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A new class of almost complex structures on tangent bundle of a Riemannian manifold

Amir Baghban, Esmaeil Abedi (2018)

Communications in Mathematics

In this paper, the standard almost complex structure on the tangent bunle of a Riemannian manifold will be generalized. We will generalize the standard one to the new ones such that the induced ( 0 , 2 ) -tensor on the tangent bundle using these structures and Liouville 1 -form will be a Riemannian metric. Moreover, under the integrability condition, the curvature operator of the base manifold will be classified.

Complex structures on S O g M

Tommaso Pacini (1999)

Bollettino dell'Unione Matematica Italiana

Data una varietà Riemanniana orientata M , g , il fibrato principale S O g M di basi ortonormali positive su M , g ha una parallelizzazione canonica dipendente dalla connessione di Levi-Civita. Questo fatto suggerisce la definizione di una classe molto naturale di strutture quasi-complesse su M , g . Dopo le necessarie definizioni, discutiamo qui l'integrabilità di queste strutture, esprimendola in termini della struttura Riemanniana g .

Filtration de Harder-Narasimhan pour des fibrés complexes ou des faisceaux sans torsion

Laurent Bruasse (2003)

Annales de l’institut Fourier

On généralise dans cet article la notion de filtration de Harder-Narasimhan au cas des fibrés complexes sur une variété presque complexe compacte d'une part, et au cas des faisceaux cohérents sans torsion sur une variété holomorphe d'autre part. On démontre, dans les deux cas, l'existence d'un déstabilisant maximal. On obtient un théorème de convergence en famille et par là-même l'ouverture de la stabilité en déformation.

Geometry of universal embedding spaces for almost complex manifolds

Gabriella Clemente (2024)

Archivum Mathematicum

We investigate the geometry of universal embedding spaces for compact almost-complex manifolds of a given dimension, and related constructions that allow for an extrinsic study of the integrability of almost-complex structures. These embedding spaces were introduced by J-P. Demailly and H. Gaussier, and are complex algebraic analogues of twistor spaces. Their goal was to study a conjecture made by F. Bogomolov asserting the “transverse embeddability” of arbitrary compact complex manifolds into foliated...

Invariants of complex structures on nilmanifolds

Edwin Alejandro Rodríguez Valencia (2015)

Archivum Mathematicum

Let ( N , J ) be a simply connected 2 n -dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on N compatible with J to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving...

J -holomorphic discs and real analytic hypersurfaces

William Alexandre, Emmanuel Mazzilli (2014)

Annales de l’institut Fourier

We give in 6 a real analytic almost complex structure J , a real analytic hypersurface M and a vector v in the Levi null set at 0 of M , such that there is no germ of J -holomorphic disc γ included in M with γ ( 0 ) = 0 and γ x ( 0 ) = v , although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J , we give sufficient conditions under which there exists such a germ of disc.

Non-split almost complex and non-split Riemannian supermanifolds

Matthias Kalus (2019)

Archivum Mathematicum

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. For almost complex structures, the existence of a splitting is equivalent to the existence of local coordinates in which the almost complex structure can be represented by a purely numerical matrix, i.e. containing no Grassmann variables. For Riemannian metrics, terms up to degree 2 are allowed in...

Normal pseudoholomorphic curves

Fathi Haggui, Adel Khalfallah (2011)

Annales Polonici Mathematici

First, we give some characterizations of J-hyperbolic points for almost complex manifolds. We apply these characterizations to show that the hyperbolic embeddedness of an almost complex submanifold follows from relative compactness of certain spaces of continuous extensions of pseudoholomorphic curves defined on the punctured unit disc. Next, we define uniformly normal families of pseudoholomorphic curves. We prove extension-convergence theorems for these families similar to those obtained by Kobayashi,...

On the automorphism group of strongly pseudoconvex domains in almost complex manifolds

Jisoo Byun, Hervé Gaussier, Kang-Hyurk Lee (2009)

Annales de l’institut Fourier

In contrast with the integrable case there exist infinitely many non-integrable homogeneous almost complex manifolds which are strongly pseudoconvex at each boundary point. All such manifolds are equivalent to the Siegel half space endowed with some linear almost complex structure.We prove that there is no relatively compact strongly pseudoconvex representation of these manifolds. Finally we study the upper semi-continuity of the automorphism group of some hyperbolic strongly pseudoconvex almost...

On the Calabi-Yau equation in the Kodaira-Thurston manifold

Luigi Vezzoni (2016)

Complex Manifolds

We review some previous results about the Calabi-Yau equation on the Kodaira-Thurston manifold equipped with an invariant almost-Kähler structure and assuming the volume form T2-invariant. In particular, we observe that under some restrictions the problem is reduced to aMonge-Ampère equation by using the ansatz ˜~ω = Ω− dJdu + da, where u is a T2-invariant function and a is a 1-form depending on u. Furthermore, we extend our analysis to non-invariant almost-complex structures by considering some...

Partial integrability on Thurston manifolds

Hyeseon Kim (2013)

Annales Polonici Mathematici

We determine the maximal number of independent holomorphic functions on the Thurston manifolds M 2 r + 2 , r ≥ 1, which are the first discovered compact non-Kähler almost Kähler manifolds. We follow the method which involves analyzing the torsion tensor dθ modθ, where θ = ( θ ¹ , . . . , θ r + 1 ) are independent (1,0)-forms.

Schwarz-type lemmas for solutions of ¯ -inequalities and complete hyperbolicity of almost complex manifolds

Sergey Ivashkovich, Jean-Pierre Rosay (2004)

Annales de l'Institut Fourier

The definition of the Kobayashi-Royden pseudo-metric for almost complex manifolds is similar to its definition for complex manifolds. We study the question of completeness of some domains for this metric. In particular, we study the completeness of the complement of submanifolds of co-dimension 1 or 2. The paper includes a discussion, with proofs, of basic facts in the theory of pseudo-holomorphic discs.

Some characterizations of hyperbolic almost complex manifolds

Fathi Haggui, Adel Khalfallah (2010)

Annales Polonici Mathematici

First, we give some characterizations of the Kobayashi hyperbolicity of almost complex manifolds. Next, we show that a compact almost complex manifold is hyperbolic if and only if it has the Δ*-extension property. Finally, we investigate extension-convergence theorems for pseudoholomorphic maps with values in pseudoconvex domains.

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