 15, Выпуск 3, С. 67-76
УДК 514.765
GEODESIC ORBIT RIEMANNIAN METRICS ON SPHERES1 Yu. G. Nikonorov
In this paper, a complete classification of geodesic orbit Riemannian metrics on spheres Sn is obtained.
We also construct some explicit examples of geodesic vectors for Sp(n +1 )U (l)-invariant metrics on S 4n+3.
Mathematics Subject Classification (2010): 53C20 (primary), 53C25, 53C35 (secondary).
Key words: homogeneous spaces, homogeneous Riemannian manifolds, naturally reductive Riemannian
manifolds, normal homogeneous Riemannian manifolds, geodesic orbit spaces, symmetric spaces, weakly
symmetric spaces.
Introduction
A Riemannian manifold (M,g) is called a geodesic orbit manifold (GO-manifold) if every its geodesic is an orbit of a one-parameter group of isometries of (M, g). Every such manifold is homogeneous and can be identified with a coset space M = G/H of a transitive Lie group G of isometries. A Riemannian homogeneous space (M = G/H, g) of a Lie group G is called a space with homogeneous geodesies (or a geodesic orbit space, shortly, GO-space), if any its geodesic is an orbit of a one-parameter subgroup of the group G. This terminology was introduced by O. Kowalski and L. Vanhecke in the paper [17]. We discuss some properties of geodesic orbit Riemannian manifolds and related results in Section 1.
The main goal of this paper is a complete classification of geodesic orbit Riemannian metrics on spheres Sn. The classification of all transitive and effective actions of connected compact Lie groups on spheres is obtained in [18]. We collect in Table 1 all variants to represent Sn as a homogeneous space G/H. In this table, by dim(G/H) we denote the dimension of the corresponding space (and the corresponding sphere), dim(M) (respectively, dim(MGo)) means the dimension of the space of G-invariant (the space of G-invariant geodesic orbit) Riemannian metrics on the homogeneous space G/H.
Note that Riemannian metrics of constant sectional curvature constitute a one-parameter family of metrics on each sphere Sn, n ^ 2 (there is no such notion for the trivial case n = 1). These are exactly the metrics invariant under the action of the orthogonal group O(n + 1) and under its connected unit component SO(n + 1). These groups are respectively the full isometry group and the full connected isometry group of each metric of constant curvature on Sn
We list all inclusions between the isometry groups in Table 1: G2 С SO(7), SU(k) С U(k) С SO(2k), Sp(k) С Sp(k)U(1) С Sp(k)Sp(1) С SO(4k), Sp(k) С Sp(k)U(1) С U(2k), SU(4) С Spin(7) С SO(8), Spin(9) С SO(16) (see details e. g. in Chapter 4 of [20]).
It should be noted that geodesic orbit metrics on some spaces in Table 1 are well known. Below we describe all known results and emphasize the cases that should be studied.
© 2013 Nikonorov Yu. G.
1 The project was supported in part by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation, grant № NSh-921.2012.1, and by Federal Target Grant «Scientific and educational personnel of innovative Russia» for 2009-2013, agreement № 8206, application № 2012-1.1-12-000-1003-014.
Table 1
Invariant metrics on spheres
g H dim (G/H) dim(^) dim(^Go) Cond.
1 SO(n+ 1) SO(n) n 1 1 n ^ 1
2 g2 SU( 3) 6 1 1
3 Spin(7) g2 7 1 1
4 SU( 2) M 3 6 1
5 SU(n+ 1) SU(n) 2n+ 1 2 2 n ^ 2
6 U(n + 1) U(n) 2n+ 1 2 2 n ^ 1
7 Spin( 9) Spin(7) 15 2 2
8 Sp(n+l)Sp(l) Sp(n) diag(S'p(l)) 4n + 3 2 2 n ^ 1
9 Sp(n+l)U(l) Sp(n) diag(f/(l)) 4n + 3 3 3 n ^ 1
10 Sp(n + 1) Sp(n) 4n + 3 7 2 n ^ 1
Case 1). The homogeneous space SO(n + 1)/SO(n) is irreducible symmetric, all SO(n + l)-invariant Riemannian metrics are SO(n + 1)-normal homogeneous (hence, geodesic orbit) and constitute the set of Riemannian metrics of constant sectional curvature on Sn. This set is a part of any other family of invariant metrics from Table 1.
Cases 2) and 3). The spaces G2/SU(3) and Spin(7)/G2 are isotropy irreducible. All invariant metrics on these spaces are normal homogeneous (hence, GO-metrics) and have constant sectional curvature (see Section 7 in [12]).
Case 4). All left-invariant metrics on a compact Lie group G, that are geodesic orbit with respect to G, should be biinvariant (see Proposition 8 in [2]). Since the group SU(2) is simple, then all biinvariant Riemannian metrics on SU(2) constitute a one-parameter family of metrics. Since SU(2)2/diag(SU(2)) = SO(4)/SO(3), then these metrics are exactly metrics of constant curvature on S3 = SU(2).
Cases 5) and 6). Note that the set of U(n + 1)-invariant metrics on S2n+1 coincides with the set of SU(n + 1)-invariant metrics and constitutes a 2-parametric family of metrics. Every such metric is naturally reductive and weakly symmetric [26, 27, 28]. Therefore, in both these cases we have a two-parameter family of geodesic orbit metrics.
Case 7). The family of invariant metrics on Spin(9)/Spin(7) is 2-parametric. All these metrics are weakly symmetric but not naturally reductive [27, 28]. Therefore, we have a two-parameter family of geodesic orbit metrics.
Case 8). The family of invariant metrics on Sp(n + 1)Sp(1)/Sp(n) diag(Sp(1)) is 2-parametric. Every such metric is naturally reductive and weakly symmetric [27, 28]. Therefore, in both these cases we have a 2-parameter family of geodesic orbit metrics. More details on this case could be found in Sections 2.
Case 9). Note that the previous family is a part of this one. The family of Sp(n+1)U(1)-invariant metrics on Sp(n + 1)U(1)/Sp(n) diag(U(1)) = S4n+3 is 3-parametric. Every such metric is weakly symmetric (see e. g. 12.9.2 in [24] or Table 1 in [25]). Therefore, in this case we have a three-parameter family of geodesic orbit metrics. Details on the normality and the natural reductivity of Sp(n + 1)U(1)-invariant metrics could be found in Remark 1. Some explicit form of geodesic vectors for Sp(n + 1) x U(1)-invariant metric could be found in Section 4.
Case 10). In the last case we get a 7-dimensional space of Sp(n + 1)-invariant metrics on Sp(n + 1)/Sp(n) = S4n+3. This is a unique case that we should study in details. We
deal with this case in Sections 2 and 3. By Theorem 1, a homogeneous Riemannian space (S4n+3 = Sp(n + 1)/Sp(n),g) is geodesic orbit if and only if the metric g is invariant under Sp(n + 1)Sp(1).
The structure of the paper is the following. We recall in Section 1 some useful facts on the class of geodesic orbit Riemannian manifolds and some related classes of Riemannian manifolds. In Section 2 we discuss actions of the groups Sp(n + 1), Sp(n + 1)U(1), Sp(n + 1)Sp(1) (and corresponding invariant metrics) on the sphere S4n+3. In the next section we classify Sp(n+1)-invariant geodesic orbit metrics on S4n+3. The final section is devoted to an explicit description of geodesic vectors for Sp(n + 1)U(1)-invariant metrics on the sphere S4n+3.
Acknowlegment. The author is grateful to V. N. Berestovskii, W. Ziller, and O. S. Yakimova for helpful discussions.
1. On geodesic orbit manifolds
There are some important subclasses of geodesic orbit manifolds. Indeed, GO-spaces may be considered as a natural generalization of Riemannian symmetric spaces, introduced and classified by E. Cartan in [13]. On the other hand, the class of GO-spaces is much larger than the class of symmetric spaces. Any homogeneous space M = G/H of a compact Lie group G admits a Riemannian metric g such that (M, g) is a GO-space. It suffices to take the metric g induced by a biinvariant Riemannian metric go on the Lie group G such that (G, go) ^ (M = G/H, g) is a Riemannian submersion with totally geodesic fibres. Such geodesic orbit space (M = G/H, g) is called a normal homogeneous space (in the sense of M. Berger [10]). It should be noted also that any naturally reductive Riemannian manifold is geodesic orbit. Recall that a Riemannian manifold (M, g) is naturally reductive if it admits a transitive Lie group G of isometries with a biinvariant pseudo-Riemannian metric go, which induces the metric g on M = G/H (see [12] and [16]).
An important class of GO-spaces consists of weakly symmetric spaces, introduced by A. Selberg [21]. A homogeneous Riemannian manifold (M = G/H, g) is a weakly symmetric space if any two points p, q G M can be interchanged by an isometry a G G. This property does not depend on the particular G-invariant metric g. Weakly symmetric spaces M = G/H have many interesting properties and are closely related with spherical spaces, commutative spaces, Gelfand pairs etc. (see papers [3, 25] and book [24] by J. A. Wolf). The classification of weakly symmetric reductive homogeneous Riemannian spaces was given by O. S. Yakimova [25] on the base of the paper [3] (see also [24]). It is very important that weakly symmetric Riemannian manifolds are geodesic orbit by a result of J. Berndt, O. Kowalski, and L. Vanhecke [11].
Note that generalized normal homogeneous Riemannian manifolds (5-homogeneous manifold, in another terminology) constitute another important subclass of geodesic orbit manifold. All metrics from this subclass are of non-negative sectional curvature and have some other interesting properties (see details in [4-6]). In the paper [9], a classification of generalized normal homogeneous metrics on spheres and projective spaces is obtained. Finally, we notice that Clifford-Wolf homogeneous Riemannian manifolds constitute a partial subclass of generalized normal homogeneous Riemannian manifolds [7].
Many interesting results about GO-spaces one can find in [2, 11, 14, 15, 19, 22, 23, 28], where there are also extensive references.
Now we recall some important properties of homogeneous Riemannian spaces and geodesic orbit Riemannian spaces in particular.
Let M = G/H be a homogeneous space of a compact connected Lie group G. Let us denote by (■, ■) a fixed Ad(G)-invariant Euclidean metric on the Lie algebra g of G (for example, the minus Killing form if G is semisimple) and by
g = h e p (1)
the associated (■, -)-orthogonal reductive decomposition, where h is the Lie algebra of the group H. An invariant Riemannian metric g on M is determined by an Ad(H)-invariant inner product go = (■, ■) on the space p which is identified with the tangent space Mo at the initial point o = eH.
Recall that X G g is called a geodesic vector if the orbit of the point o = eH under the action of the one-parameter group 7(f) = exp(tX), t G R, is a geodesic in (M = G/H, g), see details in the paper [17] or in Section 5 of the book [8].
For a given inner product (■, ■), we consider a metric endomorphism A : p ^ p that is defined by the equality (X,Y) = (AX,Y) for all X,Y G p. Obviously, A is Ad(H)-equivariant, positive definite and symmetric operator (with respect to (■, ■)). It is clear also that a metric endomorphism determines a corresponding invariant Riemannian metric uniquely. The following lemma is very useful.
Lemma 1 [1]. A compact homogeneous Riemannian manifold (M = G/H, g) with reductive decomposition (1) and metric endomorphism A is GO-space if and only if for any X G p there is Z G h such that [Z + X, AX] G h • The latter condition is equivalent to the property of Z + X G g to be a geodesic vector.
2. On invariant metrics and transitive actions of groups
Sp(n +1), Sp(n + 1)U(1), and Sp(n + 1)Sp(1) on S4n+3
Let H be the field of quaternions. Denote by i, j, k the quaternionic units in H (ij = -ji = k, jk = -kj = i, ki = -ik = j, ii = jj = kk = -1). For X = x\ + ix2 + jx3 + kx4, Xi G R, define Re(X) = x\ (the real part of X), X = x\ — 1x2 — JX3 — kx4 and ||X|| = Vxx. If Re(X) = x1 =0, then the quaternion X is called pure imaginary.
Let us consider a (left) vector space HP+1 over HI. For X = (Xi,X2,... ,Xn+i) G Hn+1 and Y = (Fi, Y2, ■ ■ ■, Yn+1) G we define (X, Y) 1 = XSYS. Then Sp{n + 1) is the
group of all H-linear operators A : Hn+1 ^ Hn+1 with the property (A(X), A(Y))1 = (X, Y)1 for every X, Y G Hn+1. If we choose some (■, ■)1 -orthonormal quaternionic basis in Hn+1, then we can identify Sp(n + 1) with a group of matrices A = (a j), aj G H, with the property A= A*, where a*^ = a^ for 1 ^ i, j ^ n + 1. In this case 5p(n + 1) consists of quaternionic ((n + 1) x (n + 1))-matrices A with the property A* = -A. Later on we shall use these identifications.
We have a natural embedding H ^ R4 via x1 + ix2 + jx3 + kx4 ^ (x1,x2,x3,x4) and the induced embedding Hn+1 ^ R4n+4. It is well known that the group G := Sp(n + 1) acts transitively on the sphere
S4n+3 = { (X1,X2,... ,X„+1) G Hn+1 : ||X1 II2 + ||X2II2 + ■ ■ ■ + yX„+1y2 = 1}.
Let us consider natural embedding diag(Sp(1), Sp(n)) C Sp(n + 1), and let K and H be the images of Sp(1) and Sp(n) respectively under this embedding. Then H is the isotropy subgroup of a point (1,0,..., 0) G Hn+1 under the above action of Sp(n+1). Since K = Sp(1)
is a normal subgroup in the group diag(Sp(1), Sp(n)), then we have the following (almost effective) transitive action of G x K on S4n+3 = G/H:
(a, b)(cH) = acHb-1 = acb-1H, a, c G G, b G K. (2)
The isotropy group of this action at the point (1,0,..., 0) G is
Sp(n - 1) x Sp(1) = Sp(n - 1) x diag(Sp(1)) C Sp(n) x Sp(1) = G x K.
We also get an effective representation S4n+3 = Sp(n + 1)Sp(1)/Sp(n) diag(Sp(1)) (after dividing by the noneffectiveness kernel).
Let L be any subgroup U(1) = S1 in K = Sp(1). Then we get transitive (and almost effective) action of G x L on S4n+3 = G/H:
(a, b)(cH) = acHb-1 = acb-1H, a, c G G, b G L, (3)
that is a part of the action (2). In this case we get the following isotropy group at the point (1, 0,..., 0) G Hn+1:
Sp(n) x U(1) = Sp(n) x diag(U(1)) C U(1) x Sp(n) x U(1) C Sp(n + 1) x U(1) = G x L.
We also get an effective representation S4n+3 = Sp(n + 1)U(1)/Sp(n) diag(U(1)). For A, B G sp(n + 1) we define
{A,B) = l-{Re(AB*)). (4)
It is easy to see that (•, ■) is a Ad(Sp(n + 1))-invariant inner product on the Lie algebra g = sp(n + 1).
We write Eij for the skew-symmetric matrix with 1 in the ij-th entry and —1 in the ji-th entry, and zeros elsewhere. We denote by Fj the symmetric matrix with 1 in both the ij-th and ji-th entries, and zeros elsewhere. Denote also by Gi the matrix with \/2 in w-th entry, and zeros elsewhere.
It is easy to check that the matrices iGj, jG^, kG^, Ejj, iFj, jF^j, kF^j, where 1 ^ i, j ^ n + 1 and i < j, constitute a (•, -)-orthonormal (see (4)) basis in sp(n + 1). Let us consider the following (•, -)-orthogonal decomposition:
sp(n + 1) = k esp(n) e p1 = sp(n) e p, k = ie p2, (5)
where k and l are the Lie algebras of the Lie subgroups K and L (see above). Therefore, the embedding of kesp(n) = sp(1) esp(n) in sp(n + 1) is defined by (A, B) ^ diag(A, B), where A G sp(1) and B G sp(n).
Without loss of generality we may suppose that the Lie subalgebra l = u(1) (u(1)esp(n) C sp(1) e sp(n) C sp(n + 1)) is spanned on the vector iG1. Then p2 = Lin{jG1, kG1}.
Any Sp(n + 1)-invariant metric on S4n+3 is defined by an Ad(Sp(n))-invariant inner product (■, ■) on p. Note that Ad(Sp(n)) acts irreducibly on p1 and trivially on k. Therefore, any such inner product is generated by the metric endomorphism of the type
A = Ai e x1 Id |P1 (6)
for some x1 > 0 and some symmetric and positive definite operator A : k ^ k. In particular, it implies that the space of Sp(n + 1)-invariant Riemannian metric on S4n+3 is 7-dimensional.
If A = x2 Id |p2 e x3 Id |[, then the inner product (■, ■) generates Sp(n + 1) x L-invariant metric on S4n+3 (see Section 4). If A = x2 Id |k, then the inner product (■, ■) generates Sp(n + 1) x K-invariant metric on S4n+3, see e. g. [27].
Remark 1. Let us note that for x1 = x2 = x3 we get Sp(n + 1)U(1)-naturally reductive metrics on the sphere S4n+3 (they are even Sp(n + 1)U(1)-normal homogeneous for x3 < x1 = x2). All these metrics are also U(2n + 2)-invariant and U(2n + 2)-naturally reductive. By analogy, for x3 = x2 = x1 we get Sp(n + 1)Sp(1)-naturally reductive metrics on the sphere S4n+3 (for x3 = x2 < x1 these metrics are even Sp(n + 1)Sp(1)-normal homogeneous). Obviously, for x1 = x2 = x3 we get Sp(n + 1)-normal homogeneous metrics. For all other values of parameters x^ i = 1,2,3, Sp(n + 1)U(1)-invariant metrics are not naturally reductive. See details in [27] and [28].
Remark 2. Consider the homogeneous spaces Sp(n + 1)/Sp(n) ■ U(1), where n ^ 1, U(1) C Sp(1), and Sp(1) is the first factor in the group Sp(1) x Sp(n) C Sp(n + 1). The Lie algebra of the group Sp(n) ■ U(1) is l e sp(n) C sp(n + 1) in the decomposition (5). It is known that the homogeneous space Sp(n + 1)/Sp(n) ■ U(1) is diffeomorphic to CP2n+1, hence we get a representation of an odd-dimensional complex projective space. It is also known that the space Sp(n + 1)/Sp(n) ■ U(1) admits a two-parameter family of Sp(n + 1)-invariant Riemannian metrics [27]. All these metrics are weakly symmetric [28]. It is interesting that only Sp(n + 1)-normal and SU(2n + 2)-normal metrics in this family are naturally reductive. Note that explicit expressions of geodesic vectors for Sp(n + 1)-invariant Riemannian metrics on CP2n+1 = Sp(n + 1)/Sp(n) ■ U(1) could be found in paper [4].
3. Sp(n + 1)-invariant geodesic orbit metrics on the sphere S4n+3
Here we classify all geodesic orbit metrics on the sphere S4n+3 with respect to the group Sp(n + 1). At first, we should establish some auxiliary results. By direct calculation we get the following lemma (see the definitions of Eij, Fij, and Gi in the previous section).
Lemma 2. For any 1 ^ i < j ^ n +1 the following relations are fulfilled:
[aGfc,Ejj] = [aGfc,^Fjj] =0 (Vk G {i, j}, VG {i,j,k});
[aG,, Eij] = V2aFtj, [aGj, Etj\ = -V2aFtj (Va £ {i, j, k}); [aG^aFij] = -y/2Eijt [aG^aF^] = V2Etj (Va G {i,j,k}); [aGi,/3Fij] = [aGj,/3Fij] = V2(a ■ P)Ftj (Va,/3 £ {i,j,k}, (3 + a),
where ■ means the quaternion multiplication.
The following lemma is very important for our goals.
Lemma 3. For any 2 ^ s ^ n + 1 and given vectors U = a1iG1 + a2jG1 + a3kG1 and V = b1E1 s + 62iF1s + 63jF1s + 64kF1s, there is a vector W — C1iGs + C2jGs + C3kGs such that [U, V] = [W,V].
< All is clear when V is trivial. Now we suppose that V = 0. Using Lemma 2, we see that the equality [U, V] = [W, V] is equivalent to u ■ v = v ■ w for the quaternions u = a1 i + a2j + a3k, v = 61 + 62i + 63j + 64k, and w = c1i + c2j + c3k. But now we can define w by the formula w = v-1 ■ u ■ v. This definition is correct, since for a given nonzero quaternion q, a map of the type a ^ q ■ a ■ q-1 (that is an automorphism of the field of quaternions) preserves a subspace of pure imaginary quaternions. Indeed, if a = 0
is a pure imaginary, then a 1 = ||a|| 2a = —||a|| 2a. Therefore, if b = q ■ a ■ q 1, then ||6||_26 = b~l = q ■ a~l ■ q~l = —||a||_26, and b is also pure imaginary. >
Theorem 1. A homogeneous Riemannian space (S4n+3 = Sp(n + 1)/Sp(n),g) is geodesic orbit with respect to Sp(n +1), if and only if the metric g is invariant under Sp(n+1) x Sp(1).
< Suppose that a metric endomorphism A (see (6)) generates a Riemannian geodesic orbit metric g with respect to Sp(n +1). Then by (6) and Lemma 1 we get that for any X e k = sp (1) there is Z e sp(n) such that [Z + X, AX] e sp(n). But [Z, AX] e [sp(n), k] = 0, therefore, [X, AX] e sp(n). On the other hand, [X, AX] e [k, k] = k. Hence, we get [X, AX] = 0 for all X e k = sp(1). Now, it is easy to see that A = x2 Id |k for some x2 > 0 and (■, ■) generates Sp(n + 1) x Sp(1)-invariant metric on S4n+3 (see the discussion in the previous section). Indeed, the centralizer of any nontrivial X e sp(1) in sp(1) is exactly RX. Therefore, A preserves all 1-dimensional subspaces in sp(1) and should be a multiple of Id on k = sp (1).
Now, consider any Sp(n + 1) x Sp(1)-invariant metric on S4n+3. It is generated by a metric endomorphism of the type A = x2 Id |k © x1 Id |Pl. For any X = Xi + X2 (Xi e pi, X2 e k) we get AX = x1X1 + x2X2. In order to prove that this metric is geodesic orbit with respect to Sp(n + 1), it suffices to find Z e sp(n) such that [Z, X1 ] = (x2/x1 — 1)[X2,X1 ] (see Lemma 1). If x1 = x2 then we can choose Z = 0. Consider now the case x1 = x2.
Let X1 = X2 + Xf +-----+ Xn+1, where Xf e Lin|E1S, iF^, jFis, kF1S}, 2 ^ s ^ n +1. By
Lemma 3, there is a vector e Lin{iGs,jGs,kGs} C sp(n) such that [Us,Xf] = (x2/x1 — 1)[X2,Xf]. Now by Lemma 2, we get
n+1
(x2/x1 — 1)[X2 ,X1 ] = ¿(x2/x1 — 1)[X2, Xf]
s=2
n+1 n+1 p n+1
^[Uf,Xf]^[Uf,X1]= Uf,X1 .
s=2 s=2 L s=2
Therefore, we can choose Z = ^n+2 Us. By Lemma 1 we get that A = x2 Id |k © x1 Id |Pl does generate a GO-metric with respect to Sp(n + 1). >
4. Geodesic vectors for Sp(n + 1)U(1)-invariant metrics on the sphere S4n+3
We already know (see Case 9) in Introduction) that all Sp(n + 1)U(1)-invariant metrics (they constitute a 3-parametric family) on Sp(n + 1)U(1)/Sp(n) diag(U(1)) = S4n+3 are geodesic orbit because of the weak symmetry. On the other hand, sometimes it is useful to have explicit forms of suitable geodesic vectors. In this section, we get such description for all Sp(n + 1)U(1)-invariant metrics on S4n+3.
At first, we give more details on Sp(n + 1)U(1)-invariant metrics on the sphere S4n+3. We suppose that l is supplied with the inner product (•, •) as a subalgebra of sp(n + 1) (see the decomposition (5)). Then we can extend (•, ■) to the Lie algebra sp(n + 1) © l assuming (sp(n + 1), l) = 0. Let us consider the following (•, -)-orthogonal decompositions:
Sp(n + 1) © l = t) © p1 © p2 © p3, h = f)1 © f)2,
where p1 = {(X, 0) e sp(n + 1) © 11 X e p1}, p2 = {(X, 0) e sp(n + 1) © 11 X e J2}, P3 = {(X, —X) e sp(n + 1) © 11X e l}, t)1 = {(X,0) e sp(n + 1) © 11X e sp(n)}, ifj2 =
{(X,X) e sp(n +1) © l1 X e l}. It is easy to see that the modules pj, i = 1,2,3, are ad( j)-irreducible. Then every Sp(n + 1)U(l)-invariant metric g on S4n+3 is determined by the metric endomorphism
A = xi Id |il © X2 Id |p2 © X3 Id |p3
for some positive x1, x2, x3.
Now, we shall find for every X e p1 © p2 © p3 a vector Z eh such that X + Z is a geodesic vector on the homogeneous Riemannian space (Sp(n + 1)U(1)/Sp(n) diag(U(1)) = S4n+3,g).
Let us consider any X = X1 + X2 + X3, where X1 e p1, X2 e p2, X3 e p3. Then AX = x1X1 + x2X2 + x3X3 and
[AX, X] = (x1 - x2)[X1,X2] + (x1 - x3)[X1,X3] + (x2 - x3)[X2, X3].
Obviously, [X1, X2] e p1, [X1, X3] e p1, [X2, X3] ej2
By Lemma 1, it suffices to find a vector Z e j such that [Z, AX] = [AX, X]. Consider Z = Z1 + Z2, where Z1 e jh and Z2 e Since [Z1 ,X1] e P1, [Z1,X2] = [Z1 ,X3] = 0, [Z2,X1] e p1, [Z2,X2] e p2 and [Z2,X3] = 0 we get that [Z, AX] = [AX, X] is equivalent to the following two equations:
x2[Z2,X2] = (x2 - x3)[X2,X3], (7)
x1[Z1,X1 ]+ x1 [Z2,X1 ] = (x1 - x2)[X1 ,X2] + (x1 - x3)[X1, X3]. ( )
It is clear that X1 = (Y, 0) e j for some Y e p1, Z1 = (U, 0) e j1 for some U e sp(n),
X3 = (aiG1, -aiG1) e j, X2 = (A)G1 + 7kG1,0) e j, Z2 = (5iG1,5iG1) e j2
for some real numbers a, 7, 5.
Since [Z2, X2] = (-7fiG1 + , 0) e j and [X2,X3] = MG1 - kG, 0) e j2, then for 5 = (x3/x2 - 1)a the equality (7) holds.
Substituting 5 = (x3/x2 - 1)a into equality (7) and using the inclusions
[Z2,X1] = ((x3/x2 - 1)a[iG1, Y],0) e P1, [X2 ,X1] = (j 1 + 7kG1 ,Y], 0) e P1, [X3, X1 ] = (a[iG1, Y], 0) e j,
we see that the equality (7) is equivalent to the following one:
[U, Y] = [(x3/x1 - x3/x2)aiG1 + (x2M - 1)#jG1 + (x2/x1 - 1)YkG1 ,Y]. (8)
Consider the sum Y = Y2 + Y3 + ••• + Yn+1, where Ys e Lin{E1s, iF1s, jF1s, kF1s}, 2 ^ s ^ n + 1. By Lemma 3 there is a vector Us e Lin{iGs, jGs, kGs} C sp(n) such that
[Us, Ys] = [(x3/x1 - x3/x2)aiG1 + (x2/x1 - 1)^jG1 + (x2/x1 - 1)YkG1, Ys].
Now, by Lemma 2, we get
[(x3/x1 - x3/x2)aiG1 + (x2/x1 - 1)^jG1 + (x2/x1 - 1)YkG1 ,Y]
n
y] [(x3/x1 - x3/x2)aiG1 + (x2/x1 - 1)^jG1 + (x2/x1 - 1)YkG1, Ys]
s=2
n n r n
]T[Us,Ys]^[Us,Y]= £ Us, Y
s s s
s=2 s=2
s)
L s=2
Hence, if we choose U = ^n=2 Us, then the equality (8) holds. Therefore, the vector X + Z1 + Z2, where Z1 = (u = ElS Us, 0) and Z2 = (x3/x2 - 1)a (iG1, iG1), is a geodesic vector by Lemma 1. In particular, the metric endomorphism A = x1 Id |p1 e x2 Id |p2 e x3 Id |p3 does generate a GO-metric with respect to Sp(n + 1) x U(1) for every positive x1,x2,x3.
Remark 3. In particular, this proves that every Sp(n + 1)U(1)-invariant metric on the sphere S4n+3 is geodesic orbit with respect to the group Sp(n + 1)U(1).
The conclusion
It is clear that Theorem 1 completes the classification for Case 10) in Introduction. Therefore, we have verified completely all data from Table 1.
All geodesic orbit Riemannian metrics from Table 1 induce geodesic orbit Riemannian homogeneous metrics on corresponding real projective spaces RPn. The metrics obtained in such a way, metrics from Remark 2 together with the normal homogeneous metrics on the projective spaces CPn = SU(n + 1)/S(U(n) x U(1)), HPn = Sp(n + 1)/Sp(n) x Sp(1), and CaP2 = F4/Spin(9) exhaust all geodesic orbit Riemannian metrics on projective spaces (see details in [27] and [28]).
References
1. Alekseevsky D. V., Arvanitoyeorgos A. Riemannian flag manifolds with homogeneous geodesies // Trans. Amer. Math. Soe.-2007.-Vol. 359.—P. 3769-3789.
2. Alekseevsky D. V., Nikonorov Yu. G. Compact Riemannian manifolds with homogeneous geodesies // SIGMA Symmetry Integrability Geom. Methods Appl.—2009.—№ 5 (093).—16 p.
3. Akhiezer D. N., Vinberg E. B. Weakly symmetric spaces and spherical varieties // Transf. Groups.— 1999.—Vol. 4.—P. 3-24.
4. Berestovskii V. N., Nikitenko E. V., Nikonorov Yu. G. Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic // Dif. Geom. Appl.—2011.—Vol. 29, Issue 4.— P. 533-546.
5. Berestovskii V. N., Nikonorov Yu. G. On 5-homogeneous Riemannian manifolds // Dif. Geom. Appl.— 2008.—Vol. 26, № 5.—P. 514-535.
6. Berestovskii V. N., Nikonorov Yu. G. On 5-homogeneous Riemannian manifolds, II // Siber. Math. J.—2009.—Vol. 50, № 2.—P. 214-222.
7. Berestovskii V. N., Nikonorov Yu. G. Clifford-Wolf homogeneous Riemannian manifolds // J. Dif. Geom.—2009.—Vol. 82, № 3.—P. 467-500.
8. Berestovskii V. N., Nikonorov Yu. G. Riemannian Manifolds and Homogeneous Geodesics.— Vladikavkaz: SMI VSC RAS, 2012.—412 p.—[Russian].
9. Berestovskii V. N., Nikonorov Yu. G. Generalized normal homogeneous Riemannian metrics on spheres and projective spaces // Ann. Glob. Anal. Geom.—2013.—D0I 10.1007/s10455-013-9393-x.—URL: http://arxiv.org/abs/1210.7727.
10. Berger M. Les varietes riemanniennes homogenes normales a courbure strictement positive // Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser.—1961.—Vol. 15.—P. 179-246.
11. Berndt J., Kowalski O., Vanhecke L. Geodesics in weakly symmetric spaces // Ann. Global Anal. Geom.—1997.—Vol. 15.—P. 153-156.
12. Besse A. L. Einstein Manifolds.—Berlin: Springer-Verlag, 1987.
13. Cartan E. Sur une classe remarquable d'espaces de Riemann // Bull. Soc. Math. de France.—1926.— Vol. 54.—P. 214-264; 1927.—Vol. 55.—P. 114-134.
14. Ducek Z., Kowalski O., Nikcevic S. New examples of Riemannian g.o. manifolds in dimension 7 // Diff. Geom. Appl.—2004.—Vol. 21.—P. 65-78.
15. Gordon C. Homogeneous Riemannian manifolds whose geodesics are orbits // Progress in Nonlinear Differential Equations. Vol. 20. Topics in geometry: in memory of Joseph D'Atri.—Birkhauser, 1996.— P. 155-174.
16. Kobayashi S., Nomizu K. Foundations of differential geometry. Vol. I.—N.Y.: John Wiley & Sons, 1963; Vol. II.—N.Y.: John Wiley & Sons, 1969.
17. Kowalski O., Vanhecke L. Riemannian manifolds with homogeneous geodesics // Boll. Unione Mat. Ital. Ser. B.—1991.—Vol. 5, № 1.—P. 189-246.
18. Montgomery D., Samelson H. Transformation groups on spheres // Ann. of Math.—1943.—Vol. 44.— P. 454-470.
19. Nikonorov Yu. G. Geodesic orbit manifolds and Killing fields of constant length // Hiroshima Math. J.—2013.—Vol. 43, № 1.—P. 129-137.
20. Onishchik A. L. Topology of Transitive Transformation Groups.—Leipzig, Berlin, Heidelberg: Johann Ambrosius Barth, 1994.
21. Selberg A. Harmonic Analysis and discontinuous groups in weakly symmetric riemannian spaces, with applications to Dirichlet series // J. Indian Math. Soc.—1956.—Vol. 20.—P. 47-87.
22. Tamaru H. Riemannin geodesic orbit metrics on fiber bundles // Algebra Groups Geom.—1998.— Vol. 15.—P. 55-67.
23. Tamaru H. Riemannin g. o. spaces fibered over irreducible symmetric spaces // Osaka J. Math.—1999.— Vol. 36.—P. 835-851.
24. Wolf J. A. Harmonic Analysis on Commutative Spaces.—Providence (R. I.): Amer. Math. Soc., 2007.— XVI+387 p.
25. Yakimova O. S. Weakly symmetric Riemannian manifolds with a reductive isometry group // Mat. Sb.—2004.—Vol. 195, № 4.—P. 143-160.—[Russian]; English transl. in Sb. Math.—2004.—Vol. 195, № 3-4.—P. 599-614.
26. Ziller W. The Jacobi equation on naturally reductive compact Riemannian homogeneous spaces // Comment. Math. Helv.—1977.—Vol. 52.—P. 573-590.
27. Ziller W. Homogeneous Einstein metrics on Spheres and projective spaces // Math. Ann.—1982.— Vol. 259.—P. 351-358.
28. Ziller W. Weakly symmetric spaces // Progress in Nonlinear Dif. Eq. Vol. 20. Topics in geometry: in memory of Joseph D'Atri.—Birkhauser, 1996.—P. 355-368.
Received November 6, 2012.
Nikonorov YuriI Gennadievich South Mathematical Institute of VSC RAS Vladikavkaz, Markus str., 22, 362027, RUSSIA E-mail: nikonorov2006@mail.ru
ГЕОДЕЗИЧЕСКИ ОРБИТАЛЬНЫЕ МЕТРИКИ НА СФЕРАХ
Никоноров Ю. Г.
В данной работе получена полная классификация геодезически орбитальных римановых метрик на сферах Яп. Также найдены явные выражения геодезических векторов для Яр(и + 1 )и(1)-инвари-антных метрик на S4"+3.
Ключевые слова: однородные пространства, однородные римановы многообразия, естественно ре-дуктивные римановы многообразия, нормальные однородные римановы многообразия, геодезически орбитальные пространства, симметрические пространства, слабо симметрические пространства.
