Distributions on almost contact manifolds

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to determine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G₂-structure.     

A Note on Donaldson's “Tamed to Compatible” Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed to compatible" question(Draghici T, Zhang W. A note on exact forms on almost complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011).That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure? They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson's question in dimension 4. In this note, we give another simpler proof of their result.     

A Remark on the Global Existence of a Third Order Dispersive Flow into Locally Hermitian Symmetric Spaces

We prove time-global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation under consideration generalizes two-sphere-valued completely integrable systems modeling the motion of vortex filament. Unlike one-dimensional Schrödinger maps, our third order equation is not completely integrable under the curvature condition on the target manifold in general. The idea of our proof is to exploit two conservation laws and an “almost conserved quantity” which prevents the formation of a singularity in finite time.     

A Note on Donaldson＇s ＂Tamed to Compatible＂, Question

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson＇s ＂tamed to compatible＂ question （Draghici T, Zhang W. A note on exact forms on almost complex manifolds, arXiv： 1111. 7287vl [math. SC]. Submitted on 30 Nov. 2011）. That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure？ They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson＇s question in dimension 4. In this note, we give another simpler proof of their result.     

Geometric methods for construction of quantum gates

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. These methods are shown to be very useful for the problem of constructing a universal set of gates for quantum computations: the well-known result that the set of all one-bit gates together with almost any one two-bit gate is universal is considered from the control theory viewpoint. Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold, and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action, and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over the Grassmannian and its relation with the Berry phase is considered and investigated for the trajectories of Hamiltonian dynamical systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 44, Quantum Computing, 2007.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

The sharp Jackson inequality in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> on the segment [−1, 1] with the power weight

It is proved that there exists an equivariant almost complex structure on any quasitoric manifold that admits a positive omniorientation. This gives an answer to the question raised by M. Davis and T. Januszkiewicz: Find a criterion for the existence of an equivariant almost complex structure on a quasitoric manifold in terms of its characteristic function.     

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.     

Differential operators on complex manifolds with a flat protective structure

Some natural differential operators on a complex manifold equipped with a flat projective structure have been constructed. As an application, a higher dimensional analog of the Schwarzian derivative has been defined. This higher dimensional analog shares the characteristic properties of the usual one dimensional Schwarzian derivative with respect to the projective transformationsA canonical decomposition of the space of all differential operators between certain line bundles over a Riemann surface equipped with a projective structure has been described.     

Dirichlet problem for Hermitian-Einstein equation over almost Hermitian manifold

In this paper, we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold, and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.     

Sur l'existence du nombre de Lelong d'un courant positif psh défini sur une variété presque complexe

The goal of this Note is to extend to an almost complex manifold the existence of the Lelong number of a positive plurisubharmonic (psh) current. In this way, we generalize results of Lelong and Skoda established in the case of an integrable complex structure, and of Haggui in the non-integrable case, but only for a closed positive current. The main point is to establish a Lelong–Jensen formula for a positive psh current defined on an almost complex manifold, which generalizes a formula proved by Demailly when the structure is integrable. To cite this article: F. Elkhadhra, S.K. Mimouni, C. R. Acad. Sci. Paris, Ser. I 344 (2007).     

On the existence of almost contact structure and the contact magnetic field

We give a simple proof of the existence of an almost contact metric structure on any orientable 3-dimensional Riemannian manifold (M ³, g) with the prescribed metric g as the adapted metric of the almost contact metric structure. By using the key formula for the structure tensor obtained in the proof this theorem, we give an application which allows us to completely determine the magnetic flow of the contact magnetic field in any 3-dimensional Sasakian manifold.     

On the Neumann function and the method of images in spherical and ellipsoidal geometry

The invention of an image system for a boundary value problem adds to a significant understanding of the structure of the problem, both at the mathematical and at the physical level. In this paper, the interior and exterior Neumann functions for the Laplacian in the cases of spherical and ellipsoidal domains are represented in terms of images. Besides isolated images, the presence of the normal derivative in the Neumann condition demands an additional continuous distribution of images, which in the spherical cases, can be restricted to a one‐dimensional manifold, whereas for the ellipsoid, both a one‐dimensional and a two‐dimensional distribution of images is needed. Copyright © 2012 John Wiley & Sons, Ltd.     

Almost Hermitian structures induced from a Kähler structure which has constant holomorphic sectional curvature

We obtain a non-Kähler almost Hermitian manifold of constant holomorphic sectional curvature by changing the almost complex structure in a Kähler manifold of constant holomorphic sectional curvature.

     

A new class of almost complex structures

The purpose of this paper is to introduce a new class of almost complex structures J on a Riemannian manifold M by using a certain identity for the relationship between the tensor F _i ^j of J and the Riemann curvature tensor R _hijk of M. This class contains the Kählerian structures, and its relationship with some known classes of almost Hermitian structures defined by similar identities is discussed. For convenience we call each structure of this new class an almost C-structure, and a manifold with an almost C-structure an almost C-manifold. We obtain an analogue of F. Schur's theorem concerning the holomorphic sectional curvature of an almost Hermitian C-manifold, and some sufficient conditions for an almost Hermitian C-manifold to be Kählerian. We show that these results are also true for a manifold with a complex structure.     

The f-invariant and index theory

In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the stable homotopy groups of spheres.     

Sur l'existence du nombre de Lelong d'un courant positif fermé défini sur une variété presque complexe

Let (M,J) be an almost complex manifold. By using local coordinate system adapted to the structure J, we prove that every closed positive current on M possesses a Lelong number at any point. In case the manifold is equipped with an integrable complex structure, this Lelong number coincides with the usual Lelong number of a closed positive current.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

Some pseudo-Kähler Einstein 4-symmetric spaces with a “twin” special almost complex structure

On 4-symmetric symplectic spaces, invariant almost complex structures -up to sign- arise in pairs. We exhibit some 4-symmetric symplectic spaces, with a pair of “natural” compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non-integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non-integrable one is Ricci Hermitian (in the sense that the almost complex structure preserves the Ricci tensor of the associated Levi Civita connection) and special in the sense that the associated Chern Ricci form is proportional to the symplectic form.     

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.