512.542+517.551 GEOMETRY OF CROSS RATIO Zelikin M.I. Generalizationofthecrossratiotopolarizationsoflinearfiniteandinfinite-dimensional spaces (in particular to Sato Grassmannian) is given and explored. This cross ratio ap- 7 0 pears to be a cocycle of the canonical (tautalogical) bundle over the Grassmannian with 0 coefficients in the sheaf of its endomorphisms. Operator analog of the Schwarz differen- 2 tial is defined. Its connections to linear Hamiltonian systems and Riccati equations are n established. These constructions aim to obtain applications to KP-hierarchy. a J 8 Keywords: Grassmannian Sato, Kadomzev-Petviashvily hierarchy (KP-hierarchy), 1 Crossratio,Hamiltoniansystem, Riccatiequation, Schwarzderivative, Cohomologieswith ] coefficients in sheafs. P A MSC: 14M15,34H05 . h Sato Grassmanian and KP-hierarchy t a m We shall use the following standard notations. As a field of constants k one may [ consider both R or C. For a basic ring B = k[[x]] = { a xi|a ∈ k} – (the formal i≥0 i i 1 Taylor series) let E = k[[x]]((∂−1)) – be the ring of formal pseudodifferential operators v P 0 with coefficients in B. 0 Consider the direct sum decomposition E = E ⊕ E , where E = B[∂] is the 5 + − + 1 ring of differential operators, and E− = B[[∂−1]]∂−1 – is the ring of operators with the 0 negative order (Volterra type operators). So that for any P ∈ E one have P = P +P , + − 7 where P is a differential part and P is a Volterra part of P . A decomposition of an 0 + − / infinite-dimensional space on a direct sum of its two infinite-dimensional subspaces shall h t be called polarization. a m Let H be a Hilbert space, equipped with a polarization, i.e. with a decomposition on : two closed orthogonal subspaces v i X H = H ⊕H . (1) + − r a It is convenient to define this decomposition by an operator of complex structure, i.e. by the unitary operator J : H → H which is equal to +Id on H and to −Id + on H . The General restricted group GL (H) is defined as a subgroup of GL(H) − res consisted of operators A such that the commutator [J,A] is a Hilbert-Schmidt operator. Equvivalently, if A ∈ GL(E) is represented as (2×2)-matrix a b A = c d ! relative to the decomposition (1), then A ∈ GL (H) iff b and c are Hilbert-Shmidt res operators. It follows that a and d are Fredholm operators. U (H) is a subgroup of res GL (H), which consists of unitary operators. res 0ThisresearchwassupportedinpartbytheRussianFoundationforBasicResearchgrant04-01-00735, and Mintechprom RF grant NSh-304.2003.1 1 Sato Grassmannian SGr(H) is the set of all closed subspaces W ∈ H such that 1. The orthogonal projection pr : W → H is a Fredholm operator + + 2. The orthogonal projection pr : W → H is a Hilbert-Shmidt operator. − − It is known [1] that the subgroup U (H) of the group GL (H) acts on SGr(H) res res transitively, and the stabilizer of the subspace H is U(H )×U(H ). Hence SGr(H) is + + − represented asa homogeneous space SGr(H) = U (H)/((U(H )×U(H )). SGr(H) is res + − aninfinite-dimensionalmanifoldmodelledbyHilbertspace J (H ,H ) ofHilbert-Shmidt 2 + − operators T : H → H where the norm T ∈ J (H ,H ) is defined as ( k Te k2)1/2, + − 2 + − i i and {e } is a full orthonormal basis on H . The set of such operators (more exactly, i + P a chart on SGr(H) corresponding to this set) is called the big cell and is denoted by SGr . The group GL (H) acts on SGr(H) by M¨obius transforms + res a b ·T = (c+dT)(a+bT)−1. c d! The tangent space to the manifold SGr(H) at a point W is T = Hom(W,H/W). W Consider theaffinesubspace E′ = ∂+H ofthelinear space H andthemultiplicative − group G = 1 + H . Both of them have the tangent space H . Define the mapping − − ξ : G → E′ for S ∈ G bytheformula ξ : S 7→ L = S∂S−1 andthemapping η : G → Gr + by the formula η : S 7→ S−1w , w ∈ SGr being a fix subspace. Differentials of 0 0 + these mappings at a point S acts on A ∈ H by formulae dξ : A 7→ [AS−1,L] and − dη : A 7→ −S−1A accordingly. The main result (known as Sato’s correspondence) [1], [2] is: The operator of mul- tiplication by z−n (which induces S ∈ Hom(W,H/W)) transfers by dξ(dη)−1 into the commutator [Ln,L], i.e. into the right hand side of KP equation. Given n, the operator + z−n : H → H defines a fix vector field on SGr(H), i.e. the right hand side of ”constant coefficient” Riccati equation [3]. Thus KP-hierarchy corresponds to a countable polysys- tem of mutually commuting flows of operator Riccati equations on Sato Grassmannian. Analogues theory for noncommutative rings of coefficients was developed in [5], and for multidimensional x in [6]. To investigate KP-hierarchy it is supposed to use the following generalization of the cross ratio [4]. Operator cross-ratio Considerfirstthefinite-dimensionalcaseandsubspacesofhalfdimension. Let P ,P ,P ,P 1 2 3 4 be four n-dimensional subspaces in R2n which corresponds to four points with matrix coordinates P ,P ,P ,P of the big cell of the manifold Gr (R2n). Suppose that P ,P 1 2 3 4 n 1 2 and P ,P define polarizations of R2n, i.e. P ⊕P = P ⊕P = R2n. The polarization 3 4 1 2 3 4 P ,P will be denoted by Π . The class of matrices which are similar to the matrix i j ij (P −P )−1(P −P )(P −P )−1(P −P ) (invers matrices are defined since Π and Π 1 2 2 3 3 4 4 1 12 34 are polarizations) is an invariant of the ordered four points of Grassmannian relative to M¨obius transformations [3]. It is called matrix cross-ratio. Theprojectionparalleltoasubspace P willbedenotedby π orbythefigure i above i i the arrow which gives the corresponding mapping. If P and P defines a polarization, i j then the image of π in P is uniquely defined. i j heorem 1 Let Π and Π be polarizations. Then DV(P ,P ;P ,P ) is the matrix of 12 34 1 2 3 4 4 2 the composition mapping P −→ P −→ P of the space P on itself. 1 3 1 1 2 Proof. Let (g,P g) be the projection of an element (f,P f) of the space P on P 3 1 1 3 parallel to P . This means that (f −g,P f −P g) ∈ P , i.e. P f −P g = P (f −g). 4 1 3 4 1 3 4 Hence, g = (P −P )−1(P −P )f . Similarly, the projection of an element (g,P g) on 4 3 4 1 3 P parallel to P is (h,P h), where h = (P − P )−1(P − P )g = (P − P )−1(P − 1 2 1 2 1 2 3 1 2 2 P )(P −P )−1(P −P )f . 2 3 3 4 4 1 Note 1 The τ -function in the theory of integrable systems [7] is in essence only the de- terminant of an operator cross-ratio; remaining invariants of the corresponding operators was not used before. Hence, in [7] was proved less general assertion concerning only about the determinant of the cross ratio. Our theorem allows to remove the restriction of half-dimension and to define cross- ratio for a pair of polarizations of the space Rm with similar dimensions, i.e. dimP = 1 dimP = k; dimP = dimP = m − k. Moreover, this gives the possibility to define 3 2 4 an operator cross-ratio for infinite-dimensional case. In so doing, the invariance of the operatorcross-ratio relative totheM¨obius groupisinherited by theconstruction itself due to the linearity of projection operators. Note that the composition mapping is invariant but its matrix is defined up to conjugation Let P ,P ,P ,P be subspaces of linear space H. The operator DV(P ,P ;P ,P ) 1 2 3 4 1 2 3 4 defined in theorem 1 will be denoted by DV . 12;34 We need another expression for the cross ratio. Let H = E ⊕E be a polarization of 1 2 H. We shall call E the horizontal subspace and E the vertical one. We suppose that 1 2 P ,P ∈ E , P ,P ∈ E . Let the subspaces P , P have as its coordinates matrices P 1 3 1 2 4 2 1 3 1 and P correspondingly. For the subspaces P and P we change the role of the vertical 3 2 4 and the horizontal subspaces. Let in this new system of coordinates the subspaces P 2 and P have as its coordinates the matrices P and P correspondingly. 4 2 4 Proposition 1 Let Π and Π are polarizations and let the projections parallel to P 12 34 2 and P are isomorphisms of the spaces P and P . Then the matrix DV(P ,P ;P ,P ) 4 1 3 1 2 3 4 4 2 of the composition mapping P −→ P −→ P of the space P on itself has the form 1 3 1 1 (P P −I)−1(P P −I)(P P −I)−1(P P −I). 2 1 2 3 4 3 4 1 Proof. Let (g,P g) be the projection of an element (f,P f) of the space P on P parallel 3 1 1 3 to P . This means that (f −g,P f −P g) ∈ P . As soon as the roles of of the vertical 4 1 3 4 and the horizontal subspaces interchange for P we have P (P f −P g) = (f −g). The 4 4 1 3 matrix (P P −I) is invertible hence we have g = (P P −I)−1(P P −I)f . Analogousely 4 3 4 3 4 1 the projection of the element (g,P g) on P parallel to P is (h,P h), where h = 3 1 2 1 (P P −I)−1(P P −I)g = (P P −I)−1(P P −I)(P P −I)−1(P P −I)f . 2 1 2 3 2 1 2 3 4 3 4 1 2 Lemma 1 Let projections parallel to P and P be isomorphisms of the spaces P and 2 4 1 P . Then DV = DV . 3 12;34 34;12 Proof. Cosider the mappings 4 2 DV : P −→ P −→ P ; 12;34 1 3 1 2 4 DV : P −→ P −→ P . 34;12 3 1 3 3 If one identifies P and P by using projection π , then both maps will coinside. 2 1 3 4 Thus the cross-ratio does not depend on the order of polarization pairs. Lemma 2 Let P ⊕P = P ⊕P = P ⊕P = P ⊕P = H. Then DV = DV−1 . 1 2 3 4 1 4 3 2 12;34 14;32 Proof. Consider the mappings 4 2 DV : P −→ P −→ P ; 12;34 1 3 1 2 4 DV : P −→ P −→ P . 14;32 1 3 1 We have P −→4 P = (P −→4 P )−1 P −→2 P = (P −→2 P )−1, which follows the 1 3 3 1 1 3 3 1 2 lemma. Lemma 3 Let P ⊕P = P ⊕P = H. Then DV = Id−DV . 1 2 3 4 12;43 12;34 Proof. Design the image of h ∈ P under the mapping DV : P −→3 P −→2 P . 1 1 12;43 1 4 1 We have π h = h +h , where h ∈ P , and h +h ∈ P . Further on, π (h +h ) = 3 1 1 3 3 3 1 3 4 2 1 3 h + h + h , where h ∈ P , and h + h + h ∈ P . Let us rewrite the image of 1 3 2 2 2 1 3 2 1 DV in the form h + h + h = h − (h − (h + h + h )). Since h + h ∈ P , 12;43 1 3 2 1 1 1 3 2 1 3 4 and h −(h + h ) ∈ P , then h −(h + h ) is the image of h under the projection 1 1 3 3 1 1 3 1 π on P . The subtraction of h gives the image of projection π on P . Hence 4 3 2 2 1 (h −(h +h +h )) = DV h . Consequently, DV = Id−DV . 2 1 1 3 2 12;34 1 12;43 12;34 Lemmas 1-3 define generators of the representation of the group of permutations for four subspaces in the group generated by identity and D = DV . Let us formulate 12,34 corollaries from these lemmas first for various composition mappings of the space P on 1 itself. DV = Id−D; DV = DV = (Id−D−1)−1, DV = (Id−D)−1, DV = D−1. 12,43 13,24 14,23 13,24 14,32 It remains to consider the case when the images (and the preimages) of composi- tion mappings are nonisomorphic (say have different dimensions). The corresponding operators in this case differs only by direct summonds of bigger subspace which maps identically. Reduction to the case of isomorphic subspaces is realized as follows. Let dimP > dimP , and one wishes to compare DV with DV . Under the mapping 2 1 12;34 21;43 DV the subspace (P ∩P ) maps on itself identically because its vectors belong both 21;43 2 4 to P and P . The subspace P = (P ⊕P ) is invariant also because both mappings 2 4 13 1 3 π and π are projections parallel to this subspace. P is invariant relative to DV 1 3 13 12;34 also because it contained both subspaces P , and P . Hence, the question is reduced to 1 3 the structure of restriction of mappings DV and DV on P . In general case the 12;34 21;43 13 intersection of subspaces P and P with P have the same dimension as P . Hence, 2 4 13 1 one can apply lemmas 1-3 in the space P : 13 DV = Id−DV = Id−DV = DV = DV . 21,43 21,34 34,21 34,12 12,34 Let us show that operator cross-ratio defines cocycle with values in operators. 4 Lemma 4 Let two subspaces P , i = 1,2 be equivalent relative to the group U and i res three subspaces Q , j = 1,2,3 complete them to the whole space (that is P ⊕ Q = j i j H, i = 1,2, j = 1,2,3). Then DV(P Q ,P Q )DV(P Q ,P Q )DV(P Q ,P Q ) = Id, (2) 1 1 2 2 1 3 2 1 1 2 2 3 i.e. the product of these three operators is the identity. Proof. Consider the chain of mappings P −Q→2 P −Q→1 P −Q→1 P −Q→3 P −Q→3 P −Q→2 P , 1 2 1 2 1 2 1 which defines the left hand side of the formula (2). The composition of the second and the third mappings, as well as the composition of the fourth and the fifth ones, are identities. 2 After its reduction the remaining composition gets identity too. Let us clarify the geometrical sense of the lemma 4. Consider the canonical (tautalogical) bundle γ on the manifold Gr(H), i.e. the bundle whose fiber at any point W ∈ Gr(H) is the linear space W . Introduce the following trivialization of γ. Fix a point W ∈ SGr(H). The chart U on γ is + V defined by a plane V ∈ H that complete W , i.. W ⊕ V = H, and besides the + + projecting operator π : H → W parallel to the space V is assumed to be bounded, V + in other words, for the decomposition h = w + v, where h ∈ H, w ∈ W , v ∈ V the + following estimate with a constant C has to be valid ||w|| ≤ C||h|| (the space V does not have ”infinitesimally small” angles with W ). The coordinates of a point (W,x) ∈ γ, + where x ∈ W , in the chart U will be (W,π x) ∈ Gr(H)×W . Let us calculate the V V + transformation formulas of coordinates from a chart U to that of U . Let coordinates V1 V2 of a point (W,x) ∈ γ in the chart U be (W,y), where y ∈ W . Then x = π−1y, and V1 + V1 the same point has in the chart U coordinates (W,π ◦π−1y). But π = π−1, since V2 V2 V1 V V π is a projector. Hence, the transition function is defined by formula V W −V→1 W −V→2 W , + + which coincides with the cross-ratio of four subspaces DV(W ,V ;W,V ). By lemma 4, + 2 1 the transition from U to U , further on, to U , and finaly, back to U gives the V1 V2 V3 V1 cocycle property (2). So, the transition from a chart U to a chart U is defined by the transform V1 V2 DV(W ,V ;W,V ) which acts on coordinates x ∈ W as a linear fractional function + 2 1 from operator coordinates of a plane W . These transforms are defined on intersections of charts U . But the set of these charts does not cover all the Grassmann manifold. In V contrastwiththefinite-dimensional case, two isomorphicsubspaces ofinfinite-dimensional Hilbert manifolds does not have in general a common complementary subspace V . The exact necessary and sufficient conditions of existence a common complement for two given subspaces was found in ([8]). To overcome this difficulty let us exchange the cross ratio DV(W ,V ;W,V ) by DV(W,V ;W ,V ). In view of lemma 1 the cross ratio remains + 2 1 1 + 2 the same but now it defines a transformation of W instead of W . So we can change + W and the corresponding charts cover all Grassmann manifold. + Hence, coordinate transformations of the canonical bundle γ are endomorphisms DV(W,V ;W ,V ) that can be regarded as endomorphisms of γ itself. In view of lemma 1 + 2 5 4 the cross ratio defines on the Grassmannian a cocycle {DV} with coefficients in the sheaf of endomorphisms of the canonical bundle γ, i.e. {DV} ∈ H1(SGr(H),End(γ)). Following the scheme ofAtijah [10]andTurin [9] consider the principal bundle P with the group G corresponding to the vector bundle γ.Take the exact sequence of bundles over SGr(H) f g 0−→L −→ Q −→ T−→0, (3) where T is the tangent bundle to SGr(H); Q is the bundle of invariant tangent vector fields on P ; L is the bundle of Lie algebras corresponded to left invariant vector fields on G (vector fields tangent to fibers). The exact sequence (3) defines an extension of T by L. Classes of equivalent extensions are in one-to-one correspondence with the elements of H1(SGr(H),Hom(T,L)) — the one-dimensional cohomology group with coefficients in the sheaf Hom(T,L). As in the finite-dimensional case, we say that the sequence (3) is split if there exists a homomorphism h : T → Q such that gh = Id : T → T . Aconnection inthe principal bundle isa splitting of the corresponding exact sequence. Let us denote by a(γ) ∈ H1(SGr(H),Hom(T,L)) an element corresponding to the extension (3). The coordinate transformations ϕ = ij u−1u froma chart u : U ×G → P| to a chart u : U ×G → P| of the bundle P are i j i i Ui j j Uj definedbythecrossratio ϕ = DV(W,V ;W ,V ). Thechart u induces anisomorphism ij i + j i of tangent bundles and since it commute with the action of G we obtain the isomorphism uˆ : T ⊕ L → Q . Here T ,Q ,L are the restrictions of the corresponding bundles to i i i i i i i the neighbourhood U . The sequence (3) splits on this neighbourhood hence there exists i a lifting of the identity endomorphism of T that gives an element a : T → Q , namely i i i a (t) = uˆ (t ⊕ 0). Put a = (a − a ) : T → Q . Then {a } represents the cocycle i i ij j i ij ij ij a(P). Let Ω1 be the sheaf of germs of differential 1-forms on the manifold SGr(H). We have Ω1 = Hom(T,1). Hence H1(SGr(H),Hom(T,L)) = H1(SGr(H),(L×Ω1)). Thus a()P ∈ H1(SGr(H),(L× Ω1)). For compact K¨ahler manifolds in view of isomorphism Dolbeault H1(SGr(H),(L×Ω1)) correspondestocohomologiesof H1,1-type. Itgenerates the ring of characteristic classes of the corresponding bundle. In our case (the manifold SGr(H) isnoncompact)wewillbydefinitionconsider a(P) asananalogofthegenerating Chern class of the canonical bundle γ. The group U transitively acts onthebig cell ofSato’s Grassmannianbut notdouble res transitively, as it is known even in finite-dimensional case. It is natural to find invariants of pairs of points relative to the group U . These invariants are designed as in finite- res dimensional case [3] and are closely related with the operator cross-ratio. Let A; B ∈ Gr (H). Its coordinates are Hilbert-Shmidt operators A and B respec- + tively. The classe of operator DV := (Id+A∗A)−1(Id+A∗B)(Id+B∗B)−1(Id+B∗A) AB will be called the operator angle between subspaces A and B. Let us remark that the operator (Id+A∗A) is positive definite and invertible. 6 Two pairs of subspaces S, T and P, Q shall be called equivalent relative to U , if res there exists an element g ∈ U , such that gS = P, gT = Q. res Two operators V,W : H → H shall be called comparable if there exist two unitary + − operators α ∈ U(H ) and β ∈ U(H ), such that W = αVβ. − + heorem 2 Two pairs of points P, Q and S, T of Sato’s Grassmannian are equivalent iff the classes DV and DV coincide. PQ ST First we prove the following lemma. Lemma 5 The operator angle DV between subspaces A and B is the operator of the AB composite mapping B⊥ A⊥ A −→ B −→ A. Proof. Let us decompose an element (h,Ah) of the subspace A relative to the basis B,B⊥. If (x,Bx) is its projection on B, then (h,Ah) − (x,Bx) is orthog- onal to B, i.e. h(h − x,Ah − Bx),(y,By)i = 0 for any y where angle brackets stand for scalar product in H . It follows that x = (Id+B∗B)−1(Id+B∗A)h. Mak- ing the decomposition relative to the basis A,A⊥, one obtains the element (z,Az) with z = (Id+A∗A)−1(Id+A∗B)(Id+B∗B)−1(Id+B∗A)h. It remains to apply the proposition 1. 2 Proof of the Theorem 2. Necessity follows from the lemma 5 in view of invariance of the composite mapping Sufficiency. Since U transitively acts on the big cell, let us transfer S into P and put the coor- res dinate of the obtaining subspace equal to zero. Let the coordinate of T became W and the coordinate of Q became V . Then DV = (Id+V∗V)−1, DV = (Id+W∗W)−1. PQ ST Since these operators are similar then operators V∗V and W∗W are similar also. These operators correspond to the quadratic forms hVx,Vxi and hWx,Wxi correspondinly on the space H . + To continue the proof we need the following lemma. Lemma 6 Let V,W : H → H are Hilbert-Shmidt operators. Two quadratic forms + − hVx,Vxi and hWx,Wxi on the space H are similar iff operators V and W are + comparable. Sufficiency. Let W = αVβ; α∗α = Id; β∗β = Id. Then (W∗W) = β∗V∗α∗αVβ = β∗(V∗V)β. Necessity. Let (W∗W) = β∗(V∗V)β. Then KerW = β(KerV). Multiply V from the right by the unitary operator β such that KerW = KerV , where V = Vβ. In view of proved 1 1 sufficiency the operators V∗V and W∗W are similar. Multiply V from the left by the 1 1 1 unitary operator γ : H → H , which transfers the image of ImV into ImW . Denote − − γV = V . Operators V∗V and W∗W remain similar. 1 2 2 2 7 The operator V˜ : (KerW)⊥ → ImW , being restriction of the operator V , is invert- 2 ible. Denote by W˜ the restriction of the operator W to the subspace (KerW)⊥. The operators V˜ and W˜ are equivalent. Hence, there exists a unitary transform q of the subspace (KerW)⊥ such that W˜ ∗W˜ = q∗V˜∗V˜q. (4) Take α = q∗, α = W˜ q∗V˜−1. Then, firstly, α V˜α = W˜ , i.e. V˜ and W˜ are compara- 1 2 2 1 ble, and, secondly, α∗α = (V˜∗)−1qW˜ ∗W˜ q∗V˜−1 = Id, in view of (4). It follows that the 2 2 operator α is unitary. If we extend α and α by identity to the kernel we obtain that 2 1 2 V and W are comparable. 2 To complete the proof of the theorem it remains to note that the stabilizer of the point 0 in U are block diagonal operators res α 0 1 , 0 α (cid:18) 2(cid:19) transferring coordinate V of the subspace Q into the comparable operator. Hence, the pair of subspaces P,Q transforms into the pair S,T . 2 Integrals of KP-Hierarchy While finding τ -function[1]oneobtainsaninfinite matrixwhich wasupper-triangular with units on the main diagolal, except for a finite-dimensional block. That was the reason for existing its determinant. It is easily seen that it is the matrix of operator cross-ratio. If we use the lemma 3 and change the order of four subspaces we obtain the infinite matrix A which will be ”asymptotically nilpotent” (i.e. upper-triangular with zeros on the main diagolal) except for a finite-dimensional block. Such matrix A we shall call almost asymptotically nilpotent. Almost asymptotically nilpotent matrix and all its powers have a trace which gives us the possibility to define ζ-function of a cross-ratio. The cross-ratio relates to four spaces P ,P ,P ,P ∈ Gr the class of operators 1 2 3 4 + similar to DV(P ,P ,P ,P ). Each subspace P defines, due to Sato’s correspondence, a 1 2 3 4 i solutionofKP-Hierarchytherewiththeimageofeach P isobtainedbyM¨obiustransform. i Hence, the cross-ratio of four spaces remains invariant in the flow of KP. It means that it changes isospectrally. Invariants of almost asymptotically nilpotent matrix relative to isospectral deformations are traces of its powers. It is natural to describe these invariants by ζ-function. Let we have three solutions W (t), i = 1,2,3 of Riccati equation on Sato’s Grass- i mannian. Using Sato’s correspondence we obtain three solutions L (t), i = 1,2,3 of i KP-hierarchy on the space E′. Take any subspace W and designe the cross-ratio DV(W,W (t),W (t),W (t)). We obtain the operator, spectrum of which does not de- 1 2 3 pend on t. Its characteristic numbers gives the set of integrals and ζ-function remains invariant. It is convenient for calculations to take, as some of W (t), stationary solutions i of the corresponding Riccati equations. Schwarz operator. 8 Operator cross ratio allows us to define operator analog of Schwarz derivative for curves taking values in Grassmann manifolds [3]. Let H be a Hilbert space equiped with a polarization H = H ⊕H . Consider the + − Grassmann manifold Gr(H) defined by the subspace H and the dual Grassmannian + G˜r(H) consisted of complementary subspaces. We will consider the case when comple- mentary subspaces are isomorphic to that of Gr(H). In the finite-dimensional case this means that dimensions of H and H coincide. + − Consider a smooth family of subspaces of H depending of onedimensional parameter s such that pairs (z(s),z(σ)) for s > 0 and σ ≤ 0 give a polarization of the space H. Our nearest aim is to build an analog of the Schwarz derivative of the curve z(s) at s = 0. Take s < 0 < s < s . Then pairs of spaces (z(s ),z(s )) and (z(0),z(s )) give po- 2 1 3 2 1 3 larizations of H. From now on we will for brevity write z ,z′... instead of z(s ),z′(s )...; i i i i values of z(0),z′(0)... will be denoted by z,z′... (without indices). Consider the mapping f depending on four parameters and giving by the cross ratio f(s ,s ,0,s ) = DV(z ,z ;z,z ) = (z −z )−1(z −z)(z −z )−1(z −z ). 2 1 3 2 1 3 2 1 1 3 3 2 Let s → 0. At s = 0 the cross ratio f gives the identity mapping. The derivative 2 2 of f at s = 0 is equal to ∂f (0,s ;0,s ) = −(z −z )−1z′ +(z −z )−1z′. 1 3 1 3 ∂s 2 Now let s → s . At s = s the function f and its derivative relative to s are 3 1 3 1 2 equal to zero. We have ∂2f (0,s ;0,s ) = −(z −z )−1z′(z −z )−1z′. (5) ∂s ∂s 1 1 1 1 1 3 2 The right hand side of (5) is defined at s > 0 and has a singularity at s = 0. 1 1 Consider the situation when subspaces of a polarization tend to be bound together. Find the asymptotic of (5) when s → 0. 1 z′ = z′ +s z′′ + s21z′′′ +o(s2), 1 1 2 1 z −z = s z′ Id+s1(z′)−1z′′ + s21(z′)−1z′′ +o(s2) , 1 1 2 6 1 (z −z)−1 = s(cid:16)−1 Id−s1(z′)−1z′′ − s21(z′)−1z′′′ + s(cid:17)21((z′)−1z′′)2 +o(s2) . 1 1 2 6 4 1 (cid:16) (cid:17) By substituting these expressions for (5) we obtain ∂2f s2(z′)−1z′′′ s2((z′)−1z′′)2 (0,0;0,0) = (s )−2 Id+ 1 − 1 +o(s2) . ∂s3∂s2 1 6 4 1 ! In accordence with this formula define the differential operator 3 S(z) = (z′)−1z′′′ − ((z′)−1z′′)2. (6) 2 This expression is ananalogof the Schwarz derivative. Inthe finite-dimensional case it was introduced and explored in [3]. The above given deduction shows that the expression 9 (6), as a cross ratio with the help of which it is built, defines the same class of conjugate operators independently of M¨obius transforms in the ambient space. Let us show that (6) is closely relates with Hamiltonian systems and Riccati equations. Let q′ = A(t)q +p (7) p′ = −B(t)q −A∗(t)p ( be a linear Hamiltonian system with the Hamiltonian 1 H = (< p,p > +2 < p,Aq > + < Bq,q >), (8) 2 where A(t) and B(t) are n×n-matrices; the matrix B(t) is symmetric; angle brackets stand for scalar product in Rn. Hamiltonian (8) corresponds to minimization problem of the quadratic functional t1 (< q′,q′ > −2 < Aq,q′ > + < (A∗A−B)q,q >)dt. (9) Zt0 The identity coefficient of the first summond is the result of reduction of variational problems with the strong Legendre condition. The system (7) is equivalent to the Euler equation for the functional (9): q′′ +(A∗ −A)q′ +(B −A′ −A∗A)q = 0. (10) We will consider fundamental systems of solutions to (7) and hence p and q are n×n-matrices also. The coeffitions matrix of the system (7) is symplectic thus (7) defines a flow on the Lagrange-Grassmann manifold Λ of the space (p,q). Local coordinates of points of Λ aregiven by matrices W = pq−1. The evolution of coordinates W is described by Riccati equation W′ = (−Bq −A∗p)q−1 −pq−1(Aq +p)q−1 = −B −A∗W −WA−W2. (11) Let us return to our analog of Schwarz operator. S(z(t)) = [(z′(t))−1z′′]′ − 1[(z′(t))−1z′′]2 = 2 (12) (z′(t))−1z′′′ − 3[(z′(t))−1z′′]2. 2 It is convenient to consider t as changing on a projective line (real or complex) or on a Rimanian surface on which acts M¨obius group of linear fractional mappings. It was shown in [3] that the equivalent class of the image of Schwarz derivative S(z(·)) isinvariant relative to theM¨obius group. Namely if onedenote by M a M¨obius transform M : z 7→ (C z+C )(C z+C )−1, where C is (n×n)-matrices, then thereexists a matrix 1 2 3 4 i K(t) such that S(M(z(t))) = K(t)S(z(t))K−1(t). In other words, M¨obius transforms of preimage of the Schwarz operator lead to isospectral change of the image. Let us describe a connection of the operator S with the Hamiltonian system (7) or (it is the same) with the Riccati equation (11). Suppose that the matrix A is symmetric. We find a connection between a solution to Riccati equation W and the function z given by the formula 1 W = − [(z′(t))−1z′′]−A. (13) 2 10