WITH NON SELF-ADJOINT BOUNDARY CONDITIONS
PAOLO PICCIONE AND DANIEL V. TAUSK
ABSTRACT. We present an extension of some results of the classical Sturm theory in the case of linear systems of differential equations of the form−J00+RJ = 0. We assume that, for eacht,R(t)is a linear operator onIRn which is symmetric with respect to a fixed nondegen- erate symmetric bilinear formg; the corresponding differential operator is not self-adjoint, unlessgis positive definite. Motivated by a problem in semi-Riemannian geometry, we also consider non self-adjoint bound- ary conditions for the differential equation. We concentrate our attention to the set of zeroes of solutions for the initial value problem; in partic- ular, we study the problem of the stability of zeroes. We also present some generalizations of the Sturm Oscillation Theorem to the case of Morse–Sturm–Liouville systems by the introduction of afocal indexand aspectral index, which are the analogue of the number of zeroes of non trivial solutions and the number of negative eigenvalues of the Sturm dif- ferential operator. The results are obtained using topological techniques, by developing an intersection theory in the compact manifold of the La- grangian subspaces of a suitable symplectic space. The theory leads to the introduction of the Maslov index, which is a topological invariant for the differential problem.
Date: May 1999.
1
CONTENTS
1. Introduction 2
2. The differential problem 6
2.1. The Morse–Sturm–Liouville system and its focal index 6 2.2. The set of zeroes of the solutions is discrete 8
3. The symplectic structure 9
4. Geometry of the Lagrangian Grassmannian 10
4.1. Generalities on symplectic spaces 11
4.2. The Lagrangian Grassmannian 12
5. Intersection Theory: the Maslov Index 20
5.1. The fundamental group of the Lagrangian Grassmannian 20 5.2. The intersection theory and the construction of the Maslov
index 22
5.3. Computation of the Maslov index 29
6. Applications of the Maslov Index 31
6.1. The Maslov index of a differential problem 32
6.2. Stability of the indexes 34
6.3. Comparison between zeroes of solutions
and of their derivatives 36
7. The Spectral Index.
An Extension of the Sturm Oscillation Theorem 38 7.1. Eigenvalues of the differential problem and the spectral index 38 7.2. A generalized Sturm Oscillation Theorem 43
References 46
1. INTRODUCTION
The classical Sturm theory ([6, 14, 15]) deals with second order linear differential equations of the form−(px0)0+rx= 0, withp >0; the main re- sults of the theory, namely the comparison, the separation and the oscillation theorem, give information about the distribution of the zeroes of solutions of the equation. This kind of equations arises naturally in many problems of calculus of variations and classical mechanics, like for instance in the problem of the vibrating string(see [14]). A classical result of the theory, known as the Oscillation Theorem, says that there is a positively diverging sequenceλ0 < λ1 < . . . < λk < . . .of eigenvalues for the boundary value problem
(1.0.1) −(px0)0+rx=λx, x(a) =x(b) = 0,
and that, for each k ∈ IN, the eigenfunction xk corresponding to thek-th eigenvalueλk has precisely k zeroes in(a, b). The differential equation in (1.0.1) is a special equation in the class of the so called Sturm–Liouville equations, which are of the form
(1.0.2) −(px0)0+ (r−λh)x= 0,
with p, h > 0in [a, b]. The oscillation theorem holds more in general for this kind of equations withself-adjoint boundary conditions; however, the original statements of the theory cannot be extended to case of systems of differential equations.
The Sturm theory can be reformulated in terms of the calculus of varia- tions, and a statement of the Sturm Oscillation Theorem may be given as the equality of the sum of the dimensions of the kernels of a continuous family of bounded linear operators At, t ∈]a, b[ and the number of nega- tive eigenvalues of the operator Ab. More precisely, denoting by Co1[α, β]
the space of C1-functions on [α, β] vanishing atα and β, the index of the symmetric bilinear formB(x, y) = Rb
a[px0y0 +rxy] dtin Co1[a, b]is equal to the sum over t ∈]a, b[ of the dimensions of the kernels of the bilinear formsBt=Rt
a[px0y0+rxy] dtinCo1[a, t]. The Sturm equation is theEuler–
Lagrange equation associated to the quadratic functionalQ(x) = B(x, x) in Co1[a, b], which is satisfied by the stationary points ofQ. Among oth- ers, the variational approach to the Sturm theory has the advantage of being extendible to the case of systems of differential equations, which was orig- inally done by Morse. By a Morse-Sturm-Liouville system of differential equations we mean a system of the form (1.0.2), withp(t) symmetric and r(t), h(t) symmetric with respect to the inner product on IRn defined by p(t). For the Morse’s extension of the Sturm theory to systems, one re- quires thatp(t)be apositive linear operator onIRn; in this paper we drop this assumption. For simplicity, in this exposition we will only consider the case that p(t) ≡ g is aconstant symmetric bilinear form onIRn. We em- phasize that virtually all the results proven in this paper for this special case can be extended to the general case by minor adaptations of the proofs.
For the extension to systems, the statements of the results of the Sturm theory relate the eigenvalues of a boundary value problem with theconju- gate pointsof the differential equation, rather than with the zeroes of a given solution.
In the context of Riemannian geometry (see [7]), the Morse-Sturm equa- tions are obtained as the Jacobi equations for vector fields along a given geodesicγ, whose solutions representinfinitesimalvariations ofγby a one- parameter family of geodesics. When stated in geometrical terms, the Sturm
oscillation theorem gives precisely the statement of theMorse Index Theo- rem, that relates the number (with multiplicity) of conjugate points along a geodesic with the index of the second variation of the energy functional.
Similar results holds for geodesics with endpoints varying in submanifolds, which corresponds to studying the Morse–Sturm equations withself-adjoint boundary conditions. By this, we mean that the differential operator dtd22 is self-adjoint with respect to these boundary conditions.
We now turn our attention to semi-Riemannian manifolds, i.e., manifolds endowed with a non degenerate (indefinite) metric tensor, like for instance Lorentzian manifolds, that are the mathematical model for spacetimes in the sense of General Relativity. The Jacobi equations for vector fields along geodesics in this kind of metrics are of Morse-Sturm type, but the coefficient p fails to be positive. In this case, some new and interesting phenomena arise:
• the differential operator isnotself-adjoint
• the index of the associated quadratic form is in general infinite
• the boundary conditions corresponding to the case of variable end- points are no longer self-adjoint
• the conjugate points mayaccumulate.
It is the goal of this paper to show how the Morse-Sturm-Liouville theory should be modified to extend the classical results to this situation. Besides the geometrical and analytical motivation, there is a genuine interest for the kind of equations considered in this paper arising from some problems in mathematical physics. As we have observed, manifolds with non pos- itive definite metric tensors describe the gravitational field in a relativistic spacetime; the geodesics in these manifolds represent trajectories of masses and photons. More precisely, a timelike geodesic, i.e., a geodesic whose tangent vector has negative length at each point, represents the trajectory of a freely falling massive object. The lightlike geodesics, i.e., those for which the tangent vectors have null length, give the trajectories of the light rays. Conjugate points along acausal(timelike or lightlike) geodesic have a physical meaning, for instance, a conjugate point along a lightlike geodesic is a focal point of an optical system similar to a lens which is caused by the gravitational field. At such a conjugate point there is a concentration of the light rays emitted from a light source. Hence, there is a natural inter- est in studying the location of the conjugate points, and also in determining theirstability. Namely, the trajectory of a light ray can be approximated in a suitable sense by the trajectory of massive objects whose speed tendsto the speed of light, i.e., lightlike geodesics can be approximated by timelike geodesics. In this situation, it is a natural problem to understand whether
there is convergence of the conjugate points and their multiplicity. Using the theory described in this paper, one gets a positive answer to this prob- lem in a large variety of circumstances.
For a better understanding of the ideas lying behind the theory that will be discussed, it is useful to recall the ideas of the classical proofs of the Sturm theory. The classical proof of the Sturm oscillation theorem ([6, Chapter 8]) is obtained bytopological methods, showing that the two quantities involved in the thesis can be obtained as thewinding numberof two homotopic curves in the real projective line. Namely, for allλ ∈ IR, ifxλ(t) is any solution on the interval[a, b]of the equation−(px0)0+rx=λxwithxλ(a) = 0, let c(t, λ)be the line inIR2 through the origin and the point(xλ(t), x0λ(t)); let alsopdenote the line through the origin and(0,1). In such a way, we obtain a two-parameter continuous curve in the projective line that has the property that sum of the dimensions of the kernels ofBt is equal to the number of times that the curvet7→c(t,0)passes through the pointp, while the number of negative eigenvalues ofQis equal to the number of times that the curve λ7→c(b, λ)passes throughpforλ <0.
When passing to the case of vector valued differential equations, the pro- jective line is replaced by the Lagrangian Grassmannian Λ of a suitable symplectic space which is naturally associated to the differential problem.
Similarly, the pointpis replaced by a codimension one compact subvariety ofΛ. The generalization of the map c(t, λ) is given by a map`(t, λ)with values inΛ. The conjugate points of the system correspond to the intersec- tions of the curvet7→`(t,0)with the compact subvariety and the eigenval- ues of the system correspond to the intersections of the curveλ 7→ `(b, λ) with the compact subvariety. Theintersection numberof these two curves are topological invariants for the differential problem corresponding to the winding numbers of the curvecin the proof of the Sturm Oscillation Theo- rem. For the entire theory, the symmetry of the coefficients of the equation plays a fundamental role.
We briefly outline the contents of this article.
In Section 2 we introduce the differential problem, we give the basic defi- nitions concerning conjugate points and we introduce the notion offocal in- dex, which is an algebraic count of the multiplicities of the conjugate points.
In Section 3 we describe the symplectic structure associated to the dif- ferential problem, and in Section 4 we give the manifold structure of the Lagrangian GrassmannianΛ, and we give a useful description of its tangent bundle. We also describeΛas a homogeneous space, which is used in the next section for the computation of its fundamental group, and we describe the structure of the mentioned compact subvariety denoted byΛ≥1(L0).
In Section 5 we define the intersection number between a continuous curve inΛand the subvarietyΛ≥1(L0); such number will be defined as the Maslov indexof the curve. For its definition we will use the first relative ho- mology groupH1(Λ,Λ\Λ≥1(L0)), therefore this number is a homotopical invariant of the curve.
In Section 6 we define theMaslov index of the differential problemas the Maslov index of the corresponding curve of Lagrangians`(t). Under suit- able nondegeneracy assumptions, this number is proven to be equal to the focal index of the problem. Under this hypotheses, one concludes automat- ically the stability of the focal index, which can be applied to the general relativistic problem presented.
Finally, in Section 7 we introduce the spectral index of the differential problem, which is an algebraic count of the negative eigenvalues of the Morse–Sturm differential operator. Since this operator is unbounded andnot self-adjoint, the discreteness and the lower boundedness of its real eigenval- ues does not follow directly from abstract theory. The spectral index is proven to be equal to the Maslov index of the curveλ 7→ `(b, λ), provided that some nondegeneracy assumptions are satisfied. By a homotopy argu- ment, we then prove the equality between the spectral index and the focal index of the differential problem, obtaining the aimed generalization of the Sturm Oscillation Theorem.
The main references for all the material exposed in this paper are [12, 17].
2. THE DIFFERENTIAL PROBLEM
Letg be a non degenerate symmetric bilinear form inIRn, and letR(t), t∈ [a, b], be a continuous curve in the space of linear operators inIRnsuch thatR(t)isg-symmetric for allt ∈[a, b], i.e.,g(R(t)[v], w) =g(v, R(t)[w]) for allv, w∈IRn.
LetP ⊂ IRn be a subspace such that the restriction ofg toP is non de- generate, and let S be a symmetric bilinear form on P; then, there exists a g-symmetric linear operator on P, which we also denote by S, satisfy- ing S(v, w) = g(S[v], w) for all v, w ∈ P. We will say that the family (g, R, P, S)is anadmissible quadruple for the differential probleminIRn. 2.1. The Morse–Sturm–Liouville system and its focal index
We consider the following linear differential equation inIRn: (2.1.1) J00(t) =R(t)[J(t)], t∈[a, b];
we will consider solutionsJ of (2.1.1) that satisfy in addition the following initial conditions:
(2.1.2) J(a)∈P, J0(a) +S[J(a)]∈P⊥,
whereP⊥ is the g-orthogonal complement of P inIRn; such vector fields will be called(P, S)-solutions. Note that, if P = {0}(and thusS = 0), a (P, S)-solution is simply a solution of (2.1.1) vanishing att=a.
LetJbe the space of all(P, S)-solutions:
(2.1.3) J=
n
J :J satisfies (2.1.1) and (2.1.2) o
; and, fort ∈ [a, b], we setJ[t] =
J(t) :J ∈J .
Observe that dim(J) = n since the subspace of IRn ⊕IRn determined by (2.1.2) is n-dimensional. We give some general definitions concerning symmetric bilinear forms.
Definition 2.1.1. LetV be any real vector space andB : V ×V 7→ IRa symmetric bilinear form. Thenegative type number (orindex)n−(B)ofB is the possibly infinite number defined by
n−(B) = sup n
dim(W) :
W subspace ofV on whichB is negative definite o
. (2.1.4)
Thepositive type numbern+(B)is given byn+(B) = n−(−B); if at least one of these two numbers is finite, thesignaturesgn(B)is defined by:
sgn(B) = n+(B)−n−(B).
The kernelof B, Ker(B), is the setV⊥ of those vectorsv ∈ V such that B(v, w) = 0 for all w ∈ V; the degeneracy dgn(B)ofB is the (possibly infinite) dimension ofKer(B).
IfV is finite dimensional, then the numbersn+(B), n−(B)anddgn(B) are respectively the number of1’s,−1’s and0’s in the canonical form ofB as given by the Sylvester’s Inertia Theorem. In this case,n+(B) +n−(B) is equal to the codimension of Ker(B), and it is also called therankofB, rk(B).
Definition 2.1.2. An instantt0 ∈]a, b] is(P, S)-focal if there exists a non null(P, S)-solutionJsuch thatJ(t0) = 0. Themultiplicitymul(t0)oft0is the dimension of the subspace ofJconsisting of such solutions; ift0 is not (P, S)-focal we setmul(t0) = 0:
(2.1.5) mul(t0) = dim
J ∈J:J(t0) = 0 .
Sincedim(J)is equal ton, then the multiplicitymul(t0)is the codimen- sion ofJ[t0]inIRn:
(2.1.6) mul(t0) = codim(J[t0]) = dim(J[t0]⊥).
We now give the following:
Definition 2.1.3. Thesignaturesgn(t0)of the(P, S)-focal instantt0 is de- fined to be the signature of the restriction ofgto the spaceJ[t0]⊥. Ift0is not (P, S)-focal, we setsgn(t0) = 0; if the set of(P, S)-focal instants is finite, we define thefocal indexifoc of the quadruple(g, R, P, S)to be the sum of the signatures of the(P, S)-focal instants:
ifoc = X
t∈]a,b]
sgn(t).
2.2. The set of zeroes of the solutions is discrete
We give some conditions that guarantee the discreteness of the set of(P, S)- focal instants.
Proposition 2.2.1. Let(g, R, P, S)be an admissible quadruple for the dif- ferential problem in IRn, and let t0 be a (P, S)-focal instant. Ifg is non degenerate onJ[t0], then there are no(P, S)-focal instants other thant0 in some neighborhood of t0. Moreover, there are no(P, S)-focal instants in some neighborhood of the initial instanta.
Proof. Letmul(t0) = n−k >0be the multiplicity of the focal instantt0. LetJ1, J2, . . . , Jnbe a basis ofJsuch thatJ1(t0), . . . , Jk(t0)are a basis for J[t0]andJi(t0) = 0fori≥k+ 1.
The vectorsJk+10 (t0), . . . , Jn0(t0)are a basis ofJ[t0]⊥. To prove this, we first observe that they belong to J[t0]⊥; namely, as it will be shown in the next Section (see formula (3.0.4)), ifi∈ {k+ 1, . . . , n}andj ∈ {1, . . . , k}, we have
g(Ji0(t0), Jj(t0)) =g(Ji(t0), Jj0(t0)) =g(0, Jj0(t0)) = 0.
To prove the claim, we need to show that the vectors Jk+10 (t0), . . . , Jn0(t0) are linearly independent, becausedim(J[t0]⊥) = n−k, by (2.1.6). To see this, observe that the fieldsJk+1, . . . , Jnare linearly independent inJ, hence the pairs
(Jk+1(t0), Jk+10 (t0)), . . . ,(Jn(t0), Jn0(t0))
are linearly independent inIR2n. The conclusion follows from the fact that Jk+1(t0) =. . .=Jn(t0) = 0.
We now define a family of continuous vector fields J˜1, . . . ,J˜n alongγ, by setting:
J˜j =Jj, forj = 1, . . . , k;
and
J˜i(t) =
Ji(t)
t−t0, ift6=t0, Ji0(t0), ift=t0,
fori=k+ 1, . . . , n.
The vectorsJ˜1(t0), . . . ,J˜n(t0)are now a basis forIRn.
Namely, the firstkvectorsJ˜1(t0), . . . ,J˜k(t0)are a basis forJ[t0], and the remainingn−k vectorsJ˜k+1(t0), . . . ,J˜n(t0)are a basis forJ[t0]⊥; more- over,gis non degenerate onJ[t0], which implies thatIRn=J[t0]⊕J[t0]⊥.
By continuity, the vectors J˜1(t), . . . ,J˜n(t)are a basis for IRn for t suf- ficiently close to t0. But that implies that, for tsufficiently close tot0 and t 6= t0 the vectors J1(t), . . . , Jn(t)are a basis for IRn, which implies that there are no(P, S)-focal instants aroundt0.
The caset0 =ais treated similarly, observing thatJ[a] =P and consid-
ering thatgis non degenerate onP.
Remark2.2.2. The(P, S)-focal instants coincide precisely with the zeroes of the function r(t) = det(J1(t), . . . , Jn(t)), where J1, . . . , Jn is a basis of J. If(g, R, P, S)is an admissible quadruple with R(t) real analytic on [a, b], thenr(t)is also analytic, and so its zeros are isolated. Observe indeed that r(t) cannot vanish identically on [a, b]because, by Proposition 2.2.1, r(t)is non zero for t sufficiently close toa, t 6= a. It follows easily that, if(M,g, γ,P) is an admissible quadruple for the geometric problem with (M,g)analytic, then the set ofP-focal points alongγis finite.
3. THE SYMPLECTIC STRUCTURE
Given two solutions J1 and J2 of the differential equation (2.1.1), the quantity
(3.0.1) σ(t) =g(J1(t), J20(t))−g(J10(t), J2(t))
is constant in [a, b]. Namely, a straightforward calculation using equation (2.1.1) shows thatσ0vanishes identically. This motivates the following def- inition:
Definition 3.1. The symplectic formω onIR2nassociated togis given by:
ω[(x1, x2),(y1, y2)] =g(x1, y2)−g(x2, y1).
The nondegeneracy ofωfollows easily from the nondegeneracy ofg.
The initial conditions (J(a), J0(a)) ∈ IR2n determine uniquely a solu- tion of (2.1.1), therefore the space of solutions of (2.1.1) can be identified with IR2n. For allt ∈ [a, b], we have a linear automorphism Ψ(t) of IR2n satisfying
(3.0.2) Ψ(t)[(J(a), J0(a))] = (J(t), J0(t)),
for every solution J of (2.1.1). This automorphisms are implemented by what is usually called the fundamental matrix of the first order linear dif- ferential system associated to (2.1.1). Observe that t 7→ Ψ(t)is a curve of classC1 in the general linear groupGL(2n, IR)which satisfiesΨ(0) = Id.
Using the fact that the quantity (3.0.1) is constant, it is also easy to ob- serve thatΨ(t)preserves the symplectic formωfor allt:
ω[Ψ(t)x,Ψ(t)y] =ω[x, y], ∀x, y ∈IR2n,
hence,Ψ(t)is a curve in the symplectic group ofIR2ncorresponding toω.
The important observation here is thatω vanishes on the n-dimensional subspace of IR2n determined by the initial conditions (2.1.2). Namely, if J1, J2 ∈J, thenJi(a)∈P andJi0(a) +S[Ji(a)]∈P⊥fori= 1,2, and:
ω[(J1(a), J10(a)),(J2(a), J20(a))] =
=g(J1(a), J20(a))−g(J10(a), J2(a)) =
=g(J1(a),−S[J2(a)])−g(−S[J1(a)], J2(a)) = 0, (3.0.3)
where the last equality follows from theg-symmetry ofS.
Summarizing the facts that (3.0.1) is constant and thatωvanishes on the space of initial conditions of(P, S)-solutions, we have the following iden- tity:
(3.0.4) g(J10(t), J2(t)) =g(J1(t), J20(t)), ∀J1, J2 ∈J,
for allt∈[a, b]. Formula (3.0.4) is the analog of theAbel’s identitysatisfied by solutions of a Sturm equation (see [15]).
4. GEOMETRY OF THELAGRANGIAN GRASSMANNIAN
We have seen in Section 3 that the set J can be identified with a La- grangian subspace of the symplectic space (IR2n, ω), i.e., a maximal sub- space of IR2n on which ω vanishes. In view to future applications, in this Section we present the main properties and we discuss the geometrical struc- ture of the collection of all Lagrangian subspaces of a symplectic space.
Throughout this section we will assume thatV is a 2n-dimensional real vector space, equipped with a symplectic form ω, i.e., a skew symmetric non degenerate bilinear form onV.
4.1. Generalities on symplectic spaces
A symplectic basis of (V, ω) is a vector space basis e1, . . . , e2n of V such that
ω[en+j, ej] =−ω[ej, en+j] = 1
for allj = 1, . . . , n, andω[ei, ej] = 0otherwise; the existence of a symplec- tic basis in(V, ω)is standard. We recall that acomplex structure forV is a linear operatorI : V 7→V such thatI2 =−Id. A complex structureI on V induces a complex vector space structure onV, andIbecomes the scalar multiplication by the imaginary uniti. A complex structureIiscompatible with the symplectic form ω if the bilinear form ω[I·,·] is symmetric and positive definite onV.
If(Vi, ωi), i = 1,2, are symplectic spaces of the same dimension2n, a linear map T : V1 7→ V2 is called a symplectomorphism if ω2(T x, T y) = ω1(x, y)for allx, y ∈ V1. Observe that a symplectomorphism T is always an isomorphism; namely, then-th exterior powersωni are volume formsin Vi,i= 1,2, which are preserved byT.
We identifyIR2nwithCnby considering the firstn coordinates to be the real part, and the remaining coordinates to be the imaginary part. Therefore, we get a complex structureI0 given byI0(ej) = en+j,I0(en+j) =−ej, for j = 1, . . . , n, where{ei}2ni=1 is the canonical basis ofIR2n. Forx, y ∈IR2n, we denote byx·ythe Euclidean inner product, and byhx, yithe Hermitian product in Cn ' IR2n whose real part is x·y and which is conjugate in the second variable. Thecanonicalsymplectic formω0 inIR2nis the imag- inary part of the Hermitian product. Observe that the canonical basis is a symplectic basis forω0andI0is compatible withω0.
A subspaceW of V will be calledisotropic ifω vanishes identically on W (by this we mean on W × W); an n-dimensional isotropic subspace W will be called a Lagrangian subspace of (V, ω). It is easy to see that the Lagrangian subspaces coincide with themaximalisotropic subspaces of (V, ω).
Given a Lagrangian direct sum decompositionV = L0 ⊕L1, i.e., both subspacesL0andL1are Lagrangian, we denote byDL0,L1 the isomorphism fromL1 to the dual spaceL∗0 given by:
(4.1.1) DL0,L1(v) =ω[v,·]
L0
, ∀v ∈L1.
The injectivity ofDL0,L1 follows immediately from the non degeneracy of ω. We observe that, by the anti-symmetry ofω, the following identity holds:
(4.1.2) DL1,L0 =−(DL0,L1)∗.
Remark4.1.1. The existence of a complex structure compatible with(V, ω) is proven easily. Namely, a complex structure compatible with(V, ω)is ob- tained as the pull-back ofI0 by the symplectomorphismV 7→IR2ndefined by a symplectic basis of (V, ω). Using a compatible complex structure I, we can now prove that every Lagrangian subspaceL0ofV admits a comple- mentary Lagrangian subspaceL1. Namely, just defineL1 =I(L0). Given any Lagrangian direct sum decomposition V = L0 ⊕ L1, we construct a symplectic basis {e1, . . . , en, f1, . . . , fn} of V by taking any linear basis {e1, . . . , en}ofL0 and the linear basis{f1, . . . , fn}ofL1 whose image by DL0,L1 is the dual basis of{e1, . . . , en}. This implies that every linear iso- morphismψ : L0 7→ IRn⊕ {0} extends to a symplectomorphismψ from (V, ω)to(IR2n, ω0)which carriesL1 to{0} ⊕IRn.
Thesymplectic groupSp(V, ω)is the Lie subgroup ofGL(V)consisting of symplectomorphisms of (V, ω); its Lie algebra sp(V, ω) consists of all linear mapsH :V 7→V such that:
(4.1.3) ω(Hx, y) +ω(x, Hy) = 0, ∀x, y ∈V.
Equation (4.1.3) is equivalent to the symmetry of the bilinear formω(H·,·) onV.
The group Sp(IR2n, ω0) is also denoted by Sp(n, IR); the subgroup of GL(2n, IR)consisting of unitary transformations with respect to the canon- ical Hermitian product is denoted by U(n). Sinceω0 is the imaginary part of the Hermitian product which is preserved by elements in U(n), we see thatU(n)is a subgroup ofSp(n, IR).
By O(n)we mean the orthogonal group inIRn, and bySO(n) the sub- group of O(n) consisting of matrices with determinant equal to 1. Ev- ery linear map ψ : IRn 7→ IRn has a unique C-linear extension to a map ψC :Cn 7→ Cn. Ifψ ∈O(n), thenψC ∈ U(n), which identifiesO(n)with the subgroup ofU(n)consisting of those maps that preserve the subspace IRn⊕ {0}inIR2n.
It is well known thatU(n),O(n)andSO(n)are compact Lie groups, and Sp(n, IR), or more in generalSp(V, ω), is a non compact Lie group.
4.2. The Lagrangian Grassmannian
For k = 0, . . . ,2n, we denote by Gk(V) the Grassmannian of all the k- dimensional subspaces of V. We will be interested in the subset Λ = Λ(V, ω)⊂Gn(V)consisting of all the Lagrangian subspaces of(V, ω):
Λ = Λ(V, ω) =n
L:Lis a Lagrangian subspace of(V, ω)o .
For simplicity, we will omit the argument (V, ω)whenever there is no risk of confusion, and we will write simplyΛ.
We recall thatGk(V)has the structure of a real analytic manifold of di- mension k(2n − k); given a direct sum decomposition V = W0 ⊕ W1, wheredim(W0) =k, a local chart ofGk(V)is defined in an open neighbor- hood ofW0 taking values in the vector spaceL(W0, W1)of all linear maps T :W0 7→W1. Namely, to everyW ∈ Gk(V)which istransversaltoW1, i.e.,W∩W1 ={0}, one associates the uniqueT ∈ L(W0, W1)whose graph inW0⊕W1 =V isW. We now give a description of the restrictions to Λ of the local charts defined onGn(V)by this construction.
Given any real vector spaceZ, we denote by B(Z, IR)and Bsym(Z, IR) respectively the space of bilinear forms and symmetric bilinear forms onZ. There is an identification ofB(Z, IR)withL(Z, Z∗)obtained by associating to eachB ∈B(Z, IR)the mapv 7→B(v,·).
Definition 4.2.1. Given a Lagrangian direct sum decompositionV =L0⊕ L1, for all W ∈ Gn(V) transverse to L1, i.e., W ∩L1 = {0}, we define φL0,L1(W)∈B(L0, IR)' L(L0, L∗0)by
φL0,L1(W) =DL0,L1 ◦T,
whereT is the unique linear operatorT : L0 7→ L1 whose graph in V = L0⊕L1 isW.
The mapφL0,L1 is a diffeomorphism from the open set ofGn(V)consist- ing of subspaces transverse toL1 ontoB(L0, IR).
Observe thatφL0,L1 is simply one of the local charts onGn(V)described above, up to the composition with the linear isomorphismDL0,L1. We now show how the mapsφL0,L1 induce a submanifold structure onΛ.
Proposition 4.2.2. The set Λ is an analytic embedded submanifold of the Grassmannian Gn(V) having dimension 12n(n+ 1); each map φL0,L1 re- stricts to a local chart on Λ which maps the open set of Lagrangian sub- spaces transverse toL1 ontoBsym(L0, IR).
For all L0 ∈ Λ the tangent space TL0Λ is canonically isomorphic to Bsym(L0,IR); more precisely, this isomorphism is given by the differential atL0 of any coordinate mapφL0,L1, and this isomorphism does not depend on the choice of the complementary LagrangianL1.
Moreover, the isomorphisms TL0Λ ' Bsym(L0, IR) are natural in the sense that, given a symplectomorphismψ of(V, ω), we have the following
commutative diagram:
(4.2.1)
TL0Λ −−−→d ˆψL0 Tψ(L0)Λ
y
y
Bsym(L0, IR) −−−→ψ∗ Bsym(ψ(L0), IR),
where the vertical arrows are the canonical isomorphisms,ψˆ: Λ7→Λis the diffeomorphism given byL 7→ ψ(L), andψ∗ is the push-forward operator given byB 7→B(ψ−1., ψ−1.).
Proof. Let L ∈ Gn(V) be transverse to L1, and letT : L0 7→ L1 be the linear operator whose graph inV =L0⊕L1 isL. Then,Lis Lagrangian if and only ifω[v+T(v), w+T(w)] = 0for allv, w∈L0, i.e., if and only if
ω[v, T(w)] +ω[T(v), w] = 0, ∀v, w∈L0.
This is just the symmetry of the bilinear formφL0,L1(L) = DL0,L1 ◦T. We now prove that the differential dφL0,L1(L0) does not depend on the choice of the complementary LagrangianL1; observe that, by Remark 4.1.1, we can always find complementary Lagrangians toL0. To prove the claim, let L1 and L2 be two complementary Lagrangians to L0; the two charts φL0,L1 andφL0,L2 map L0 to the zero bilinear map. We have to prove that the differential of the transition map fromφL0,L1 toφL0,L2 at0is the identity ofBsym(L0, IR). The transition map is given by:
(4.2.2) L(L0, L∗0)3B 7−→B◦(Id +ρ◦D−1L0,L1 ◦B)−1 ∈ L(L0, L∗0), whereρis the restriction toL1of the projectionL0⊕L2 7→L0andIdis the identity onL0. The differential of (4.2.2) atB = 0is easily computed to be the identity.
It remains to prove the commutativity of (4.2.1). Let Ω and Ω0 be the domains of the chartsφL0,L1 andφψ(L0),ψ(L1) respectively. Then, it is easy to check the commutativity of the diagram:
(4.2.3)
Λ⊃Ω −−−→ψˆ Ω0 ⊂Λ
φL0,L1
y
yφψ(L0),ψ(L1) Bsym(L0, IR) −−−→ψ∗ Bsym(ψ(L0), IR).
The conclusion follows by differentiating (4.2.3).
The action ofSp(V, ω)on Λinduces a map sp(V, ω) 7→ TL0Λ for every L0 inΛ. This map is described in the following:
Proposition 4.2.3. Let L0 ∈ Λ; define the map κL0 : Sp(V, ω) 7→ Λ by κL0(ψ) = ψ(L0). The differential dκL0(Id) of κL0 at the neutral element Id ∈ Sp(V, ω) maps each H ∈ sp(V, ω) to the symmetric bilinear form dκL0(Id)[H]∈Bsym(L0, IR)given by the restriction ofω(H·,·)toL0. Proof. LetL1 be any complementary Lagrangian toL0,V =L0 ⊕L1, and letφL0,L1 be the corresponding coordinate map aroundL0. Recall that the differential dφL0,L1 at L0 is the isomorphism used to identify TL0Λ with Bsym(L0, IR)(see Proposition 4.2.2). Letπ0 :V 7→L0andπ1 :V 7→L1be the projections onto the summands.
In the chartφL0,L1, the mapκL0 is given by:
(4.2.4) ψ 7−→φL0,L1 ◦κL0(ψ) = DL0,L1 ◦ψ10◦ψ00−1,
whereψ00 = π0◦(ψ|L0)andψ10 =π1◦(ψ|L0). Formula (4.2.4) holds for ψ in a neighborhood ofId∈Sp(V, ω), whereψ00is invertible.
The differential of (4.2.4) is then easily computed as:
sp(V, ω)3H 7−→dκL0(Id)[H] =DL0,L1 ◦H10, whereH10=π1◦(H|L0).
The conclusion follows at once from the definition ofDL0,L1. We recall that, if f : M 7→ N is a smooth map between differentiable manifolds, two smooth vector fieldsXandY onM andN respectively are said to bef-related ifdf(p)[X(p)] = Y(f(p))for allp ∈ M. IfX andY aref-related, thenf maps integral curves ofXinto integral curves ofY.
If a Lie group Gacts on the left on the manifold M, then to eachX in the Lie algebra ofGwe associate a vector fieldX∗inM given byX∗(p) = dκp(1)[X], where κp : G 7→ M is the mapg 7→ g ·p and1is the neutral element ofG. For allp ∈ M, the vector fieldX∗ isκp-related to theright invariantvector field onGassociated toX.
ConsideringG= Sp(V, ω)andM = Λ, we are not motivated to give the following definition:
Definition 4.2.4. Let H ∈ sp(V, ω), the vector field H∗ in Λ associate to eachL∈Λthe vectorH∗(L)∈TLΛ'Bsym(L, IR)given by the restriction ofω(H·,·)toL.
The vector fields H∗ will be used to project differential equations in Sp(V, ω)to differential equations inΛ.
Using group actions, we now give a description of the geometrical struc- ture ofΛas a homogeneous space.
Proposition 4.2.5. Λ is diffeomorphic to U(n)/O(n); in particular, Λ is compact and connected.
Proof. By choosing a symplectic basis for (V, ω), we reduce the problem to the caseV = IR2n and ω = ω0. The group Sp(n, IR) acts smoothly on Λ; we show that the restriction of this action toU(n)is transitive onΛ. Let L0, L1 ∈Λbe fixed; we consider basesB0andB1ofL0andL1respectively, which are orthonormal relatively to the Euclidean inner product of IR2n. Since the imaginary part of the Hermitian product isω0, andω0vanishes on bothL0 andL1, then B0 and B1 are orthonormal basis ofCn ' IR2n with respect to the Hermitian product. Hence, there exists an element of U(n) that carriesB0 toB1, andU(n)acts transitively onΛ.
Obviously, the isotropy group of L0 = IRn ⊕ {0} is O(n), which con-
cludes the proof.
We now give the following definition:
Definition 4.2.6. Let L0 ∈ Λ andk = 0,1, . . . , nbe fixed. We denote by Λk(L0)the subset ofΛconsisting of LagrangiansLwithdim(L∩L0) = k.
We also define the setsΛ≤k(L0)andΛ≥k(L0)by:
Λ≤k(L0) =
k
[
i=0
Λi(L0), Λ≥k(L0) =
n
[
i=k
Λi(L0).
Remark4.2.7. Clearly, Λ0(L0)is precisely the set of all Lagrangians com- plementary to L0. It is an open set of Λ, since it is the domain of any coordinate map φL1,L0; moreover, it is diffeomorphic to a vector space by Proposition 4.2.2. Fork = 0, . . . , n, we observe thatΛ≤k(L0)is open, and so Λ≥k(L0) is closed in Λ. Namely, let L ∈ Λ≤k(L0); we prove that L admits a neighborhood inGn(V)consisting only of subspacesW such that dim(W∩L0)≤k. For, simply consider a subspaceW1 ofV which is com- plementary to both L0 andL; then, given a linear operator T : L0 7→ W1, its graph in L0 ⊕W1 = V intercepts L0 in a subspace of dimension less than or equal tokif and only ifdim(Ker(T))≤ k. The conclusion follows easily by observing that the set of linear operatorsT ∈ L(L0, W1)such that dim(Ker(T))≤kis open.
GivenL0 ∈Λ, we denote bySp(V, ω, L0)the closed subgroup ofSp(V, ω) consisting of elementsψ such thatψ(L0) =L0; bySp+(V, ω, L0)we mean the subgroup ofSp(V, ω, L0)consisting of thoseψ whose restriction toL0 is orientation preserving. The Lie algebrasp(V, ω, L0)of bothSp(V, ω, L0) andSp+(V, ω, L0)is the subalgebra ofsp(V, ω)consisting of thoseHsuch thatH(L0)⊂L0.
Clearly,Sp(V, ω, L0)andSp+(V, ω, L0)act on all the spacesΛ∗(L0)in- troduced in Definition 4.2.6. These actions are transitive on eachΛk(L0), as we prove in the following:
Proposition 4.2.8. For allk= 0, . . . , n, the groupSp+(V, ω, L0)acts tran- sitively onΛk(L0).
Proof. By choosing a symplectic basis of(V, ω), we can reduce to the case V =IR2n,ω=ω0andL0 =IRn⊕{0}(see Remark 4.1.1); let{e1, . . . , e2n} be the canonical basis ofIR2n. LetLbe any Lagrangian such thatdim(L∩ L0) = k; we show that there is an element ψ ∈ Sp+(V, ω, L0) such that ψ(L)∩L0 = IRk⊕ {0}. Letψ ∈ SO(n)be a linear isometry ofIRnsuch thatψ(L∩L0) =IRk⊕ {0}; now consider the complex linear extension of ψ toCn'IR2n. Such a map has the required property.
LetL1be the subspace generated by{e1, . . . , ek, en+k+1, . . . , e2n}. Then, L1 is Lagrangian, and L1 ∩ L0 = IRk ⊕ {0}. It remains to prove that, given a Lagrangian L with L∩L0 = IRk ⊕ {0}, there exists an element ψ ∈Sp+(V, ω, L0)such thatψ(L) = L1.
To prove this claim, we define the following spaces. LetV1 be the space generated by the family {e1, . . . , ek, en+1, . . . , en+k}; V2 be generated by the family{ek+1, . . . , en, en+k+1, . . . , e2n}andSbe generated by the family {e1, . . . , en, en+k+1, . . . , e2n}. Observe thatSis the orthogonal complement ofIRk⊕ {0}with respect toω0; also,IR2n =V1⊕V2, andω0restricts to the canonical symplectic forms ofV1 'IR2kand ofV2 ' IR2(n−k), that will be still denoted byω0. Letπ:S 7→V2 be the restriction toSof the projection V1 ⊕V2 7→ V2. It is easy to check thatω0(π(x), π(y)) = ω0(x, y)for all x, y ∈ S. Since L is Lagrangian, we haveL ⊂ S; moreover, it is easily seen thatπ(L)is Lagrangian in V2. Since L∩L0 = IRk ⊕ {0}, we have that π(L) is complementary to IRn−k ⊕ {0}in V2 ' IRn−k⊕IRn−k. By Remark 4.1.1, there exists a symplectomorphism ϕ of (V2, ω0) that is the identity onIRn−k⊕ {0}and carriesπ(L)into{0} ⊕IRn−k =π(L1).
Finally, the required elementψ ∈Sp+(V, ω, L0)is given by:
ψ V1
= Id, ψ V2
=ϕ.
Indeed,ψ(L) =L1, becauseψ(L)andL1 are both subspaces ofScontain- ingKer(π)that have the same image underπ. This concludes the proof.
Corollary 4.2.9. Given any two Lagrangians L0 and L in Λ, there exists L1 ∈Λwhich is complementary to bothL0andL. In particular, the domain of the coordinate mapφL0,L1 contains bothL0 andL.
Proof. By choosing a symplectic basis of(V, ω), we can reduce to the case V =IR2n,ω =ω0 andL0 =IRn⊕ {0}(see Remark 4.1.1).
Let{e1, . . . , e2n}be the canonical basis ofIR2n andL2 be the subspace generated by{e1, . . . , ek, en+k+1, . . . , e2n}, wherek = dim(L0∩L). Since L2andLare both inΛk(L0), Proposition 4.2.8 gives a symplectomorphism
ψ of (IR2n, ω0) such that ψ(L0) = L0 and ψ(L2) = L. Observe that the diagonal
∆ ={(v, v) :v ∈IRn}
is a Lagrangian subspace of IR2n which is complementary to both L0 and L2; the desired LagrangianL1 is, for instance,ψ(∆).
Although we will not need it, we observe that the existence of com- plementary Lagrangians can be proven in a much more general situation.
Namely, using Baire’s Theorem, one proves that, given a sequence{Lk}k∈IN of Lagrangians in Λ, the set T
k∈IN
Λ0(Lk) of their common complementary Lagrangians is dense inΛ. EachΛ0(Lk)is open dense because its comple- ment inΛis a finite union of embedded submanifolds of lower dimension, as we will see next.
Proposition 4.2.10. For all k = 0, . . . , n and all L0 ∈ Λ, Λk(L0) is a connected embedded analytic submanifold of Λhaving codimension equal to 12k(k+ 1). ForL∈Λk(L0), the tangent spaceTLΛk(L0)⊂Bsym(L, IR) is equal to the space of symmetric bilinear forms onLthat vanish inL∩L0. The submanifold Λ1(L0), that has codimension 1 in Λ has a transverse orientationinΛ, namely, for L∈Λ1(L0), a vectorB ∈Bsym(L)'TLΛis positiveifBis positive definite on the one-dimensional spaceL∩L0. More- over, the transverse orientation of Λ1(L0)inΛ isnaturalin the sense that, givenψ ∈Sp(V, ω, L0), the diffeomorphismL7→ ψ(L)ofΛis orientation preserving.
Proof. To prove thatΛk(L0)is an embedded submanifold ofΛ, observe first that, by Proposition 4.2.8,Λk(L0)is an orbit of the action ofSp+(V, ω, L0).
It follows thatΛk(L0)is an immersedsubmanifold, i.e., it does not neces- sarily have the relative topology. By [23, Theorem 2.9.7], an orbit is em- bedded if and only if it is locally closed, i.e., it is the intersection of an open and a closed set. Now, recall Remark 4.2.7 and simply observe that Λk(L0) = Λ≥k(L0)∩Λ≤k(L0).
We now compute the codimension ofΛk(L0)inΛ.
LetL1be any Lagrangian complementary toL0; the Lie groupSp(V, ω, L0) is diffeomorphic toGL(L0)×Bsym(L1, IR). Namely, we have a diffeomor- phism:
F : GL(L0)×Bsym(L1, IR)7−→Sp(V, ω, L0)
that associates to each pair(α, β)the symplectomorphismψ = F(α, β)of (V, ω)whose restriction toL0isαand whose restriction toL1is equal to:
ψ L
1 =α◦D−1L
1,L0 ◦β+D−1L
0,L1 ◦(α∗)−1◦DL0,L1,
whereα∗ ∈GL(L∗0, IR)denotes the transpose map ofα, andβ is seen as a linear mapβ :L1 7→L∗1.
It follows that the dimension ofSp(V, ω, L0)is equal ton2+12n(n+ 1).
The groupSp+(V, ω, L0)is the image under F of the productGL+(L0)× Bsym(L1, IR), whereGL+(L0)is the group of orientation preserving isomor- phisms ofL0. It follows thatSp+(V, ω, L0)and henceΛk(L0)is connected.
Now, we choose an elementL∈Λk(L0)and we calculate the dimension of its isotropy group in Sp(V, ω, L0). To this aim, let S ⊂ L0 be any k- dimensional subspace and let S0 ⊂ L1 be the image under D−1L
0,L1 of the annihilator of S inL∗0. Then, L = S⊕S0 is a Lagrangian inV and L ∈ Λk(L0).
The isotropy group of Lis the image under F of the set of pairs(α, β) such thatα(S)⊂ S andβ vanishes onS0. It follows that the dimension of this isotropy group isn2 + 12k(k + 1). Hence, using Proposition 4.2.2, the codimension ofΛk(L0)inΛis computed as 12k(k+ 1).
We now compute the tangent spaceTLΛk(L0)at any pointL ∈ Λk(L0).
Such a space is given by the image of sp(V, ω, L0) under the differential dκL(Id), defined in Proposition 4.2.3:
TLΛk(L0) =n
ω(H·,·)
L:H ∈sp(V, ω, L0)o .
The elements ofTLΛk(L0)vanish onL0∩L. A simple dimension counting shows that TLΛk(L0)consists precisely of those elements. This completes the proof of the first part of the statement.
We now consider the submanifold Λ1(L0); by the formula computed above, its codimension in Λ is equal to 1. The transverse orientation is well defined in the statement of the Proposition, and the naturality follows easily from the commutative diagram (4.2.1) in Proposition 4.2.2.
Remark4.2.11. In Proposition 4.2.10 we have given a description of a tan- gent space TLΛk(L0) as a subspace of Bsym(L, IR), where Bsym(L, IR) is identified withTLΛby means of a coordinate mapφL,L1 (Proposition 4.2.2).
In many situations we will have to deal with curvesL(t)of Lagrangians, and to study the tangent space TL(t)Λ it will be more convenient to work with afixedcoordinate mapφL0,L1 rather than using variable chartsφL(t),L1. For this reason, we now describe the transition map from a coordinate mapφL0,L1 to φL,L1, where L1 is a complementary Lagrangian to bothL0 andL(see Corollary 4.2.9).
Letη : L0 7→ Lbe the isomorphism obtained by the restriction to L0 of the projectionL⊕L1 7→L. The transition map fromφL0,L1 toφL,L1 is now
easily computed as:
Bsym(L0, IR)3B 7−→φL,L1(L0) +η∗(B),
where η∗ is the push-forward operator given by η∗(B) = B(η−1., η−1.).
Thus, the transition map isη∗ plus a translation by a fixed element, and so its differential at any point is given byη∗.
Observe thatηis the identity inL0∩L, therefore we get (4.2.5)
dφL0,L1(L)
TLΛk(L0)
= n
B ∈Bsym(L0, IR) :B vanishes onL∩L0
o . The reader should compare formula (4.2.5) with the description of the tan- gent spaceTLΛk(L0)given in the statement of Proposition 4.2.10. Observe also that, for L ∈ Λ1(L0), since the push-forward operator does not affect the positivity of a bilinear form, a given B ∈ Bsym(L0, IR) is such that dφL0,L1(L)−1[B]is a positive vector in the transverse orientation ofΛ1(L0) if and only ifBis positive definite onL∩L0.
5. INTERSECTIONTHEORY: THEMASLOVINDEX
The purpose of this Section is to associate an integer number to each pair (`, L0), where ` is a continuous curve in the Lagrangian Grassmannian Λ studied in the previous section, and L0 ∈ Λ. Such a number, that will be defined to be theMaslov Index of` with respect toL0, in the generic case will count (algebraically) the number of intersections of`withΛ≥1(L0).
We will assume throughout the Section that(V, ω) is a fixed symplectic space of dimension2n, and thatΛis the associated Lagrangian Grassman- nian.
5.1. The fundamental group of the Lagrangian Grassmannian
We begin with an easy result on the fundamental group of homogeneous spaces:
Lemma 5.1.1. LetGbe a connected Lie group andKbe a closed subgroup of G; we denote by p : G 7→ G/K the quotient map. Let q : ˜G 7→ Gbe the universal covering group ofG,K˜ = q−1(K)andK˜0 be the connected component of the neutral element 1 ∈ K. Then, the fundamental group˜ π1(G/K)is isomorphic to the quotientK/˜ K˜0. The isomorphism
ζ : ˜K/K˜0 7→π1(G/K, p(1))
is defined as follows. If gK˜0 is any element of K/˜ K˜0, letc : [0,1] 7→ G˜ be any continuous curve such that c(0) = 1 ∈ G˜ and c(1) = g−1. Then,
