Singular Manakov Flows and Geodesic Flows on Homogeneous Spaces

SINGULAR MANAK O V FLO WS AND GEODESIC FLO WS ON HOMOGENE OUS SP A CES VLADIMIR DRAGO VI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANOVI ´ C Abstract. W e prov e complete integrabilit y of the Manako v-t ype S O ( n )-inv a- riant ge odesic flows on homog eneous spaces S O ( n ) /S O ( k 1 ) × · · · × S O ( k r ), for an y c hoice of k 1 , . . . , k r , k 1 + · · · + k r ≤ n . In particular, a new pro of of the int egrability of a Manak ov symmetric r igid b o dy motion around a fixed point is presente d. Al so, the pro of of integrabilit y of the S O ( n )-inv ar ian t Einstein metrics on S O ( k 1 + k 2 + k 3 ) /S O ( k 1 ) × S O ( k 2 ) × S O ( k 3 ) and on the Stiefel manifolds V ( n, k ) = S O ( n ) /S O ( k ) is gi v en. 1. Introduction It was F r ahm who gav e the first four-dimensiona l genera lization of the E ule r top in the second half of XIX century , [15]. U nfortunately , his pap er was for gotten for more than a century . A mo dern history of higher -dimensional genera liz ations of the Euler top has more than thirty years after the pap er of Manakov in 1976 [18]. Manako v used Dubrovin’s theory of matrix Lax op eartor s (see [10, 11]) to prov e that the so lutio ns of F ra hm-Manako v’s top can be giv en in terms of theta- functions. Although the sub ject has had intensiv e development since then, there are still few questions which in our opinion deser ve additional trea tmen t. 1.1. Liouvill e In tegrabilit y. Let ( M , {· , ·} ) b e a Poisson manifold. The equations (1) ˙ f = { f , H } , f ∈ C ∞ ( M ) are calle d Hamiltonian e quations with the Hamiltonian function H . A function f is an integral of the Hamilto nia n system (constant alo ng tra j ectories of (1)) if and only if it commutes with H : { f , H } = 0 . One of the central pro blems in Hamiltonian dynamics is whether the equations (1) a re co mpletely integrable or not. The e q uations (1) are c ompletely inte gr able or Liouvil le inte gr able if there are l = 1 2 (dim M + corank {· , ·} ) Poisson-commuting smo oth int egrals f 1 , . . . , f l whose differentials ar e independent in an op en dense subset of M . The set of in tegrals F = { f 1 , . . . , f l } is called a c omplete involutive set of functions o n M . T o distinguish the situa tion from the cas e o f non-co mm utative int egrability , the last s e t will b e calle d c ommutative as well. If the s ystem is completely integrable, by the Liouv ille -Arnold theor e m there is an implicitly given set of c o ordinates in which the sys tem trivialize s . Moreov er, from the Liouville- Arnold theorem [1] follows that reg ular compact connected in- v ariant submanifolds given by integrals F are Lagra ngian tori within appropr ia te 1 2 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C symplectic leaves of the Poisson bracket {· , ·} a nd the dynamics ov e r the inv ariant tori is qua si-p erio dic. 1.2. Noncommutativ e In tegrabili t y. Let ( M , { · , ·} ) be a Poisson ma nifold, Λ be the asso cia ted bivector field on M { f , g } ( x ) = Λ x ( d f ( x ) , dg ( x )) and let F b e a Poisson subalg ebra of C ∞ ( M ) (or a collection of functions closed under the Poisson bra ck et). Consider the linear spaces (2) F x = { d f ( x ) | f ∈ F } ⊂ T ∗ x M and supp ose that w e can find l functionally indep endent functions f 1 , . . . , f l ∈ F whose differentials span F x almost everywhere on M and that the corank of the matrix { f i , f j } is equal to s ome constant k , i.e., dim ker Λ x | F x = k . The n umbers l and k are called differ ential dimension and differ ential index of F a nd they are denoted by ddim F and dind F , resp ectively . The set F is ca lled c omplete if (see [2 5, 24, 8]): ddim F + dind F = dim M + corank {· , ·} . The Hamiltonian system (1) is c ompletely inte gr able in the nonc ommutative sense if it p ossesses a co mplete set of first integrals F . Then (under compac tnes s con- dition) M is almost ev er ywhere folia ted b y (dind F − co rank {· , ·} )-dimensional inv aria n t tori. As in the Liouville-Arnold theorem, the Hamiltonian flow restricted to reg ular inv a riant tori is qua s i-p erio dic (see Nekhoro shev [25] and Mishchenk o and F omenko [24]. 1.3. Mishchenk o–F omenko Conject ure. Let F b e any Poisson clos ed subset of C ∞ ( M ), then a subset F 0 ⊂ F is a c omplete subset if ddim F 0 + dind F 0 = ddim F + dind F . In pa rticular, a comm utative subset F 0 ⊂ F is complete if ddim F 0 = 1 2 (ddim F + dind F ) . Mishchenk o and F omenko stated the conjecture that non-c ommutative int e gr able systems ar e inte gr able in the usual c ommut ative sense by me ans of inte gr als that b elong to the same funct ional class as the original non-c ommutative inte gr als . In other words, if F is a complete set, then we can alwa ys construct a co mplete commutativ e set F 0 ⊂ F . Let us ment ion tw o cases in which the Mishchenk o-F omenko conjecture has bee n proved. The finite-dimensional version of the co njecture is recently prov ed by Sadetov [2 9] (see also [6, 32]): for every finite-dimensional Lie algebr a g one c an find a c omplete c ommu tative set of p olynomials on the dual sp ac e g ∗ with r esp e ct to the usu al Lie-Poisson br acket . The second case where the conjecture was proved is C ∞ –smo oth cas e for infinite-dimensio nal algebr as (see [8]). Consider the homogeneous spaces G/H of a compact Lie group G . Fix some Ad G -inv ar ia nt pos itive definite s calar pro duct h· , ·i on the Lie algebra g = Lie ( G ). SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 3 Let h = Lie ( H ) and let g = h ⊕ v b e the o rthogona l decomp os itio n with res pe c t to h· , ·i . The scalar pro duct h· , ·i induces a normal G -inv ariant metr ic on G/H via ( · , · ) 0 = h· , ·i| v , where v is identified with the tang e n t space at the class of the iden- tit y element. If G is se misimple and h· , ·i is negative K illing form, the normal metric is called standar d [4]. The geo desic flow of the no rmal metric is completely inte- grable in the non-commutativ e s ense by means of int egrals p oly nomial in momenta [7, 9] and the Mishc henko-F omenko conjecture can b e reduced to the following o nes: Conjecture 1. ([9]) F o r ev ery homogeneous space G/H of a compact Lie group G there ex is t a complete commut ative set of Ad H -inv ar ia nt p olynomials on v . Here the Poisson s tructure is defined by (33). F or example if ( G, H ) is a spherical pair, the set of Ad H -inv ar ia nt polynomials is already co mmutative. In many examples, such as Stiefel manifolds, flag manifolds, orbits of the adjoint actions, co mplete commutativ e algebr a s are obtained (se e [7, 9, 2 1]), but the genera l problem is still unsolved. Note that solving the problem of commutativ e integrability of g eo desic flows of normal metrics would allow to constr uc t new examples o f G -inv ar iant metrics o n homogeneous spaces G/ H with integrable geo des ic flows as well. 1.4. The Manak ov Flo ws. The E ule r eq ua tions o f a left-inv ariant geo desic flow on S O ( n ) have the form (3) ˙ M = [ M , Ω] , Ω = A ( M ) where Ω ∈ so ( n ) is the ang ular veloc ity , M ∈ so ( n ) ∗ ∼ = so ( n ) angular momentum and I = A − 1 the p ositive definite op erator which defines the left inv ar iant met- ric (see [1]). Here we iden tify so ( n ) and so ( n ) ∗ by means o f the scalar pro duct prop ortiona l to the Killing form (4) h X, Y i = − 1 2 tr( X Y ) , X , Y ∈ s o ( n ). The Euler equations (3) are Hamiltonian with resp ect to the Lie- Poisson bracket (5) { f , g } ( M ) = − h M , [ ∇ f ( M ) , ∇ g ( M )] i , M ∈ so ( n ) with the Hamiltonian function H = 1 2 h M , A ( M ) i . The inv ariant p olynomials tr( M 2 k ), k = 1 , . . . , rank so ( n ) are central functions, determining the reg ular sym- plectic le aves (adjoint or bits) o f the Lie-Poisson brackets (5). Thus we need ha lf of the dimension of the g e ne r ic adjoint orbit ( 1 2 (dim so ( n ) − r a nk so ( n ))) additiona l independent co mmu ting integrals for the integrability of Euler equa tio ns (3). F or a generic op er ator A and n ≥ 4 the system is no t integrable. Manako v found the Lax representation with ratio nal para meter λ (see [18 ]): (6) ˙ L ( λ ) = [ L ( λ ) , U ( λ )] , L ( λ ) = M + λ A, U ( λ ) = Ω + λB , 4 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C provided M and Ω are connected by (7) [ M , B ] = [Ω , A ] , where A and B a r e diagonal ma trices A = diag ( a 1 , . . . , a n ), B = diag( b 1 , . . . , b n ). In the case the eigenv alues of A and B a re distinct, we hav e (8) A = a d − 1 A ◦ a d B = ad B ◦ a d − 1 A ⇐ ⇒ Ω ij = b i − b j a i − a j M ij , where M ij = h M , E i ∧ E j i . Here a d A ( M ) = [ A, M ] and ad B ( M ) = [ B , M ] are considered as linear transformatio ns from so ( n ) to the zer o-diago nal subspa ce of the space of symmetric matrice s Sym( n ). The y ar e in vertible since the eig e n v alues of A and B a re dis tinct. Note that we take A and B such that A is po sitive definite. F ormally , we can take singular B (i.e., B with some equa l eigenv alue s ), but then A is not inv er tible and r epresents the op era tor which determines the left-inv ariant sub-Riemannian metric on S O ( n ). The left inv aria nt metric giv en b y the op er a tor (8) is us ually called the Manakov metric . In this c ase, Manako v pr ov ed that the solutions of the Euler equations (3) are expressible in terms of θ -functions by using the algebro- geometric integration pro cedure developed by Dubrovin in [1 0, 11] (see [18]). The explicit verification that integrals arising from the Lax representation (9) L = { tr( M + λ A ) k | k = 1 , 2 , . . . , n, λ ∈ R } , form a complete Poisson-co mm utative set on so ( n ) was given by Mishchenk o a nd F omenko in [23] in the case when the eigenv alues of A are distinct (see also Bo lsinov [5]). F urthermore , the system is algebra ically co mpletely in tegrable. Conv ersely , if a diago nal metrics Ω ij = A ij M ij with distinct A ij has algebr a ically completely int egrable geo desic flow then A has the for m (8) for certain A and B (see [16 ]). An another Lax pa ir of the sys tem can b e fo und in [12]. 1.5. Singular M anak ov Flows. W e shall describ e op era tors A satisfying the con- dition (7) when the eigenv a lue s o f A a nd B ar e not all distinct. Supp ose a 1 = · · · = a k 1 = α 1 , . . . , a n +1 − k r = · · · = a n = α r , b 1 = · · · = b k 1 = β 1 , . . . b n +1 − k r = · · · = b n = β r , (10) k 1 + k 2 + · · · + k r = n, α i 6 = α j , β i 6 = β j , i, j = 1 , . . . , r. Let (11) so ( n ) = so ( n ) A ⊕ v = so ( k 1 ) ⊕ so ( k 2 ) ⊕ · · · ⊕ so ( k r ) ⊕ v be the orthogo nal decompo sition, where so ( n ) A = { X ∈ s o ( n ) | [ X , A ] = 0 } . By M so ( n ) A and M v we denote the pro jectio ns of M ∈ so ( n ) with r esp ect to (11). F urther, let B : so ( n ) A → so ( n ) A be an ar bitrary p ositive definite op erato r. W e take A and B s uch that the sectional op erator A : s o ( n ) → so ( n ) de fined via (12) A ( M v + M so ( n ) A ) = ad − 1 A ad B ( M v ) + B ( M so ( n ) A ) , SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 5 is p ositive definite. Now ad A and ad B are considered as inv ertible linear transfor - mations from v to [ A, v ] ⊂ Sym( n ). F or the g iven A we have [Ω , A ] = [Ω v , A ] = [ M v , B ] = [ M , B ] , and the Manako v condition (7) holds. It can b e prov ed that [ M v , a d − 1 A ad B ( M v )] so ( n ) A = 0 . Therefore, the system (3) takes the form ˙ M so ( n ) A = [ M so ( n ) A , B ( M so ( n ) A )] , (13) ˙ M v = [ M so ( n ) A , a d − 1 A ad B ( M v )] + [ M v , B ( M so ( n ) A )] . (14) If k i ≥ 4 for some i = 1 , . . . , r , t he equations (13) (and therefore the sy stem (13), (1 4 )) are not integrable for a gener ic B . On the other hand, since (7) ho lds, the system has Lax representation (6). But the integrals a rising from t he Lax representation do not provide complete integrability . W e r efer to (13), (14) as a singu lar Manakov flow . 1.6. Symmetric Rigid Bo di es. Consider the c a se A = B 2 and A = ad − 1 B 2 ad B . Then the angular momentum and v eloc it y are r elated by M = I (Ω) = ad − 1 B ad B 2 (Ω) = B Ω + Ω B , i.e., (15) Ω ij = 1 b i + b j M ij and the E uler equatio ns (3), in co ordinates M ij , rea d (16) ˙ M ij = n X k =1 b i − b j ( b k + b i )( b k + b j ) M ik M kj . The equa tions (16) describ e the motion of a free n -dimensional rigid b o dy with a mass tens or B and inertia tens o r I ar ound a fixed p oint [13 ]. Now, in addition, supp ose that (10) holds (the ca se of a S O ( k 1 ) × S O ( k 2 ) × · · · × S O ( k r )– symmetric rigid b o dy ). The op erato r A given b y (15) is well defined on the whole so ( n ) and the r estriction o f A to so ( k i ) is the multiplication by 1 / 2 β i . Thus, the system (13) is trivia l and w e hav e the No ether conserv ation law M so ( n ) A = const. Let us denote the set of linea r functions on so ( n ) A by S . These additiona l int egrals provide the integrability of the sy s tem. The co mplete integrability of the system is prov ed b y Bo lsinov by using the pe ncil of Lie algebr as on so ( n ) (see the last para graph of Section 2 ). 1.7. Outline of the Paper. In Section 2 we pro v e that Manak ov in tegrals L together with No e ther integrals S form a complete noncommutativ e set of p o ly- nomials on so ( n ), g iving a new pr o of fo r the integrabilit y of symmetric rig id bo dy motion (16). W e also pr ov e that Manak ov in tegrals induce a complete comm u- tative s et within S O ( n )-inv aria nt p olyno mia ls o n the cota ngent bundle o f the homogeneous space S O ( n ) /S O ( k 1 ) × · · · × S O ( k r ) in Section 3 . The complete S O ( n )- in v ariant commutativ e sets w ere known b efore o nly for certa in choices of nu mbers k 1 , . . . , k r (see [7, 9]). In particular, it is prov ed in Sectio n 4 that t he 6 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C construction implies the integrability of the S O ( n )-inv aria nt Einstein metrics on S O ( k 1 + k 2 + k 3 ) /S O ( k 1 ) × S O ( k 2 ) × S O ( k 3 ) and on the Stiefel manifo lds V ( n, k ). These Einstein metr ic s have bee n obtained in [17, 26, 2, 3]. 2. Integrability of a Sy m m etric Rigid Body Motion 2.1. Comple teness of Manak o v In tegrals . Since the algebra of linear functions S is not commutativ e if some o f k 1 , . . . , k r are greater than 2, the na tural framework in studying singula r Ma nako v flows is noncommutativ e integration. W e s tart with an equiv alent definition of the completeness. Let F b e a collection of functions closed under the Poisson brack et on the Poisson manifold M . W e say that F is c omplete at x if the space F x given by (2) is c o isotropic: (17) F Λ x ⊂ F x . Here F Λ x is skew-orthog onal complement o f F x with resp ect to Λ: F Λ x = { ξ ∈ T ∗ x M | Λ x ( F x , ξ ) = 0 } . The set F is c omplete if it is c o mplete at a generic p oint x ∈ M . In this ca se ddim F = dim F x and F Λ x = k er Λ x | F x implying dind F = dim F Λ x , fo r a gener ic x ∈ M . Note that o ne can consider Hamilto nia n sys tems res tricted to symplectic le av es as w ell. Let N ⊂ M b e a symplectic leaf (regular or singular). The set F is complete on the symplectic leaf N a t x ∈ N if (18) F Λ x ⊂ F x + ker Λ x and it is c omplete on the symple ctic le af N if it complete at a generic p oint x ∈ N . As ab ove, let S b e the set o f linear functions on s o ( n ) A and L b e the Lax pair int egrals (9). Theorem 1. L + S is a c omplete n onc ommutative set of functions on so ( n ) with r esp e ct to the Lie-Poisson br acket (5). Corollary 1. The symmet r ic rigid b o dy system (16), (10) is c ompletely inte gr able in the n onc ommutative sense. Mor e over, supp ose t hat the system (13) is c om- pletely int e gr able on so ( n ) A with a c omplete set of c ommuting inte gr als S 0 . Then the s ingu lar Manakov flow (13), (14) is also c ompletely inte gr able with a c omplete c ommuting set of inte gr als L + S 0 . Pr o of of t he or em 1. Let L M = {∇ M tr( M + λA ) k | k = 1 , 2 , . . . , n, λ ∈ R } . Accord- ing to (17 ), L + S is complete at M if (19) ( L M + so ( n ) A ) Λ ⊂ L M + so ( n ) A , where Λ is the ca no nical Lie-Poisson bivector on so ( n ): (20) Λ( ξ 1 , ξ 2 ) | M = −h M , [ ξ 1 , ξ 2 ] i . SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 7 Consider the Lie algebra gl ( n ) o f n × n rea l matrix es equipp ed with the scalar pro duct (4). W e ha v e the symmetric pair ortho g onal decomposition g l ( n ) = so ( n ) ⊕ Sym( n ) on the skew-symmetric and s y mmetric matrices: [ so ( n ) , Sym ( n )] ⊂ Sym( n ) , [Sym( n ) , Sym( n )] ⊂ so ( n ) . The s c a lar pro duct h· , ·i is p ositive definite on so ( n ) while it is negative definite on Sym( n ). Let us identify gl ( n ) ∗ and gl ( n ) by means of h· , ·i . On gl ( n ) we hav e the pa ir o f compatible Poisson bivectors (see Reyman [2 8]) Λ 1 ( ξ 1 + η 1 , ξ 2 + η 2 ) | X = −h X, [ ξ 1 , ξ 2 ] + [ ξ 1 , η 2 ] + [ η 1 , ξ 2 ] i , Λ 2 ( ξ 1 + η 1 , ξ 2 + η 2 ) | X = −h X + A, [ ξ 1 + η 1 , ξ 2 + η 2 ] i , (21) where X ∈ gl ( n ), ξ 1 , ξ 2 ∈ s o ( n ), η 1 , η 2 ∈ Sym( n ). In other words, the pe nc il Π = { Λ λ 1 ,λ 2 | λ 1 , λ 2 ∈ R , λ 2 1 + λ 2 2 6 = 0 } , Λ λ 1 ,λ 2 = λ 1 Λ 1 + λ 2 Λ 2 consist of Poisson bivectors on gl ( n ). The Poisson bivectors Λ λ 1 ,λ 2 , fo r λ 1 + λ 2 6 = 0 a nd λ 2 6 = 0, ar e isomorphic to the canonic a l Lie- Poisson bivector (in particular , their co rank is eq ual to n ). The union of their Casimir functions (22) F = { f λ,k ( X ) = tr( λM + P + λ 2 A ) k | k = 1 , 2 , . . . , n, λ ∈ R } where X = M + P , M ∈ so ( n ), P ∈ Sym( n ), is a co mm utative set with re s pe c t to the a ll brack ets from the p e ncil Π [28, 5]. Also, the skew-orthog onal complement F Λ X do es not depend o n the choice Λ ∈ Π. As ab ove, F X denotes the linea r subspace of gl ( n ) g enerated by the differentials of functions from F at X . W e need to ta ke all ob jects complexified (see [5]). The co mplexification of gl ( n ), so ( n ), Sym( n ), so ( n ) A , Π are g l ( n, C ), so ( n, C ), Sym( n, C ), so ( n, C ) A ∼ = so ( k 1 , C ) ⊕ · · · ⊕ so ( k r , C ) a nd Π C = { Λ λ 1 ,λ 2 = λ 1 Λ 1 + λ 2 Λ 2 , λ 1 , λ 2 ∈ C , | λ 1 | 2 + | λ 2 | 2 6 = 0 } , resp ectively . Here, we co ns ider (21) as complex v alued sk e w - symmetric bilinear forms. The complexified sca lar pro duct is simply given by (4), w he r e now X , Y ∈ gl ( n, C ). A t a generic point X ∈ gl ( n ), the only singula r bivector in Π C with a rank smaller then dim g l ( n ) − n is Λ − 1 , 1 . Mo reov er, the complex dimension o f the linear space K − 1 , 1 = { ξ ∈ ker Λ − 1 , 1 ( X ) | Λ 0 ( ξ , ker Λ − 1 , 1 ( X )) = 0 } is equa l to n . Her e Λ 0 is a ny Poisson bivector fro m the p encil, nonpr op ortional to Λ − 1 , 1 , say Λ 0 = Λ 0 , 1 . Whence, it follows fro m Pr op osition 2 .5 [5] that ( F Λ 1 X ) C = F C X + ker Λ − 1 , 1 ( X ) . Also, it can b e prov ed tha t (23) F C X + ker Λ − 1 , 1 ( X ) = F C X + so ( n, C ) A . 8 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C The ab ove relations imply (24) ( F X + so ( n ) A ) Λ 1 = F Λ 1 X ∩ s o ( n ) Λ 1 A ⊂ F X + so ( n ) A and the set of functions F + S is a co mplete no n-commutativ e se t on gl ( n ) with resp ect to Λ 1 (theorem 1.5 [5], for the detail pro ofs of the ab ove statements, given for an a rbitrary semi-s imple symmetric pair , see [31], pages 234-2 37). Now we w ant to verify the co mpleteness of F + S at the p oints M ∈ so ( n ). Note that in theorem 1.6 [5], a simila r problem hav e b een studied but for regular A a nd singular p oints M ∈ s o ( M ), in pr oving that Manako v integrals provide complete commutativ e sets on sing ula r adjoint o rbits. Since a regular matrix M ∈ so ( n ) (dim so ( n ) M = rank so ( n ) = [ n/ 2]), considered as a n element o f gl ( n, C ) is also re g ular (dim gl ( n, C ) M = n ), it can be easily pro ved that the only tw o sing ular bra ck ets in Π C are Λ − 1 , 1 and Λ 1 , 0 . W e have to es timate the complex dimensio ns of linea r spaces K − 1 , 1 = { ξ ∈ ker Λ − 1 , 1 ( M ) | h M + A, [ ξ , ker Λ − 1 , 1 ( M )] i = 0 } (25) K 1 , 0 = { ξ ∈ k er Λ 1 , 0 ( M ) | h M + A, [ ξ , ker Λ 1 , 0 ( M )] i = 0 } . (26) As for X ∈ gl ( n ), rep eating the ar g ument s of [3 1], pa ges 234-2 37, one can prov e that the dimensio n of (25) is n for a gener ic M ∈ so ( n ). F urther (27) ker Λ 1 , 0 ( M ) = ker Λ 1 ( M ) = s o ( n, C ) M + Sym( n, C ) , where so ( n, C ) M is the is o tropy algebr a of M in so ( n, C ) M . W e shall prov e b elow that dim C K 1 , 0 is also e qual to n for a generic M ∈ so ( n ) (see Lemma 1). Hence, according Pro po sition 2 .5 [5], a t a gener ic M ∈ so ( n ) we hav e (28) ( F Λ 1 M ) C = F C M + ker Λ − 1 , 1 ( M ) + ker Λ 1 , 0 ( M ) = F C M + so ( n, C ) A + ker Λ 1 ( M ) . Similarly as in equation (2 4) we get ( F M + so ( n ) A ) Λ 1 = ( F M + so ( n ) A + ker Λ 1 ( M )) ∩ so ( n ) Λ 1 A ⊂ F M + so ( n ) A + k e r Λ 1 ( M ) . Therefore the relation (18) holds for functions F + S and the brack et Λ 1 , i.e., this is a complete set on the sy mplec tic leaf thro ugh M . Notice that the symplectic leaves throug h M ∈ so ( n ) ⊂ gl ( n ) of the br ack et Λ 1 are S O ( n )-adjoint orbit in so ( n ) and that the restriction of Λ 1 to so ( n ) co incides with the Lie-Poisson brack et (2 0 ). Since the r estriction of the central functions (22) to so ( n ) ar e Manakov integrals (9), we obta in (19).  R emark 1 . F ro m the pro of of Theo rem 1 follows that the s kew-orthog onal comple- men t of L M within so ( n ) is given by (29) L Λ M = L M + so ( n ) A , for a g eneric M ∈ so ( n ). SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 9 Lemma 1. The c omplex d imension o f the line ar sp ac e (26) is e qual t o n for a generic M ∈ so ( n ) . Pr o of. F or ξ ∈ ker Λ 1 ( M ), let ξ 1 and ξ 2 be the pro jections to so ( n, C ) M and Sym( n, C ), resp ectively . Then h M + A, [ ξ , ker Λ 1 ( M )] i = h k er Λ 1 ( M ) , [ M + A, ξ 1 + ξ 2 ] i = h k e r Λ 1 ( M ) , [ M , ξ 2 ] + [ A, ξ 1 ] + [ A, ξ 2 ] i = h so ( n, C ) M , [ A, ξ 2 ] i + h Sym( n, C ) , [ M , ξ 2 ] + [ A, ξ 1 ] i Therefore ξ = ξ 1 + ξ 2 ∈ k er Λ 1 ( M ) b elo ng s to K 1 , 0 if and o nly if (30) [ M , ξ 2 ] + [ A, ξ 1 ] = 0 , pr so ( n, C ) M [ A, ξ 2 ] = 0 . The dimension of the s olutions of the sys tem (3 0), for a reg ular M ∈ so ( n ), is n . It can b e dir ectly calculated by taking the following anti-diagonal element: M = m 1 E 1 ∧ E n + m 2 E 2 ∧ E n − 1 + · · · + m k E k ∧ E k +1 , n = 2 k M = m 1 E 1 ∧ E n + m 2 E 2 ∧ E n − 1 + · · · + m k E k ∧ E k +2 , n = 2 k + 1 , when so ( n, C ) M = span C { E 1 ∧ E n , E 2 ∧ E n − 1 , . . . , E k ∧ E k +1 } , n = 2 k so ( n, C ) M = span C { E 1 ∧ E n , E 2 ∧ E n − 1 , . . . , E k ∧ E k +2 } , n = 2 k + 1 . Here m 1 , . . . , m k are gener ic distinct real num ber s. F o r example, if n = 2 k , then ξ ∈ ker Λ 1 ( M ) s a tisfies (30) if and only if it is o f the form: ξ = n X i =1 u i E i ⊗ E i + k X j =1 v j E j ∧ E n +1 − j , where para meters u i , v j are determined from the linear sys tem: − m j ( u j − u n +1 − j ) + ( a j − a n +1 − j ) v j = 0 , j = 1 , . . . , k . Thu s dim C K 1 , 0 = n .  2.2. P encil of Lie Algebras. Bolsinov has given another pro of of the integrabilit y of Ma nako v flows, related to the existence of co mpatible L ie algebra br a ck ets on so ( n ) [5 ]. The first Lie br ack et is standard one [ M 1 , M 2 ] = M 1 M 2 − M 2 M 1 , while the seco nd is [ M 1 , M 2 ] A = M 1 AM 2 − M 2 AM 1 . Then Λ and Λ A are compatible Poisson structures , where Λ is g iven by (20) and (31) Λ A ( ξ 1 , ξ 2 ) | M = −h M , [ ξ 1 , ξ 2 ] A i . Let Λ λ 1 ,λ 2 = λ 1 Λ + λ 2 Λ A . The central functions o f the bra ck et Λ λ, 1 of ma ximal rank ( λ 6 = − α 1 , . . . , − α r ) are (32) J = { tr( M ( λ I + A ) − 1 ) 2 k | k = 1 , 2 , . . . , [ n/ 2] , λ 6 = − α 1 , . . . , α r } . 10 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C According to the genera l constructio n, these functions commute with resp ect to all Poisson bra ck ets Λ λ 1 ,λ 2 . The following theor e m o btained by Bols inov can b e found in [31], pages 241-2 44: Theorem 2. (Bolsinov) The set of functions J + S is a c omplete set on so ( n ) with r esp e ct to the Lie-Poisson br acket (5). The families (9) and (32) commute betw een ea ch other (e.g., see [27]). Therefore, since bo th sets L + S and J + S ar e complete, they are equiv alent, i.e., they determine the s ame inv ar iant toric foliation of the phase space. 1 3. Geodesic Flo ws o n S O ( n ) /S O ( k 1 ) × · · · × S O ( k r ) 3.1. Geo des ic Flows o n Homogene ous Spaces. Co nsider the homogene o us spaces G/H of a compact Lie group G . Let g = h ⊕ v b e the ortho gonal deco mpo si- tion and let ds 2 0 be the nor mal G –inv aria nt metric induced by so me Ad G -inv ar ia nt scalar pro duct h· , ·i on the Lie algebra g . Let F G be the set of G inv a riant functions, polynomial in mo ment a and Φ : T ∗ ( G/H ) → g ∗ be the momentum mapping of the natural Hamiltonia n G -action. F rom the No ether theorem we hav e { F G , Φ ∗ ( R [ g ∗ ]) } = 0, where { · , ·} is the cano n- ical Poisson bra cket o n T ∗ ( G/H ). The Hamiltonian of the normal metric ds 2 0 is a central function of F G so it commutes b oth with the No ether functions Φ ∗ ( R [ g ∗ ]) and G -inv ar iant functions F G . On the other side, the set F G + Φ ∗ ( R [ g ∗ ]) is com- plete, implying the noncommutativ e integrability of the g eo desic flo w of the normal metric [7, 8 ]. The algebra ( F G , { · , ·} ) can b e naturally identified with ( R [ v ] H , { · , ·} v ), where R [ v ] H is the a lgebra of Ad H -inv ar ia nt poly nomials on v and (see Thimm [30]): (33) { f , g } v ( x ) = −h x, [ ∇ f ( x ) , ∇ g ( x )] i , f , g ∈ R [ v ] H . Within the cla ss of No ether int egrals Φ ∗ ( R [ g ∗ ]) one can always construct a co m- plete commutativ e subset. Thus the Mishchenk o– F omenko c o njecture reduces to a construction of a co mplete co mmu tative subset of R [ v ] H ∼ = F G . This lea ds to Conjecture 1 s tated in the Introduction. A commutativ e set F ⊂ R [ v ] H is co mplete if (34) ddim F = 1 2  ddim R [ v ] H + dind R [ v ] H  = dim v − 1 2 dim O G ( x ) , for a g eneric x ∈ v , where O G ( x ) is the adjoint orbit of G (see [7 , 9]). 1 It was p oint ed out by one of the referees that the equiv alence of the integ rals (9) and (32) can b e deri ve d directly , by usi ng the matrix identit y det( M ( A + α I ) − 1 + β I ) = det( M + β A + αβ I ) det( α I + A ) − 1 . SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 11 3.2. Normal Geo desic Flows on S O ( n ) /S O ( k 1 ) × · · · × S O ( k r ) . Let S O ( n ) A = S O ( k 1 ) × · · · × S O ( k r ) ⊂ S O ( n ) be the is otropy group of A within S O ( n ) with resp ect to the adjoint action. As ab ov e, consider the normal metric ds 2 0 defined by the scalar pro duct (4) and identify S O ( n )– inv aria nt p olynomia ls o n T ∗ ( S O ( n ) / S O ( n ) A ) with R [ v ] S O ( n ) A ( v is defined by (11)). W e shall use the following completeness criterium. Consider the space j M ⊂ v spanned by gradients of all p olynomia ls in R [ v ] S O ( n ) A . F or a g eneric po int M ∈ v we hav e j M = ([ M , so ( n ) A ] ⊥ ) ∩ v = { η ∈ v | h η , [ M , so ( n ) A ] i = 0 } = { η ∈ v | [ M , η ] ⊂ v } . The bracket (33) on R [ v ] S O ( n ) A corres p o nds to the res triction of the Lie- Poisson bivector (20) to j M . Denote this restriction by ¯ Λ. Then F ⊂ R [ v ] S O ( n ) A ∼ = F S O ( n ) is a complete commutativ e set if a nd only if F ¯ Λ M = F M , for a generic M ∈ v , where F M = span {∇ M f ( M ) | f ∈ F } ⊂ j M and F ¯ Λ M is the skew-orthogonal complements o f F M with resp ect to ¯ Λ within j M . Here , for simplicity , the gradient op erator with resp ect to the re striction of h· , ·i to v is also denoted by ∇ . Since all po lynomials in L commute with S , their restrictions to v (35) L v = { tr( M + λA ) k | M ∈ v , k = 1 , 2 , . . . , n, λ ∈ R } , form a co mm utative subset o f R [ v ] S O ( n ) A (see [7]). Let Φ : T ∗ S O ( n ) / S O ( n ) A → so ( n ) ∗ ∼ = so ( n ) b e the mo men tum ma pping o f the natural S O ( n )-Hamiltonian actio n on T ∗ S O ( n ) / S O ( n ) A and let A be a n y com- m utative set of po lynomial on so ( n ) that is complete o n adjoint orbits within the image Φ( T ∗ ( S O ( n ) / S O ( n ) A )) (for example one can take Mana ko v integrals with regular A [5]). Then Φ ∗ ( A ) is a complete commutativ e subs et in Φ ∗ ( R [ so ( n )]) and we hav e: Theorem 3. (i) L v is a c omplete c ommu tative subset of R [ v ] S O ( n ) A . (ii) The ge o desic flow of the normal metric ds 2 0 is Liouvil le inte gr able by me ans of p olynomial inte gr als L v + Φ ∗ ( A ) . R emark 2 . Note that, by using the ch ains of s ub algebr as metho d , the construc- tion of another complete commut ative algebras of S O ( n )-inv ar iant p olyno mials is per formed for homogeneous spaces S O ( n ) /S O ( k ) and S O ( n ) /S O ( k 1 ) × S O ( k 2 ) (see [7, 9]). Also, b y using the gener alize d chains of sub algebr as metho d , the Con- jecture 1 is solved for a cla s s of homogeneous spaces S O ( n ) /S O ( n ) A [9]. The class, say C , is obtained by induction from S O ( n ) /S O ( k 1 ) × S O ( k 2 ), k 1 ≤ k 2 ≤ [ n +1 2 ] in the following way: supp ose that S O ( n 1 ) /S O ( k 1 ) × · · · × S O ( k r 1 ) and S O ( n 2 ) /S O ( l 1 ) × · · · × S O ( l r 2 ) ( n 1 = n 2 ± 0 , 1 ) b elong in C , then a lso S O ( n 1 + 12 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C n 2 ) /S O ( k 1 ) × · · · × S O ( k r 1 ) × S O ( l 1 ) × · · · × S O ( l r 2 ) be longs to C . Note that, for example, the ho mogeneous spa ces S O ( n ) /S O ( k 1 ) × · · · × S O ( k r ), wher e some of k i is gra ter than [ n +1 2 ] do not b elong to the family C . Pr o of. Without los s o f gener ality , supp ose k 1 ≤ k 2 ≤ k 3 · · · ≤ k r . If the condition k r ≤  n + 1 2  is satisfied, then a g eneric element M ∈ v is regula r ele men t of so ( n ) and relation (29) will holds. Then it easily follows that (36) ¯ L ¯ Λ M = ¯ L M , ¯ L M = {∇ M f | f ∈ L v } ⊂ j M , for a g eneric M ∈ v . Hence L v is co mplete. Now, supp ose k r =  n + 1 2  + l , l > 0 . Let n ′ = n − 2 l , k ′ r = k r − 2 l , A ′ = diag( a 1 , a 2 , . . . , a n ′ ) and let (37) so ( n ′ ) = so ( n ′ ) A ′ ⊕ v ′ = so ( k 1 ) ⊕ so ( k 2 ) ⊕ · · · ⊕ so ( k ′ r ) ⊕ v ′ be the or thogonal decompos ition, where so ( n ′ ) A ′ is the isotropy algebr a of A ′ within so ( n ′ ). F urthermor e , we can consider Lie alge bras so ( n ′ ) and so (2 l ) embedded in so ( n ) as blo cks:  so ( n ′ ) 0 0 so (2 l )  . Then the linear s pa ce v ′ bec omes a linear subspac e of v : v ′ = so ( n ′ ) ∩ v . Moreov er, for an ar bitrary M ∈ v one can find a matrix K ∈ S O ( n ) A such that M ′ = Ad K ( M ) b elongs to v ′ . Indeed, consider M and K of the form M =  M 11 M 12 − M T 12 0  , K =  I n − k r 0 0 U  , where M 11 ∈ so ( n − k r ), M 12 is ( n − k r ) × ( k r ) matrix, I n − k r is the iden tit y ( n − k r ) × ( n − k r ) matrix and U ∈ S O ( k r ). Then M ′ = K M K − 1 =  M 11 M 12 U T − U M T 12 0  . Since k r − ( n − k r ) is equal to 2 l or 2 l + 1 , one can always find U such that the last 2 l rows o f U M T 12 , i.e., the la st 2 l columns of M 12 U T are equa l to zero, which implies that M ′ belo ngs to v ′ . Therefore, if the set L v is co mplete at the p oints of v ′ then it will b e complete on v a s well. SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 13 The Lie algebra s so ( n ′ ) and s o ( n ′ ) A ′ are centralizers of so (2 l ) in so ( n ) and so ( n ) A , res p ectively . Whence, fro m the ab ov e conside r ations, we can a pply Theo- rem A1 (see Ap endix) to get (38) j ′ M ′ = { η ∈ v ′ |h η , [ M ′ , s o ( n ′ ) A ] i = 0 } = { η ∈ v |h η , [ M ′ , s o ( n ) A ] i = 0 } = j M ′ , for a g eneric M ′ ∈ v ′ . In particular, (38) implies that Poisson tenso r s ¯ Λ of R [ v ] S O ( n ) A and ¯ Λ ′ of R [ v ′ ] S O ( n ′ ) A ′ coincides on a g eneric M ′ ∈ v ′ ⊂ v . Acco rding to the fir st part of the pro of, the set of p olynomials L v ′ = { tr( M ′ + λA ′ ) k | M ′ ∈ v ′ , k = 1 , 2 , . . . , n, λ ∈ R } , is a complete commutativ e subset of R [ v ′ ] S O ( n ′ ) A ′ , i.e, (39) ¯ L ′ ¯ Λ ′ M ′ = ¯ L ′ M ′ , ¯ L ′ M ′ = {∇ M ′ f | f ∈ ¯ L v ′ } ⊂ j ′ M ′ , for a g eneric M ′ ∈ v ′ . But from (38) and (39 ) we also get that (36) holds for a generic M ′ ∈ v ′ .  R emark 3 . An alternative pro o f of theorem 3 ca n b e p erfor med by using the com- patibilit y of Poisson brack ets (20) and (31), but now consider ed within the a lg ebra R [ v ] S O ( n ) A . 3.3. Submersio n of Manak o v Flo ws. Let A b e given by (12) wher e B is the ident it y op erato r. Then the s ing ular Manakov flow (13), (14) r epresent the geo desic flow of the left S O ( n )-inv a r iant metric on S O ( n ) that is also righ t S O ( n ) A -inv ar ia nt. By submersio n, this metric induces S O ( n )-inv aria nt metric on ho mogeneous space S O ( n ) / S O ( n ) A that we shall denote by ds 2 A,B . Spec ially , for A = B we have the normal metric. On v , identifi ed with the tang ent spa ce at the class of the identit y element, the metric is given by the S O ( n ) A -inv ar ia nt scalar pr o duct (40) ( · , · ) A,B = X 1 ≤ i ≤ j ≤ r α i − α j β i − β j h · , · i| v i,j , where v = M 1 ≤ i 0 . SINGULAR MANAKO V FLOWS AND GEODESIC FLOWS ON HOMOGENEOUS SP ACES 15 Note that Mana ko v integrals (41) are int egrals of the geo desic flow of the metric (42). Thus, if Euler equa tio ns on so ( n − k ) (43) ˙ M = [ M , I − 1 ( M )] are integrable, the geo desic flow of the metric ds 2 I ,κ will b e completely integrable. Let I = χ · Id so ( n − r ) . In [1 7], Jensen pro ved that for n − k = 2 there is a unique v alue, while for n − k > 2 there ar e exa ctly tw o v alues of ( χ, κ ) ∈ R 2 (up to homotheties), such that ds 2 I ,κ is Einstein metric. Since then equatio ns (43 ) are trivial, functions L p + K a r e integrals of the geo desic flow. Arv anitoy eorgos , Dzhepk o and Nikonorov found tw o new Einstein metrics [2, 3] within the class of metrics (42) with n − k = sl , s > 1, k > l ≥ 3. It app ears that the integrability of co rresp onding Euler equations (43) ca n b e ea sily proved by using the chain metho d developed by My k ytyuk [19]. Corollary 3. Th e ge o desic flows of Einstein metrics on Stiefel manifolds S O ( n ) / S O ( k ) and homo gene ous sp ac es S O ( k 1 + k 2 + k 3 ) /S O ( k 1 ) × S O ( k 2 ) × S O ( k 3 ) c on- structe d in [17, 2, 3, 26] ar e c ompletely inte gr able. Note that the integrability of the geo des ic flows of Einstein metrics on Stiefel manifolds V ( n, k ) can be prov e d in a different wa y , starting from the analog ue of the Neumann sys tem on V ( n, r ) (see [1 4]). Apendix: P airs of Reductive Lie Al gebras Let g b e a reductiv e rea l (or complex) Lie algebr a. T ake a fa ithful re pr esentation of g such that its asso ciated bilinear for m h· , ·i is nondeg enerate o n g . Let k ⊂ g b e a reductive in g s uba lgebra and v = k ⊥ = { η ∈ g | h η , k i = 0 } . F or a ny ξ ∈ v define the subspace j ξ ⊂ v by j ξ = { η ∈ v | [ ξ , η ] ∈ v } = { η ∈ v | h η , [ ξ , k ] i = 0 } . Consider a Zarisk i o p en subset of R - elements in v defined by R ( v ) = { ξ ∈ v | dim g ξ ≤ dim g η , dim k ξ ≤ dim k η , dim g 0 ξ ≤ dim g 0 η , η ∈ v } , where g η and k η are ce ntralizers of η in g and k , resp ectively , and g 0 η denote the set of all ζ ∈ g which s atisfy (a d η ) n ( ζ ) = 0 for sufficiently larg e n . Assume that ξ 0 ∈ R ( v ) and a is a reductive (in g ) s ubalgebra of k ξ 0 . Let g ′ and k ′ be the centralizers of a in g and k , r esp ectively . Then algebra s g ′ and k ′ are subalgebra s reductive in g and the restriction of h· , ·i to g ′ and k ′ are nondegenerate (for more details, see Myk ytyuk [20]) Let v ′ be the ortho g onal complemen t of k ′ in g ′ . The n v ′ = g ′ ∩ v [20]. As ab ove, define j ′ ξ = { ζ ∈ v ′ | [ ξ , ζ ] ∈ v ′ } , ξ ∈ v ′ and the s e t of R -elements R ( v ′ ) in v ′ . 16 VLADIMIR DRA GOVI ´ C, BORISLA V GAJI ´ C AND BO ˇ ZID AR JO V ANO VI ´ C The following result is co ntained in the pro of of theor em 11 [20] (s e e also pro p o - sition 2.3 given in [2 1]). Theorem A 1. (Mykytyuk [20]) The r elation j ξ = j ′ ξ is satisfie d for any element ξ in a Zariski op en subset R ( v ′ ) ∩ R ( v ) of v ′ . Ac knowledgmen ts. W e are grateful to Alexey V. Bo ls inov and Y ur i G. Nik onorov on useful discussio ns. 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