On the rigidity of the coisotropic Maslov index on certain rational symplectic manifolds

We revisit the definition of the Maslov index of loops in coisotropic submanifolds tangent to the characteristic foliation of this submanifold. This Maslov index is given by the mean index of a certain symplectic path which is a lift of the holonomy along the loop. We prove a Maslov index rigidity result for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds.     

The Hörmander and Maslov Classes and Fomenko's Conjecture

Some functorial properties are studied for the Hörmander classes defined for symplectic bundles. The behavior of the Chern first form on a Lagrangian submanifold in an almost Hermitian manifold is also studied, and Fomenko's conjecture about the behavior of a Maslov class on minimal Lagrangian submanifolds is considered.     

Relative flux homomorphism in symplectic geometry

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We also show that some quotients of the universal covering of the group of symplectomorphisms are stable under symplectic reduction.

     

Invariants de Maslov équivariants

We construct two cohomological invariants associated to pairs of Lagrangian sub-bundles of a symplectic bundle on a compact manifold upon which a compact Lie group is acting. The first invariant, which we call the classical equivariant Maslov H-invariant, provides an obstruction to Lagrangian transversality and lives in the Borel cohomology. The second invariant, which we call the equivariant Maslov U-invariant, generalises the author's results in K-Theory 13 (1998), 347–361 to the equivariant context and provides a necessary and sufficient condition for equivariant Lagrangian transversality, up to homotopic stability, and lives in the U-theory (intermediate between the real complex K-theories). As an application, we show that two Lagrangian sub-bundles of a symplectic bundle on a homogeneous space are always stably transverse.     

The η-form and a generalized Maslov index

Given a family of pairs of transverse Lagrangian subspaces of a hermitean symplectic vector space we define a family of Dirac operators on the unit interval and consider its η-form . To a family of pairwise transverse Lagrangian subspaces we associate the cocycle which is a closed form. We identify its cohomology class with a generalization to families of the triple Maslov index. Received: 6 March 1997     

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying symplectic Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.     

THE MODULI SPACE OF COMPLEX LAGRANGIAN SUBMANIFOLDS IN THE HYPER-KAEHLER MANIFOLD

51.IntroductionSpecialLagrangiansubmanifoldsofaCalabi-Yaumanifoldareoneoftherecentattractivesubjectsinmathematics(see[5-81).In1996,R.C.Mclean[7]obtainedthedeformationtheoremofspeciaILagrangiansubmanifold,whichshowsthat,givenonecompactspecialLagrangiansubmanifoldL,thereisalocalmodulispaceMlwhichisamanifoldandwhosetangelltspaceatLiscanonicallyidentifiedwiththespaceofharmonic1-formsonL.TheLzinnerproductonharmonicformsthengivesthemodulispaceanaturalRiemannianmetric.Strominger,YauandZaslow[1…     

An extension of a theorem of Nicolaescu on spectral flow and the Maslov index

In this paper we extend a theorem of Nicolaescu on spectral flow and the Maslov index. We do this by studying the manifold of Lagrangian subspaces of a symplectic Hilbert space that are Fredholm with respect to a given Lagrangian . In particular, we consider the neighborhoods in this manifold of Lagrangians which intersect nontrivially.

     

Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Abstract

This article is devoted to the quantization of the Lagrangian submanifolds in the context of geometric quantization. The objects we define are similar to the Lagrangian distributions of the cotangent phase space theory. We apply this to construct quasimodes for the Toeplitz operators and we state the Bohr-Sommerfeld conditions under the usual regularity assumption. To compare with the Bohr-Sommerfeld conditions for a pseudodifferential operator with small parameter, the Maslov index, defined from the vertical polarization, is replaced with a curvature integral, defined from the complex polarization. We also consider the quantization of the symplectomorphisms, the realization of semi-classical equivalence between two different quantizations of a symplectic manifold and the microlocal equivalences.     

Symplectic topology on subcritical manifolds

We introduce a new class of closed symplectic manifolds called subcritical. These manifolds are closed analogues of subcritical Stein manifolds. We study symplectic and Lagrangian embeddings into such manifolds and into their hyperplane sections. Received: November 13, 2000     

Pseudotoric Lagrangian fibrations of toric and nontoric fano varieties

We introduce the notion of a pseudotoric structure on a symplectic manifold, generalizing the notion of a toric structure. We show that such a pseudotoric structure can exist on toric and nontoric symplectic manifolds. For the toric manifolds, it describes deformations of the standard toric Lagrangian fibrations; for the nontoric ones, it gives Lagrangian fibrations with singularities that are very close to the toric fibrations. We present examples of toric manifolds with different pseudotoric structures and prove that certain nontoric manifolds (smooth complex quadrics) have such structures. In the future, introducing this new structure can be useful for generalizing the geometric quantization and mirror symmetry methods that work well in the toric case to a broader class of Fano varieties.     

Maslov index and Morse theory for the relativistic Lorentz force equation

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.Mathematics Subject Classification (2002): Primary: 58E10, 83C10; Secondary: 53D12     

On the usefulness of an index due to Leray for studying the intersections of Lagrangian and symplectic paths

Using the ideas of Keller, Maslov introduced in the mid-1960's an index for Lagrangian loops, whose definition was clarified by Arnold. Leray extended Arnold results by defining an index depending on two paths of Lagrangian planes with transversal endpoints. We show that the combinatorial and topological properties of Leray's index suffice to recover all Lagrangian and symplectic intersection indices commonly used in symplectic geometry and its applications to Hamiltonian and quantum mechanics. As a by-product we obtain a new simple formula for the Hörmander index, and a definition of the Conley–Zehnder index for symplectic paths with arbitrary endpoints. Our definition leads to a formula for the Conley–Zehnder index of a product of two paths.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.     

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.     

Self-adjoint elliptic operators and manifold decompositions Part II: Spectral flow and Maslov index

This the second part of a three-part investigation of the behavior of certain analytical invariants of manifolds that can be split into the union of two submanifolds. In Part I we studied a splicing construction for low eigenvalues of self-adjoint elliptic operators over such a manifold. Here we go on to study parameter families of such operators and use the previous “static” results in obtaining results on the decomposition of spectral flows. Some of these “dynamic” results are expressed in terms of Maslov indices of Lagrangians. The present treatment is sufficiently general to encompass the difficulties of zero-modes at the ends of the parameter families as well as that of “jumping Lagrangians.” In Part III, we will compare infinite- and finite-dimensional Lagrangians and determinant line bundles and then introduce “canonical perturbations” of Lagrangian subvarieties of symplectic varieties. We shall then use this information to study invariants of 3-manifolds, including Casson's invariant. © 1996 John Wiley & Sons, Inc.     

The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections. Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples.     

Maslov’s canonical operator in arbitrary coordinates on the Lagrangian manifold

We present the construction of the Maslov canonical operator adapted to an arbitrary coordinate system on the corresponding Lagrangian manifold. The construction does not require any additional choice of the phase function.     

An algebraic theory for generalized Jordan chains and partial signatures in the Lagrangian Grassmannian

We discuss an algebraic theory for generalized Jordan chains and partial signatures, that are invariants associated to sequences of symmetric bilinear forms on a vector space. We introduce an intrinsic notion of partial signatures in the Lagrangian Grassmannian of a symplectic space that does not use local coordinates, and we give a formula for the Maslov index of arbitrary real analytic paths in terms of partial signatures.