Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems
aa r X i v : . [ m a t h . DG ] J un HOMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS.RECENT RESULTS AND SOME OPEN PROBLEMS
ANDREAS ARVANITOYEORGOS
Abstract.
A homogeneous Riemannian manifold ( M = G/K, g ) is called a spacewith homogeneous geodesics or a G -g.o. space if every geodesic γ ( t ) of M is an orbitof a one-parameter subgroup of G , that is γ ( t ) = exp( tX ) · o , for some non zerovector X in the Lie algebra of G . We give an exposition on the subject, by presentingtechniques that have been used so far and a wide selection of previous and recentresults. We also present some open problems. Primary 53C25. Secondary 53C30.
Key words.
Homogeneous geodesic; g.o. space; invariant metric; geodesic vector;geodesic lemma; naturally reductive space; generalized flag manifold; generalizedWallach space; M -space; δ -homogeneous space; pseudo-Riemannian manifold; two-step homogeneous geodesic; Introduction
The aim of the present article is to give an exposition on developments about homo-geneous geodesics in Riemannian homogeneous spaces, to present various recent resultsand discuss some open problems. One of the demanding problems in Riemannian geom-etry is the description of geodesics. By making some symmetry assumptions one couldexpect that certain simplifications may accur. Let (
M, g ) be a homogeneous Riemann-ian manifold, i.e. a connected Riemannian manifold on which the largest connectedgroup G of isometries acts transitively. Then M can be expressed as a homogeneousspace ( G/K, g ), where K is the isotropy group at a fixed point o of M .Motivated by well known facts such that, the geodesics in a Lie group G with abi-invariant metric are the one-parameter subgroups of G , or that the geodesics in aRiemannian symmetric space G/K are orbits of one-parameter subgroups in
G/K , itis natural to search for geodesics in a homogeneous space, which are orbits. Moreprecisely, a geodesic γ ( t ) through the origin o of M = G/K is called homogeneous if itis an orbit of a one-parameter subgroup of G , that is γ ( t ) = exp( tX ) · o, t ∈ R , (1)where X is a non zero vector in the Lie algebra g of G . A non zero vector X ∈ g is called a geodesic vector if the curve (1) is a geodesic. A homogeneous Riemannianmanifold M = G/K is called a g.o. space if all geodesics are homogeneous with respectto the largest connected group of isometries I o ( M ). Since their first systematic studyby O. Kowalski and L. Vanhecke in [43], there has been a lot of research related tohomogeneous geodesics and g.o spaces and in various directions.Homogeneous geodesics appear quite often in physics as well. The equation of motionof many systems of classical mechanics reduces to the geodesic equation in an appro-priate Riemannian manifold M . Homogeneous geodesics in M correspond to “relativeequilibriums” of the corresponding system (cf. [5]). For further information about rela-tive equilibria in physics we refer to [34] and references therein. In Lorentzian geometry in particular, homogeneous spaces with the property that all their null geodesics arehomogeneous, are candidates for constructing solutions to the 11-dimensional super-gravity, which preserve more than 24 of the available 32 supersymmetries. In fact,all Penrose limits, preserving the amount of supersymmetry of such a solution, mustpreserve homogeneity. This is the case for the Penrose limit of a reductive homo-geneous spacetime along a null homogeneous geodesic ([33], [48], [52]). For a recentmathematical contribution in this topic see [26].All naturally reductive spaces are g.o. spaces ([39]), but the converse is not truein general. In [37] A. Kaplan proved the existence of g.o. spaces that are in no waynaturally reductive. These are generalized Heisenberg groups with two dimensionalcenter. Another important class of g.o. spaces are the weakly symmetric spaces. Theseare homogeneous Riemannian manifolds ( M = G/K, g ) introduced by A. Selberg in[54], with the property that every two points can be interchanged by an isometry of M . In [13] J. Berndt, O. Kowalski and L. Vanhecke proved that weakly symmetricspaces are g.o. spaces. In [40] O. Kowalski, F. Pr¨ufer and L. Vanhecke gave an explicitclassification of all naturally reductive spaces up to dimension five.The term g.o. space was introduced by O. Kowalski and L. Vanhecke in [43], wherethey gave the classification of all g.o. spaces up to dimension six, which are in no waynaturally reductive. Concerning the existence of homogeneous geodesics in a homo-geneous Riemannian manifold, we recall the following. In ([36]) V.V. Kajzer provedthat a Lie group endowed with a left-invariant metric admits at least one homoge-neous geodesic. O. Kowalski and J. Szenthe extended this result to all homogeneousRiemannian manifolds ([42]). An extension of this result to reductive homogeneouspseudo-Riemannian manifolds was obtained ([29], [52]).In [35] C. Gordon described g.o. spaces which are nilmanifolds and in [59] H. Tamaruclassified homogeneous g.o. spaces which are fibered over irreducible symmetric spaces.In [24] and [28] O. Kowalski and Z. Duˇsek investigated homogeneous geodesics inHeisenberg groups and some H -type groups. Examples of g.o. spaces in dimensionseven were obtained by Duˇsek, O. Kowalski and S. Nikˇcevi´c in [30].In [2] the author and D.V. Alekseevsky classified generalized flag manifolds whichare g.o. spaces. Further, D.V. Alekseevsky and Yu. G. Nikonorov in [3] studied thestructure of compact g.o. spaces and gave some sufficient conditions for existence andnon existence of an invariant metric with homogeneous geodesics on a homogeneousspace of a compact Lie group. They also gave a classification of compact simplyconnected g.o. spaces of positive Euler characteristic.In [38] O. Kowalski, S. Nikˇcevi´c and Z. Vl´aˇsek studied homogeneous geodesics in ho-mogneous Riemannian manifolds, and in [47], [18] G. Calvaruso and R. Marinosci stud-ied homogeneous geodesics in three-dimensional Lie groups. Homogeneous geodesicswere also studied by J. Szenthe in [55], [56], [57], [58]. Also, D. Latifi studied homoge-neous geodesics in homogeneous Finsler spaces ([44]), and the first author investigatedhomogeneous geodesics in the flag manifold SO(2 l + 1) / U( l − m ) × SO(2 m + 1) ([6]).Homogeneous geodesics in the affine setting were studied in [24] and [31] (and inparticular for any non reductive pseudo-Riemannian manifold).Finally, a class of homogeneous spaces which satisfy the g.o. property are the δ -homogeneous spaces, which were introduced by V. Berestovskiˇi and C. Plaut in [12].These spaces have interesting geometrical properties, but we will not persue here. Werefer to the paper [11] by V. Berestovskiˇi and Yu.G. Nikonorov for more informationib this direction. OMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS 3
The paper is organized as follows. In Section 1 we present the basic techniques forfinding homogeneous geodesics and detecting if a homogeneous space is a space withhomogeneous geodesics (g.o. space). In Section 2 we present the classification up todimension 6 and give examples in dimension 7. In Section 3 we discuss homogeneousg.o. spaces which are fibered over irreducible symmetric spaces and in Section 4 wepresent the classification of generalized flag manifolds which are g.o. spaces. In Section5 we present results about another wide class of homogeneous spaces, the generalizedWallach spaces, and in Section 6 we discuss results related to M -spaces. These arehomogeneous spaces G/K so that G/ ( S × K ) is a generalized flag manifold, where S atorus in a compact simple Lie group G . The pseudo-Riemannian setting is presented inSection 7. In Section 8 we discuss a generalization of homogeneous geodesics which wecall two-step homogeneous geodesics. These are orbits of the product of two exponentialfactors. Finally, in Section 9 we present some open problems. Acknowledgements.
The author was supported by Grant
Homogeneous geodesics in homogeneous Riemannian manifolds -Techniques
Let ( M = G/K, g ) be a homogeneous space of a compact, connected and semisimpleLie group and let g , k be the Lie algebras of G and K respecively. We consider anorthogonal reductive decomposition g = k ⊕ m (2)with respect to the negative of the Killing form of g , denoted by B . The canonicalprojection π : G → G/K induces an isomorphism between the subspace m and thetangent space T o M at the identity o = eK . A G -invariant Riemannian metric g definesa scalar product h· , ·i on m which is Ad( K )-invariant. Then any Ad( K )-invariant scalarproduct h· , ·i on m can be expressed as h x, y i = B (Λ x, y ) ( x, y ∈ m ), where Λ is anAd( K )-equivariant positive definite symmetric operator on m . Conversely, any suchoperator Λ determines an Ad( K )-invariant scalar product h x, y i = B (Λ x, y ) on m ,which in turn determines a G -invariant Riemannian metric g on m . We say that Λis the operator associated to the metric g , or simply the associated operator . Also,a Riemannian metric generated by the scalar product product B is called standardmetric . Definition 1.
A homogeneous Riemannian manifold ( M = G/K, g ) is called a spacewith homogeneous geodesics, or G -g.o. space if every geodesic γ of M is an orbit ofa one-parameter subgroup of G , that is γ ( t ) = exp( tX ) · o , for some non zero vector X ∈ g . The invariant metric g is called G -g.o. metric. If G is the full isometrygroup, then the G -g.o. space is called a manifold with homogeneous geodesics, or a g.o.manifold. Notice that if all geodesics through the origin o = eK are of the form γ ( t ) = exp( tX ) · o , then the geodesics through any other point a · p ( a ∈ G, p ∈ M ) is of the form aγ ( t ) = exp( t Ad( a ) X ) · ( a · p ). Definition 2.
A non zero vector X ∈ g is called a geodesic vector if the curve (1) isa geodesic. ANDREAS ARVANITOYEORGOS
All calculations for a g.o. space
G/K can be reduced to algebraic computationsusing geodesic vectors. These can be computed by using the following fundamentalresult of the subject, still call it “lemma” by tradinion:
Lemma 1 (Geodesic Lemma [43]) . A nonzero vector X ∈ g is a geodesic vector if andonly if h [ X, Y ] m , X m i = 0 , (3) for all Y ∈ m . Here the subscript m denotes the projection into m . A useful description of homogeneous geodesics (1) is provided by the following :
Proposition 1. ([2])
Let ( M = G/K, g ) be a homogeneous Riemannian manifold and Λ be the associated operator. Let a ∈ k and x ∈ m . Then the following are equivalent:(1) The orbit γ ( t ) = exp t ( a + x ) · o of the one-parameter subgroup exp t ( a + x ) throughthe point o = eK is a geodesic of M .(2) [ a + x, Λ x ] ∈ k .(3) h [ a, x ] , y i = h x, [ x, y ] m i for all y ∈ m .(4) h [ a + x, y ] m , x i = 0 for all y ∈ m . As a consequence, we obtain the following characterization of g.o. spaces:
Corollary 1 ([2]) . 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 ∈ m there exists an a ( x ) ∈ k such that [ a ( x ) + x, Λ x ] ∈ k . Therefore, the property of being a g.o. space
G/K , depends only on the reductivedecomposition and the G -invariant metric metric g on m . That is, if ( M = G/H, g ) is ag.o. space, then any locally isomorphic homogeneous Riemannian space ( M = G/H, g ′ )is a g.o. space. Also, a direct product of Riemannian manifolds is a manifold withhomogeneous geodesics if and only if each factor is a manifold with homogeneousgeodesics.In order to find all homogeneous geodesics in a homogeneous Riemannian manifold( M = G/K, g ) it suffices to find a decomposition of the form (2) and look for geodesicvectors of the form X = s X i =1 x i e i + l X j =1 a j A j . (4)Here { e i : i = 1 , , . . . , s } is a convenient basis of m and { A j : j = 1 , , . . . , l } is a basisof k . By substituting X = e i ( i = 1 , . . . , s ) into equation (3) we obtain a system oflinear algebraic equations for the variables x i and a j . The geodesic vectors correspondto those solutions for which x , . . . , x s are not all equal to zero. For some applicationsof this method we refer to [38] and [47]. Also, ( M = G/K, g ) is a g.o. space if andonly if for every non zero s -tuple ( x , . . . , x s ) there is an l -tuple ( a , . . . , a l ) satisfyingall quadratic equations.A useful technique used for the characterization of Riemannian g.o. spaces is basedon the concept of the geodesic graph, originally introduced in [55]. We first need thefollowing definition. Definition 3.
A Riemannian homogeneous space ( G/K, g ) is called naturally reductiveif there exists a reductive decomposition (2) of g such that h [ X, Z ] m , Y i + h X, [ Z, Y ] m i = 0 , for all X, Y, Z ∈ m . (5) OMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS 5
It is well known that condition (5) implies that all geodesics in
G/K are homogeneous(e.g. [51]).
Definition 4.
A homogeneous Riemannian manifold ( M, g ) is naturally reductive ifthere exists a transitive group G of isometries for which the correseponding Riemannianhomogeneous space ( G/K, g ) is naturally reductive in the sense of Definition 3. Therefore, it could be possible that a homogeneous space M = G/K is not naturallyreductive for some group G ∈ I ( M ) (the connected component of the full isometrygroup of M ), but it is naturally reductive if we write M = G ′ /K ′ for some larger groupof isometries G ′ ⊂ I ( M ).Let ( M = G/K, g ) be a g.o. space and let g = k ⊕ m be an Ad( K )-invariant decom-position. Then(1) There exists an Ad( H )-equivariant map η : m → k (a geodesic graph ) such thatfor any X ∈ m \{ } , the curve exp t ( X + η ( X )) · o is a geodesic.(2) A geodesic graph is either linear (which is equivalent to natural reductivity withrespect to some Ad( K )-invariant decomposition g = k ⊕ m ′ ) or it is non differentiableat the origin o .It can be shown ([41]) that a geodesic graph (for a g.o. space) is uniquely determinedby fixing an Ad( H )-invariant scalar product on k . Examples of g.o. spaces by usinggeodesic graphs are given in [27], [30], and [41]. Conversely, the property (1) impliesthat G/K is a g.o. space.Another technique for producing g.o. metrics was given by C. Gordon as shownbelow:
Proposition 2. ([35], [59])
Let G be a connected semisimple Lie group and H ⊃ K becompact Lie subgroups in G . Let M F and M C be the tangent spaces of F = H/K and C = G/H respectively. Then the metric g a,b = aB | M F + bB | M C , ( a, b ∈ R + ) is a g.o.metric on G/K if and only if for any v F ∈ M F , v C ∈ M C there exists X ∈ k such that [ X, v F ] = [ X + v F , v C ] = 0 . Actually, Gordon proved a more general result based on description of naturallyreductive left-invariant metrics on compact Lie groups given by J.E. D’Atri and W.Ziller in [22]. 2.
Low dimensional examples
The problem of a complete classification of g.o. manifolds is open. Even the classi-fication all g.o. metrics on a given Riemannian homogeneous space is not trivial (cf.for example [49]). A complete classification is known up to dimesion 6, given by O.Kowalski and L. Vanhecke:
Theorem 1. ([43]) 1)
All Riemannian g.o. spaces of dimension up to are naturallyreductive. Every -dimensional Riemannian g.o. space is either naturally reductive, or ofisotropy type SU(2) . Every -dimensional Riemannian g.o. space is either naturally reductive or oneof the following: (a) A two-step nilpotent Lie group with two-dimensional center, equipped with aleft-invariant Riemannian metric such that the maximal connected isotropy group isisomorphic to either
SU(2) or U(2) . Corresponding g.o. metrics depend on three realparameters.
ANDREAS ARVANITOYEORGOS (b)
The universal covering space of a homogeneous Riemannian manifold of the form ( M = SO(5) / U(2) , g ) or ( M = SO(4 , / U(2) , g ) , where SO(5) or SO(4 , is the iden-tity component of the full isometry group, respectively. In each case, all correspondinginvariant metrics g.o. metrics g depend on two real parameters. As pointed out by the authors in [43, p. 190], the g.o. spaces (a) and (b) are in noway naturally reductive in the following sence: whatever the representation of (
M, g )as a quotient of the form G ′ /K ′ , where G ′ is a connected transitive group of isometriesof ( M, g ), and whatever is the Ad( K )-invariant decomposition g ′ = k ′ ⊕ m ′ , the curve γ ( t ) = exp( tX ) · o is never a geodesic (for any X ∈ m \{ } ).The first 7-dimensional example of a g.o. manifold was given by C. Gordon in [35].This is a nilmanifold (i.e. a connected Riemannian manifold admitting a transitivenilpotent group of isometries), and it was obtained under a general construction ofg.o. nilmanifolds. It took some time until some more 7-dimensional examples weregiven. In [30] Z. Duˇsek, O. Kowalski and S. Nikˇcevi´c gave families of 7-dimensionalg.o. metrics. Their main result is the following: Theorem 2. ([30])
On the -dimensional homogeneous space G/K = (SO(5) × SO(2)) / U(2) (or
G/H = (SO(4 , × SO(2)) / U(2) ) there is a family g p,q of invariant metrics depend-ing on two parameters p, q (where the pairs ( p, q ) fill in an open subset of the plane)such that each homogeneous Riemannian manifold ( G/H, g p,q ) is a locally irreducibleand not naturally reductive Riemannian g.o. manifold. Fibrations over symmetric spaces
In the work [59] H. Tamaru classified homogeneous spaces M = G/K satisfying thefollowing properties: (i) M is fibered over irreducuble symmetric spaces G/H and (ii)certain G -invariant metrics on M are G -g.o. metrics. More precisely, for G connectedand semisimple, and H, K compact with G ⊃ H ⊃ K , he considered the fibration F = H/K → M = G/K → B = G/H and the G -invariant metrics g a,b on M determined by the scalar products h , i = aB | f + bB | b , a, b > . Here f and b are the tangent spaces of F and B respectively, so that the tangent spaceof M at the origin is identified with f ⊕ b . By using results from polar representations,he classifed all triplets ( G, H, K ) so that the metrics g a,b are G -g.o. metrics. Thetriplets of Lie algebras ( g , h , k ) so that ( g , h ) is a symmetric pair and ( g , k ) correspondsto a G -g.o. space G/K , are listed in Table 1.4.
Generalized flag manifolds
In the work [2] D.V. Alekseevsky and the author classified generalized flag manifoldswith homogeneous geodesics. Recall that a generalized flag manifold is a homoge-neous space
G/K which is an adjoint orbit of a compact semisimple Lie group G .Equivalently, the isotropy subgroup K is the centralizer of a torus (i.e. a maximalabelian subgroup) in G . We assume that G acts effectively on M . A flag mani-fold M = G/K is simply connected and has the canonically defined decomposition M = G/K = G /K × G /K × · · · × G n /K n , where G , . . . , G n are simple factors ofthe (connected) Lie group G . This decomposition is the de Rham decomposition of M equipped with a G -invariant metric g . In particular, ( M, g ) is a g.o. space if and only
OMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS 7 g h k so (2 n + 1) , n ≥ so (2 n ) u ( n )2 so (4 n + 1) , n ≥ so (4 n ) su (2 n )3 so (8) so (7) g so (9) so (8) so (7)5 su ( n + 1) , n ≥ u ( n ) su ( n )6 su (2 n + 1) , n ≥ u (2 n ) u (1) ⊕ sp ( n )7 su (2 n + 1) , n ≥ u (2 n ) sp ( n )8 sp ( n + 1) , n ≥ sp (1) ⊕ sp ( n ) u (1) ⊕ sp ( n )9 sp ( n + 1) , n ≥ sp (1) ⊕ sp ( n ) sp ( n )10 su (2 r + n ) , r ≥ , n ≥ su ( r ) ⊕ su ( r + n ) ⊕ R su ( r ) ⊕ su ( r + n )11 so (4 n + 2) , n ≥ u (2 n + 1) su (2 n + 1)12 e R ⊕ so (10) so (10)13 so (9) so (7) ⊕ so (2) g ⊕ so (2)14 so (10) so (8) ⊕ so (2) spin (7) ⊕ so (2)15 so (11) so (8) ⊕ so (3) spin (7) ⊕ so (3) Table 1.
Riemannian g.o. spaces
G/K fibered over irreducible sym-metric spaces
G/H ([59]).if each factor ( M i = G i /K i , g i = g | M i ) is a g.o. space. This reduces the problem ofthe description of G -invariant metrics with homogeneous geodesics in a flag manifold M = G/K to the case when the group G is simple.Flag manifolds M = G/K with G a simple Lie group can be classified in terms oftheir painted Dynkin diagrams . It turns out that for each classical Lie group there is aninfinite series of flag manifolds, and for each of the exceptional Lie groups G , F , E , E , and E there are 3, 11, 16, 31, and 40 non equivalent flag manifolds respectively(eg. [1], [14]). An important invariant of flag manifolds is their set of T -roots R T . Thisis defined as the restriction of the root system R of g to the center t of the stabilitysubalgebra k of K . In [2] we defined the notion of connected component of R T , namelytwo T -roots are in the same component if they can be connected by a chain of T -rootswhose sum or difference is also a T -root. The set R T is called connected if it has onlyone connected component. Theorem 3. ([2])
If the set of T -roots is connected then the standard metric on M = G/K is the only G -invariant metric (up to scalar) which is a g.o. metric. Hence, for a flag manifold M = G/K ( G simple), a G -invariant g.o. metric may exist,only when R T is not connected, so we only need to study those flag manifolds. It turnsout that the system of T -roots is not connected only for three infinite series of a classicalLie group (namely the spaces SO(2 ℓ +1) / U( ℓ − m ) · SO(2 m +1), Sp( ℓ ) / U( ℓ − m ) · Sp( m ),and SO(2 ℓ ) / U( ℓ − m ) · SO(2 m )), and for 10 flag manifolds of an exceptional Lie group.An perpective of the above theorem is given by the following theorem: Theorem 4. ([2])
Let M = G/K be a flag manifold of a simple Lie group. Then theset of T -roots is not connected if and only if the isotropy representation of M consistsof two irreducible (non-equivalent) components. Therefore, the problem of the description of G -invariant metrics on flag manifoldswith homogeneous geodesics reduces substantially to the study of this short list of ANDREAS ARVANITOYEORGOS prospective flag manifolds. To this end, we used the classification Table 1 of the workof H. Tamaru ([59]). Since any flag manifold can be fibered over a symmetric space([15]), then by using Theorem 4 we obtain that the only flag manifolds which are inTable 1 are SO(2 ℓ + 1) / U( ℓ ) and Sp( ℓ ) / U(1) · Sp( ℓ − ℓ + 1) / U( ℓ ) and C (1 , ℓ −
1) = Sp( ℓ ) /U (1) · Sp( ℓ − ℓ +1)-invariant metric g λ on SO(2 ℓ +1) / U( ℓ ) (depending,up to scale, on one real parameter λ ) is weakly symmetric, hence it has homogeneousgeodesics. Similarly for any Sp( ℓ )-invariant metric g λ on Sp( ℓ ) /U (1) · Sp( ℓ − ℓ + 1) on SO(2 ℓ + 1) / U( ℓ ) can be extented to theaction of the group SO(2 ℓ + 2) with isotropy subgroup U ( ℓ + 1), which preserves thecomplex structure and the standard invariant metric g (which corresponds to λ = 1).Hence, the Riemannian flag manifold (SO(2 ℓ +1) / U( ℓ ) , g ) is isometric to the Hermitiansymmetric space Com( R ℓ +2 ) = SO(2 ℓ + 2) / U( ℓ + 1) of all complex structures in R ℓ +2 .Similarly, the action of the group Sp( ℓ ) on Sp( ℓ ) /U (1) · Sp( ℓ −
1) can be extended tothe action of the group SU(2 ℓ ) with isotropy subgroup S(U(1) × U(2 ℓ − Theorem 5. ([2])
The only flag manifolds M = G/K of a simple Lie group G admitinga non naturally reductive G -invariant metric g with homogeneous geodesics are themanifolds SO(2 ℓ +1) /U ( ℓ ) and Sp( ℓ ) / U(1) · Sp( ℓ −
1) ( ℓ ≥ , which admit (up to scale)a one-parameter family of SO(2 ℓ + 1) (resp. Sp( ℓ ) )-invariant metrics g λ . Moreover,these manifolds are weakly symmetric spaces for λ > , and they are symmetric spaceswith respect to Isom ( g λ ) if and only if λ = 1 , i.e. g λ is a multiple of the standardmetric. Note that for ℓ = 2 we obtain Sp(2) /U (1) · Sp(1) ∼ = SO(5) / U(2), where the secondquotient is an example of g.o. space in [43] which is not naturally reductive.Finally, we mention a remarkable coincidence between Theorem 5 and a resut by F.Podest`a and G. Thorbergsson in [53], where they studied coisotropic actions on flagmanifolds. One of their theorems states that if M = G/K is a flag manifold of asimple Lie group then the action of K on M is coisotropic, if and only if M is up tolocal isomorphy either a Hermitian symmetric space, or one of the spaces obtained inTheorem 5. 5. Generalized Wallach spaces
Let
G/K be a compact homogeneous space with connected compact semisimpleLie group G and a compact subgroup K with reductive decomposition g = k ⊕ m .Then G/K is called generalized Wallach space (known before as three-locally-symmetricspaces, cf. [45]) if the module m decomposes into a direct sum of three Ad( K )-invariantirreducible modules pairwise orthogonal with respect to B , i.e. m = m ⊕ m ⊕ m , such that [ m i , m i ] ⊂ k i = 1 , ,
3. Every generalized Wallach space admits a three pa-rameter family of invariant Riemannian metrics determined by Ad( K )-invariant innerproducts h· , ·i = λ B ( · , · ) | m + λ B ( · , · ) | m + λ B ( · , · ) | m , where λ , λ , λ are positivereal numbers.A classification of generalized Wallach spaces was recently obtained by Yu.G. Niko-ronov ([50]) and Z. Chen, Y. Kang, K. Liang ([16]) as follows: OMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS 9
Theorem 6 ([50], [16]) . Let
G/K be a connected and simply connected compact ho-mogeneous space. Then
G/K is a generalized Wallach space if and only if it is one ofthe following types:1)
G/K is a direct product of three irreducible symmetric spaces of compact type.2) The group is simple and the pair ( g , k ) is one of the pairs in Table 1.3) G = F × F × F × F and H = diag ( F ) ⊂ G for some connected, compact, simpleLie group F , with the following description on the Lie algebra level: ( g , k ) = ( f ⊕ f ⊕ f ⊕ f , diag( f )) = { ( X, X, X, X ) | X ∈ f } , where f is the Lie algebra of F , and (up to permutation) m = { ( X, X, − X, − X ) | X ∈ f } , m = { ( X, − X, X, − X ) | X ∈ f } , m = { ( X, − X, − X, X ) | X ∈ f } . g k g kso ( k + l + m ) so ( k ) ⊕ so ( l ) ⊕ so ( m ) e so (8) ⊕ sp (1) su ( k + l + m ) su ( k ) ⊕ su ( l ) ⊕ su ( m ) e su (6) ⊕ sp (1) ⊕ R sp ( k + l + m ) sp ( k ) ⊕ sp ( l ) ⊕ sp ( m ) e so (8) su (2 l ) , l ≥ u ( l ) e so (12) ⊕ sp (1) so (2 l ) , l ≥ u ( l ) ⊕ u ( l − e so (8) ⊕ so (8) e su (4) ⊕ sp (1) ⊕ R f so (5) ⊕ sp (1) e so (8) ⊕ R f so (8) e sp (3) ⊕ sp (1)Table 2. The pairs ( g , k ) corresponding to generalized Wallach spaces G/K with G simple([50]). In [9] Yu Wang and the author investigated which of the families of spaces listed inTheorem 8 are g.o. spaces. By applying the method of searching for geodesics vectorsshown at the end of Section 1 we obtained the following:
Theorem 7. [9]
Let ( G/K, g ) be a generalized Wallach space as listed in Theorem 8.Then1) If ( G/K, g ) is a space of type 1) then this is a g.o. space for any Ad( K ) -invariantRiemannian 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. However, in order to find all homogeneous geodesics in
G/K it suffices to find allthe real solutions of a system of dim m + dim m + dim m quadratic equations.By Theorem 7 we only need to consider homogeneous geodesics for spaces of types2) and 3) given in Theorem 8, for the metric ( λ , λ , λ ), where at least two of λ , λ , λ are different. This is not easy in general. We obtained all homogeneous geodesics (forvarious values of the parameters λ , λ , λ for the generalized Wallach space SU (2) / { e } ,hence recovering a result on R.A. Marinosci ([47, p. 266]), and for the Stiefel manifoldsSO( n ) / SO( n − n ≥ M -spaces Let
G/K be a generalized flag manifold with K = C ( S ) = S × K , where S is atorus in a compact simple Lie group G and K is the semisimple part of K . Then the associated M -space is the homogeneous space G/K . These spaces were introducedand studied by H.C. Wang in [60]. In the work [10] Y. Wang, G. Zhao and the author investigated homogeneous geodesicsin a class of homogeneous spaces called M -spaces. We proved that for various classesof M -spaces, the only g.o. metric is the standard metric. For other classes of M -spaceswe give either necessary or necessary and sufficient conditions so that a G -invariantmetric on G/K is a g.o. metric. The analysis is based on properties of the isotropyrepresentation m = m ⊕ · · · ⊕ m s of the flag manifold G/K , in particular on thedimension of the submodules m i . We summarize these results below.Let g and k be the Lie algebras of the Lie groups G and K respectively. Let g = k ⊕ m be an Ad( K )-invariant reductive decomposition of the Lie algebra g , where m ∼ = T o ( G/K ). This is orthogonal with respect to B = − Killing from on g . Assumethat m = m ⊕ · · · ⊕ m s (6)is a B -orthogonal decomposition of m into pairwise inequivalent irreducible ad( k )-modules.Let G/K be the corresponding M -space and s and k be the Lie algebras of S and K respectively. We denote by n the tangent space T o ( G/K ), where o = eK . A G -invariant metric g on G/K induces a scalar product h· , ·i on n which is Ad( K )-invariant. Such an Ad( K )-invariant scalar product h· , ·i on n can be expressed as h x, y i = B (Λ x, y ) ( x, y ∈ n ), where Λ is the Ad( K )-equivariant positive definitesymmetric operator on n .The main results are the following: Theorem 8. ([10])
Let
G/K be a generalized flag manifold with s ≥ in the decom-position (2). Let G/K be the corresponding M -space.1) If dim m i = 2 ( i = 1 , . . . , s ) and ( G/K , g ) is a g.o. space, then g = h· , ·i = µB ( · , · ) | s + λB ( · , · ) | m ⊕ m ⊕···⊕ m s , ( µ, λ > .
2) If there exists some j ∈ { , . . . , s } such that dim m j = 2 , then ( G/K , g ) is a g.o.space if and only if g is the standard metric. Theorem 9. ([10])
Let
G/K be a generalized flag manifold with two isotropy summands m = m ⊕ m and let ( G/K , g ) be the corresponding M -space.1) If dim m = 2 then the standard metric is the only g.o. metric on M , unless G/K = SO(5) / SU(2) or Sp( n ) / Sp( n − .2) If dim m i = 2 ( i = 1 , ) and the corresponding M -space ( G/K , g ) is a g.o. space,then g = h· , ·i = µB ( · , · ) | s + λB ( · , · ) | m ⊕ m , ( µ, λ > , unless G/K = SO(2 n +1) / SU( n ) , n > . However, the spaces SO(5) / SU(2) and Sp( n ) / Sp( n −
1) are included in Tamaru’sTable 1, therefore they admit g.o. metrics.7.
Homogeneous geodesics in pseudo-Riemannian manifolds
It is well known that any homogeneous Riemannian manifold is reductive, but this isnot the case for pseudo-Riemannian manifolds in general. In fact, there exist homoge-neous pseudo-Riemannian manifolds which do not admit any reductive decomposition.Therefore, there is a dichotomy in the study of geometrical problems between reduc-tive and non reductive pseudo-Riemannian manifolds. Due to the existence of nullvectors in a pseudo-Riemannian manifold the definition of a homogeneous geodesic
OMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS 11 γ ( t ) = exp( tX ) · o needs to be modified by requiring that ∇ ˙ γ ˙ γ = k ( γ ) ˙ γ (see also rel-evant discussion in [46, pp. 90-91]). It turns out that k ( γ ) is a constant function (cf.[29].Even though an algebraic characterization of geodesic vectors (that is an analogueof the geodesic Lemma 1) was known to physicists ([33], [52]), a formal proof was givenby Z. Duˇsek and O. Kowalski in [29]. Lemma 2 ([29]) . Let M = G/H be a reductive homogeneous pseudo-Riemannian spacewith reductive decomposition g = m ⊕ h , and X ∈ g . Then the curve γ ( t ) = exp( tX ) · o is a geodesic curve with respect to some parameter s if and only if h [ V, Z ] m , V m i = k h V m , Z m i , for all Z ∈ m , where k is some real constant. Moreover, if k = 0 , then t is an affine parameter forthis geodesic. If k = 0 , then s = e kt is an affine parameter for the geodesic. This occursonly if the curve γ ( t ) is a null curve in a (properly) pseudo-Riemannian space. For applications of this lemma see [26]. The existence of homogeneous geodesics inhomogeneous pseudo-Riemannian spaces (for both reductive and non reductive) wasanswered positively only recently by Z. Duˇsek in [25].Two-dimensional and three-dimensional homogeneous pseudo-Riemannian manifoldsare reductive ([17], [32]). Four-dimensional non reductive homogeneous pseudo-Riemannianmanifolds were classified by M.E. Fels and A.G. Renner in [32] in terms of their nonreductive Lie algebras. Their invariant pseudo-Riemannian metrics, together with theirconnection and curvature, were explicitly described in by G. Calvaruso and A. Fino in[20].The three-dimensional pseudo-Riemannian g.o. spaces were classified by G. Cal-varuso and Marinosci in [19]. In the recent work [21], G. Calvaruso, A. Fino and A. Za-eim obtained explicit examples of four-dimensional non reductive pseudo-Riemanniang.o. spaces. They deduced an explicit description in coordinates for all invariantmetrics of non reductive homogeneous pseudo-Riemannian four-manifolds. For thosefour-dimensional non reductive pseudo-Riemannia spaces which are not g.o., they de-termined the homogeneous geodesics though a point.8.
Two-step homogeneous geodesics
In the work [8] N.P. Souris and the author considered a generalisation of homogeneousgeodesics, namely geodesics of the form γ ( t ) = exp( tX ) exp( tY ) · o, X, Y ∈ g , (7)which we named two-step homogeneous geodesics . We obtained sufficient conditionson a Riemannian homogeneous space G/K , which imply the existence of two-stephomogeneous geodesics in
G/K . A Riemannian homogeneous spaces
G/K such thatany geodesic of
G/K passing through the origin is two-step homogeneous is called two-step g.o. spaces .Geodesics of the form (7) had appeared in the work [61] of H.C. Wang as geodesics ina semisimple Lie group G , equipped with a metric induced by a Cartan involution of theLie algebra g of G . Also, in [23] R. Dohira proved that if the tangent space T o ( G/K ) of ahomogeneous space splits into submodules m , m satisfying certain algebraic relations,and if G/K is endowed with a special one parameter family of Riemannian metrics g c ,then all geodesics of the Riemannian space ( G/K, g c ) are of the form (7). The mainresult of [8] is the following: Theorem 10. ([8])
Let M = G/K be a homogeneous space admitting a naturally reduc-tive Riemannian metric. Let B be the corresponding inner product on m = T o ( G/K ) .We assume that m admits an Ad( K ) -invariant orthogonal decomposition m = m ⊕ m ⊕ · · · ⊕ m s , (8) with respect to B . We equip G/K with a G -invariant Riemannian metric g correspond-ing to the Ad( K ) -invariant positive definite inner product h· , ·i = λ B | m + · · · + λ s B | m s , λ , . . . , λ s > . If ( m a , m b ) is a pair of submodules in the decomposition (8) such that [ m a , m b ] ⊂ m a , (9) then any geodesic γ of ( G/K, g ) with γ (0) = o and ˙ γ (0) ∈ m a ⊕ m b , is a two-stephomogeneous geodesic. In particular, if ˙ γ (0) = X a + X b ∈ m a ⊕ m b , then for every t ∈ R this geodesic is given by γ ( t ) = exp t ( X a + λX b ) exp t (1 − λ ) X b · o, where λ = λ b /λ a . Moreover, if either λ a = λ b or [ m a , m b ] = { } holds, then γ is a homogeneous geodesic,that is γ ( t ) = exp t ( X a + X b ) · o , for any t ∈ R . The following corollary provides a method to obtain many examples of two-step g.o.spaces.
Corollary 2.
Let M = G/K be a homogeneous space admitting a naturally reductiveRiemannian metric. Let B be the corresponding inner product of m = T o ( G/K ) . Weassume that m admits an Ad( K ) -invariant, B -orthogonal decomposition m = m ⊕ m ,such that [ m , m ] ⊂ m . Then M admits an one-parameter family of G -invariantRiemannian metrics g λ , λ ∈ R + , such that ( M, g λ ) is a two-step g.o. space. Eachmetric g λ corresponds to an Ad( K ) -invariant positive definite inner product on m ofthe form h , i = B | m + λ B | m , λ > . The above Corollary 2 is a generalisation of Dohira’s result [23].The main tool for the proof of Theorem 10 is the following proposition.
Proposition 3. ([7])
Let M = G/K be a homogeneous space and γ : R → M be thecurve γ ( t ) = exp( tX ) exp( tY ) exp( tZ ) · o , where X, Y, Z ∈ m . Let T : R → Aut( g ) bethe map T ( t ) = Ad(exp( − tZ ) exp( − tY )) . Then γ is a geodesic in M through o = eK if and only if for any W ∈ m , the function G W : R → R given by G W ( t ) = h ( T X ) m +( T Y ) m + Z m , [ W, T X + T Y + Z ] m i + h W, [ T X, T Y + Z ] m +[ T Y, Z ] m i , is identically zero, for every t ∈ R . The above proposition is a new tool towards the study of geodesics consisting ofmore than one exponential factors. In fact, for X = Y = 0 we obtain Lemma 1 ofKowalski and Vanhecke.A natural application of Corollary 2 is for total spaces of homogeneous Riemanniansubmersions, as shown below. Proposition 4.
Let G be a Lie group admitting a bi-invariant Riemannian metric andlet K, H be closed and connected subgroups of G , such that K ⊂ H ⊂ G . Let B bethe Ad -invariant positive definite inner product on the Lie algebra g corresponding tothe bi-invariant metric of G . We identify each of the spaces T o ( G/K ) , T o ( G/H ) and OMOGENEOUS MANIFOLDS WHOSE GEODESICS ARE ORBITS 13 T o ( H/K ) with corresponding subspaces m , m and m of g , such that m = m ⊕ m . Weendow G/K with the G -invariant Riemannian metric g λ corresponding to the Ad( K ) -invariant positive definite inner product h· , ·i = B | m + λ B | m , λ > . Then ( G/K, g λ ) is a two-step g.o. space. Example 1. ([7] ) The odd dimensional sphere S n +1 can be considered as the total spaceof the homogeneous Hopf bundle S → S n +1 → C P n . Let g be the standard metricof S n +1 . We equip S n +1 with an one parameter family of metrics g λ , which “deform”the standard metric along the Hopf circles S . By setting G = U( n + 1) , K = U( n ) and H = U( n ) × U(1) , the Hopf bundle corresponds to the fibration
H/K → G/K → G/H .Since U( n + 1) is compact, it admits a bi-invariant metric corresponding to an Ad(U( n + 1)) -invariant positive definite inner product B on u ( n + 1) . We identifyeach of the spaces T o S n +1 = T o ( G/K ) , T o C P n = T o ( G/H ) , and T o S = T o ( H/K ) withcorresponding subspaces m , m , and m of u ( n + 1) . The desired one parameter familyof metrics g λ corresponds to the one parameter family of positive definite inner products h , i = B | m + λ B | m , λ > on m = m ⊕ m .Then Proposition 4 implies that ( S n +1 , g λ ) is a two-step g.o. space. In particular, let X ∈ T o S n +1 . Then the unique geodesic γ of ( S n +1 , g λ ) with γ (0) = o and ˙ γ (0) = X ,is given by γ ( t ) = exp t ( X + λX ) exp t (1 − λ ) X · o , where X , X are the projectionsof X on m = T o C P n and m = T o S respectively.Note that if λ = 1 + ǫ , ǫ > , then the metrics g ǫ are Cheeger deformations of thenatural metric g . By using Proposition 2 it is possible to construct various classes of two-step g.ospaces. The recipe is the following:(i) Let
G/K be a homogeneous space with reductive decomposition g = k ⊕ m admittinga naturally reductive metric corresponding to a positive definite inner product B on m .(ii) We consider an Ad( K )-invariant, orthogonal decomposition m = n ⊕ · · · ⊕ n s withrespect to B .(iii) We separate the submodules n i into two groups as m = n i ⊕· · ·⊕ n i n and m = n i n +1 ⊕ · · · ⊕ n i s , so that [ m , m ] ⊂ m . The decomposition m = m ⊕ m is Ad( K )-invariant and orthogonal with respect to B .(iv) Then Corollary 2 implies that G/K admits an one parameter family of metrics g λ so that ( G/K, g λ ) is a two-step g.o. space.In [8] we applied the above recipe to the following classes of homogeneous spaces:1) Lie groups with bi-invariant metrics, equipped with an one parameter family ofleft-invariant metrics.2) Flag manifolds equipped with certain one parameter families of diagonal metrics.3) Generalized Wallach spaces equipped with three different types of diagonal metrics(thus recovering some results from [7].4) k -symmetric spaces where k is even, endowed with an one parameter family ofdiagonal metrics. 9. Some open problems
It seems that the target for a complete classification of homogeneous g.o. spaces inany dimension greater than seven is far for being accomplished. In dimension seventhere are several examples but a complete classification is still unknown. However, asshown in the present paper, for some large classes of homogeneous spaces it is possible to obtain some necessary conditions for the g.o. property. These conditions are normallyimposed by the special Lie theoretic structure of corresponding homogeneous space.Also, the problem of an explicit description of homogeneous geodesics for spaces whichare not g.o., is not trivial either. Eventhough it is mathematically simple, it requireshigh computational complexity.Another difficulty that one faces, is to show that the g.o. property of a homogeneousspace ( M = G/K, g ) does not depend on the representation as a coset space and onthe Ad( K )-invariant decomposition g = k ⊕ m , that is the space is in no way naturallyreductive. Therefore, we often stress that we study G -g.o. spaces.Also, it would be interesting to see how various results about Riemannian manifoldscould be adjusted to pseudo-Riemannian manifolds, such as Propositions 1, 3.Concerning generalizations of the g.o. propery, we have introduced the conceptof a two-step homogeneous geodesic and two-step g.o. space. We conjecture that asearch for three-step (or more) homogeneous geodesics reduces to two-step homoge-neous geodesics. References [1] D.V. Alekseevsky:
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University of Patras, Department of Mathematics, GR-26500 Patras, Greece
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