Engouffrement symplectique et intersections lagrangiennes

Let }L _t{,t ∈ [0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension ≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L _t{ is conjugate to a path of exact submanifolds inT ^*L_o. This impliesL _t must intersectL _o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property of symplectic structures which is stated in the first part of this work.

     

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.     

Orthogonal trajectories on Riemannian manifolds and applications to Plane Wave type spacetimes

We present a result on trajectories of a Lagrangian system joining two given submanifolds of a Riemannian manifold, under the action of an unbounded potential. As an application, we consider geodesics in a class of semi-Riemannian manifolds, the Plane Wave type spacetimes.     

Symplectic duality of symmetric spaces

Let M⊂Cn be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ωhyp. Denote by (M^∗,ωFS) the compact dual of (M,ωhyp), where ωFS is the Fubini-Study form on M^∗. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism satisfying and for the pull-back of ΨM, where ω₀ is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ΨM, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of Cn. As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin.     

The Ribaucour transformation for flat Lagrangian submanifolds

We study the Ribaucour transformation for flat Lagrangian submanifolds in complex flat space and complex projective space. As a consequence, we obtain a process to generate a new family of such submanifolds from a given one. Analytically, this provides a method to construct new solutions of the corresponding systems of PDEs from a given one. We also show that such transformation always comes with a permutability formula.     

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.     

Lagrangian skeletons,periodic geodesic flows and symplectic cuttings

In this paper, we investigate symplectic manifolds endowed with a Morse–Bott function with only two critical submanifolds, one of which is Lagrangian while the other one is symplectic.     

Symmetric submanifolds associated with irreducible symmetric R-spaces

We construct examples of symmetric submanifolds in Riemannian symmetric spaces of noncompact type and obtain the classification of symmetric submanifolds in irreducible Riemannian symmetric spaces of noncompact type and rank greater than one. This finishes the classification problem of symmetric submanifolds in Riemannian symmetric spaces completely.     

Pseudo-parallel Lagrangian submanifolds in complex space forms

In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel.     

对称等仿射球和极小对称Lagrange子流形的对应

利用对称空间的对偶性,本文建立局部强凸对称等仿射球之集与某复空间形式中的极小对称Lagrange子流形之集间的对应关系,在自然定义的等价意义下,这是一一对应关系.作为这种对应关系的直接应用,本文用完全不同的方法重新证明胡泽军等人最近建立的一个重要定理.该定理对具有平行Fubini-Pick形式的局部强凸等仿射球进行了完全分类.     

Partial differential equations with differential constraints

A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).     

The Brake Orbits of Hamiltonian Systems on Positive-type Hypersurfaces

This paper deals with the brake orbits of Hamiltonian system on given energy hypersurfaces Σ = H ⁻¹(1). We introduce a class of contact type but not necessarily star-shaped hypersurfaces in ℝ²ⁿ and call them normalized positive-type hypersurfaces. By using of the critical point theory, we prove that if Σ is a partially symmetric normalized positive-type hypersurface, it must carries a brake orbit of (HS). Furthermore, we obtain some multiplicity results under certain pinching conditions. Our results include the earlier works on this subject given by P. Rabinowitz and A. Szulkin in star-shaped case. An example of partially symmetric normalized positive-type hypersurface in ℝ⁴ that is not star-shaped is also presented Partially supported by NNSF of China (10571085) and Science Foundation of Hohai University.     

Large-time behavior of solutions to the equations of a viscous polytropic ideal gas

We study Lagrangian systems with symmetry under the action of a constant generalized force in the direction of the symmetry field. After Routh's reduction, such systems become nonautonomous with Lagrangian quadratic in time. We prove the existence of solutions tending to an orbit of the symmetry group as t± . As an example, we study doubly asymptotic solutions for the Kirchhoff problem of a heavy rigid body in an infinite volume of incompressible ideal fluid performing a potential motion.Supported by GNFM and by MURST (40%: «Equazioni di evoluzione...»).Supported by Russian Foundation of Basic Research and by INTAS.     

Locally symmetric minimal affine Lagrangian surfaces in C2

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry.     

On Anosov energy levels of hamiltonians on twisted cotangent bundles

LetT ^* M denote the cotangent bundle of a manifoldM endowed with a twisted symplectic structure [1]. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by a convex HamiltonianH: T ^* M, and we consider a compact regular energy level ofH, on which this flow admits a continuous invariant Lagrangian subbundleE. When dimM3, it is known [9] that such energy level projects onto the whole manifoldM, and thatE is transversal to the vertical subbundle. Here we study the case dimM=2, proving that the projection property still holds, while the transversality property may fail. However, we prove that in the case whenE is the stable or unstable subbundle of an Anosov flow, both properties hold.     

Double structures and jets

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to construct holonomic and semiholonomic submanifolds in the manifold of double contact elements, and show that these give rise to affine bundles where a semiholonomic element has well-defined holonomic and curvature components.     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(=  1, 2, 3) distinct principal curvatures. Dedicated to Professor Hajime Urakawa on his sixtieth birthday. H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006.