Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in a toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states) and some non-monotone Lagrangian torus fibers.     

Lie algebra on the transverse bundle of a decreasing family of foliations

J. Lehmann-Lejeune in [J. Lehmann-Lejeune, Cohomologies sur le fibré transverse à un feuilletage, C.R.A.S. Paris 295 (1982), 495–498] defined on the transverse bundle V to a foliation on a manifold M, a zero-deformable structure J$J$ such that J²=0$J^2=0$ and for every pair of vector fieldsX$X$,Y$Y$ on M: [JX,JY]−J[JX,Y]−J[X,JY]+J²[X,Y]=0$[JX,JY]-J[JX,Y]-J[X,JY]+J^2[X,Y]=0$. For every open set Ω$\Omega$ of V, J. Lehmann-Lejeune studied the Lie Algebra _LJ(Ω)$L_J\left(\Omega \right)$ of vector fields X defined on Ω$\Omega$ such that the Lie derivative L(X)J$L\left(X\right)J$ is equal to zero i.e., for each vector field Y$Y$on Ω$\Omega$: [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$ and showed that for every vector field X on Ω$\Omega$ such thatX∈KerJ$X\in KerJ$, we can write X=∑[Y,Z]$X=\sum [Y,Z]$ where ∑$\sum$is a finite sum and Y,Z$Y,Z$ belongs to _LJ(Ω)∩(Ker_J|Ω)$L_J\left(\Omega \right)\cap \left(Ker{J}_{|\Omega }\right)$.     

Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold

We study calibrated complex structures on the generalized tangent bundle of a Riemannian manifold M and their relationship to the Riemannian geometry of M. In particular we introduce a concept of integrability of such structures and we prove that integrability conditions are strictly related to the existence of certain Codazzi tensors on M.     

Minimal Lagrangian isotropic immersions in indefinite complex space forms

Let Mⁿ be an n-dimensional (n≥3) minimal Lagrangian isotropic submanifold in an indefinite complex space form. We show that the dimension of Mⁿ satisfies n=3r+2 with r, a positive integer. When n<14, we give a classification of such submanifolds.     

Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group G$G$, the moduli space of flat G$G$-bundles over a closed surface Σ$\Sigma$ is known to be pre-quantizable at integer levels. For non-simply connected G$G$, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H³(G²;Z)$H^3(G^2;\mathbb{Z})$–that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G$G$.     

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.     

On the Dirac spectrum of Riemannian foliations admitting a basic parallel 1-form

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenböck-Lichnerowicz type formula which allows us to apply the classical Bochner-Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.     

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.     

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.     

Geometrical construction of a generalized Skyrme Lagrangian

A geometrical scheme is proposed that leads to a straightforward generalization of a Skyrme Lagrangian, comprising higher-order terms up to the eighth power in pion fields.Boursière DGRST.     

Curvatures,volumes and norms of derivatives for curves in Riemannian manifolds

The well-known formulas express the curvature and the torsion of a curve in R³${\mathbb{R}}^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in arbitrary Riemannian manifolds. Our motivation comes from physics. It follows that regular curves in Rⁿ${\mathbb{R}}^n$ are determined up to isometry by the norms of their n$n$ consecutive derivatives. We extend this fact to two-point homogeneous spaces.     

Prescribing singularities of maximal surfaces via a singular Björling representation formula

We derive a proper formulation of the singular Björling problem for spacelike maximal surfaces with singularities in the Lorentz–Minkowski 3-space which roughly asks whether there exists a maximal surface that contains a prescribed curve as singularities, and then provide a representation formula which solves the problem in an affirmative way. As consequences, we construct many kinds of singularities of maximal surfaces and deduce some properties of the maximal surfaces related to the singularities due to the geometry of the Gauss map.     

Lifting geometric objects to a cotangent bundle, and the geometry of the cotangent bundle of a tangent bundle

We show that the cotangent bundle T^*T of the tangent bundle of any differentiable manifold carries an integrable almost tangent structure which is generated by a natural lifting procedure from the canonical almost tangent structure (vertical endomorphism) of T . Using this almost tangent structure we show that T^*T is diffeomorphic to a tangent bundle, namely TT^* . This provides a new and geometrically instructive proof of a result of Tulczyjew, which has applications in Lagrangian and Hamiltonian dynamics and in field theory The requisite general definitions and results concerning liftings of geometric objects from a manifold to its cotangent bundle are given. As an application, we shed new light on the meaning of so-called adjoint symmetries of second-order differential equations.     

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.