On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.      

Generalized symplectic Cayley transforms and a higher order formula for the Conley–Zehnder index of symplectic paths

Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures.     

Lagrangian Grassmann manifold Λ(2)

Based on the relationship between symplectic group Sp(2) and Λ(2), we provide an intuitive explanation (model) of the 3-dimensional Lagrangian Grassmann manifold Λ(2), the singular cycles of Λ(2), and the special Lagrangian Grassmann manifold SΛ(2). Under this model, we give a formula of the rotation paths dened by Arnold.     

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.     

Gutzwiller’s semiclassical trace formula and Maslov-type index theory for symplectic paths

Gutzwiller’s famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with a view toward the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley–Zehnder index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley–Zehnder’s index theory in which we will recall all essential ingredients is the Maslov-type index theory for symplectic paths developed by Long and his collaborators. It would shed new light on the computations and understandings of the semiclassical trace formula. The insights in Gutzwiller’s work also seems plausible for the studies of Hamiltonian systems.     

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.     

Cup-length estimate for Lagrangian intersections

In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case.     

Indexing domains of instability for Hamiltonian systems

Based upon the understanding of the global topologies of the singular subset, its complement, and the hyperbolic subset in the symplectic group, in this paper we study the domains of instability for hyperbolic Hamiltonian systems and define a characteristic index for such domains. This index is defined via the Maslov-type index theory for symplectic paths starting from the identity defined by C. Conley, E. Zehnder, and Y. Long, and the hyperbolic index of symplectic matrices. The old problem of the relation between the non-degenerate local minimality and the instability of hyperbolic extremal loops in the calculus of variation is also studied via this new index for the domains of instability. Received July 4, 1997     

The Iteration Formulae of the Maslov-Type Index Theory in Weak Symplectic Hilbert Space

The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.     

The Künneth formula in Floer homology for manifolds with restricted contact type boundary

We prove the Künneth formula in Floer (co)homology for manifolds with restricted contact type boundary. We use Viterbo's definition of Floer homology, involving the symplectic completion by adding a positive cone over the boundary. The Künneth formula implies the vanishing of Floer (co)homology for subcritical Stein manifolds. Other applications include the Weinstein conjecture in certain product manifolds, obstructions to exact Lagrangian embeddings, existence of holomorphic curves with Lagrangian boundary condition, as well as symplectic capacities. Supported by ENS Lyon, école Polytechnique (Palaiseau) and ETH (Zürich).     

Symplectic spinors,holonomy and Maslov index

In this note it is shown that the Maslov index for pairs of Lagrangian paths as introduced by Leray and later canonized by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinor bundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold \(L\), along closed or non-closed paths therein. In especially, the CLM-Index mod \(4\) determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on \(L\), hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown that the CLM-Index determines parallel transport in that line-bundle along arbitrary paths when compared to the parallel transport w.r.t. to the canonical flat connection of Euclidean space, if the Lagrangian tangent planes at the endpoints either coincide or are orthogonal. This is derived from a result on parallel transport of certain elements of the dual spinor bundle along closed or endpoint-transversal paths.     

Computation of the Maslov index and the spectral flow via partial signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics. To cite this article: R. Giambò et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Index iteration theory for symplectic paths and multiple periodic solution orbits

In this paper, a survey on the index iteration theory for symplectic paths is given. Three applications of this theory are presented including closed characteristics on convex hypersurfaces and brake orbits on bounded domains.     

Geodesics of Hofer’s metric on the space of Lagrangian submanifolds

We study geodesics of Hofer’s metric on the space of Lagrangian submanifolds in arbitrary symplectic manifolds from the variational point of view. We give a characterization of length–critical paths with respect to this metric. As a result, we see that if two Lagrangian submanifolds are disjoint then we cannot join them by length-minimizing geodesics.     

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW(Ⅱ)

Based on the spectral flow and the stratification structures of the symplectic group Sp(2n, C),the Maslov-type index theory and its generalization, the w-index theory parameterized by all w on the unit circle, for arbitrary paths in Sp(2n, C) are established. Then the Bott-type iteration formula of the Maslov-type indices for iterated paths in Sp(2n, C) is proved, and the mean index for any path in Sp(2n, C) is defined. Also, the relation among various Maslov-type index theories is studied.     

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.     

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW （II）

1.IntroductionandMainResultsStartingfromthepioneeringworks[5,61ofH.AmannandE.Zehnderin1980,C.ConleyandE.Zehnderestablishedanindextheoryin1984intheircelebratedwork[if]fornondegeneratepathsinSp(Zn)startedfromtheidentitymatrixwithn22.Thisindextheorywasextendedtothenondegeneratecaseofn~1byE.Zehnderandthefirstauthorin[28]of1990.ThisindextheoryforthedegenerateHamiltoniansystemswasestablishedbythefirstauthorin[21]of1990andC.Viterboin[34]of1990viadifferentmethods,andthenextendedtoalldegenerates…     

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 index of a net subgroup of a symplectic group over a Dedekind ring

An explicit formula is derived for the index of a net congruence subgroup of a symplectic group over a Dedekind ring. A classification of symplectic D-nets over a field is obtained along the way.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 74–86, 1978.     

The unregularized gradient flow of the symplectic action

The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.