Riemannian-polarized k-symplectic manifolds

2021 ◽  
Vol 18 (12) ◽  
Author(s):  
Ismail Benali ◽  
Souhaila Elamine ◽  
Azzouz Awane
Keyword(s):  
Symplectic Structure ◽  
Complex Structure ◽  
Riemannian Metric ◽  
Symplectic Manifolds ◽  
Complex Case ◽  
Hermitian Structure ◽  
Complex Structures ◽  
Dimensional Manifold ◽  
Almost Complex

In this paper, we give an analogue of the Hermitian structure in the almost complex case, on an [Formula: see text]-dimensional manifold endowed with an almost [Formula: see text]-complex metric. Also, we study the compatibility between Riemannian metric and polarized [Formula: see text]-symplectic structure. Also, we study some properties of an almost [Formula: see text]-complex structure. Moreover, we give an equivalence between almost [Formula: see text]-complex structures, [Formula: see text]-almost tangent structures and [Formula: see text]-almost cotangent structures.

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

2018 ◽  
Vol 26 (2) ◽  
pp. 137-145
Author(s):  
Amir Baghban ◽  
Esmaeil Abedi
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Integrability Condition ◽  
Complex Structure ◽  
Riemannian Metric ◽  
Curvature Operator ◽  
Complex Structures ◽  
New Class ◽  
Almost Complex Structures ◽  
Almost Complex

AbstractIn 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.

scholarly journals BRANCHED SHADOWS AND COMPLEX STRUCTURES ON 4-MANIFOLDS

2008 ◽  
Vol 17 (11) ◽  
pp. 1429-1454 ◽  
Author(s):  
FRANCESCO COSTANTINO
Keyword(s):  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Homotopy Classes ◽  
Almost Complex Structures ◽  
Almost Complex

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spinc-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-dimensional handlebody is homotopic to a complex one.

scholarly journals LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 88-95
Author(s):  
Nikolay Smolentsev ◽  
Nikolay Smolentsev
Keyword(s):  
Lie Groups ◽  
Complex Structure ◽  
New Left ◽  
Riemannian Metrics ◽  
Nilpotent Lie Groups ◽  
Complex Structures ◽  
Structural Group ◽  
Almost Complex ◽  
Flat Structures ◽  
Simply Connected

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

On the Relation of Real and Complex Lie Supergroups

2015 ◽  
Vol 58 (2) ◽  
pp. 281-284 ◽  
Author(s):  
Matthias Kalus
Keyword(s):  
Complex Structure ◽  
Sufficient Conditions ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
New Approach ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Lie Supergroups ◽  
Necessary And Sufficient

AbstractA complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complexHarish-Chandra superpairs. A universal complexi ûcation of a real Lie supergroup is constructed

scholarly journals Extending invariant complex structures

2015 ◽  
Vol 26 (11) ◽  
pp. 1550096 ◽  
Author(s):  
Rutwig Campoamor Stursberg ◽  
Isolda E. Cardoso ◽  
Gabriela P. Ovando
Keyword(s):  
Lie Algebra ◽  
Algebraic Structure ◽  
Symplectic Structure ◽  
Complex Structure ◽  
Complex Structures ◽  
Additional Structure ◽  
Totally Real

We study the problem of extending a complex structure to a given Lie algebra 𝔤, which is firstly defined on an ideal 𝔥 ⊂ 𝔤. We consider the next situations: 𝔥 is either complex or it is totally real. The next question is to equip 𝔤 with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either 𝔥 is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of 𝔤. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

scholarly journals Symplectic and Contact Geometry with Complex Manifolds

2019 ◽  
Vol 39 ◽  
pp. 119-126
Author(s):  
AKM Nazimuddin ◽  
Md Showkat Ali
Keyword(s):  
Riemannian Manifolds ◽  
Symplectic Geometry ◽  
Contact Geometry ◽  
Symplectic Manifolds ◽  
Complex Structures ◽  
Contact Manifolds ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Real Submanifold ◽  
Complex Symplectic

In this paper, we discuss about almost complex structures and complex structures on Riemannian manifolds, symplectic manifolds and contact manifolds. We have also shown a special comparison between complex symplectic geometry and complex contact geometry. Also, the existence of a complex submanifold of n-dimensional complex manifold which intersects a real submanifold GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 119-126

scholarly journals On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

2021 ◽  
pp. 2150075
Author(s):  
Andrea Cattaneo ◽  
Antonella Nannicini ◽  
Adriano Tomassini
Keyword(s):  
Complex Structure ◽  
Kodaira Dimension ◽  
Classification Theory ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Almost Complex Manifolds ◽  
Kähler Structure ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Almost Kähler

The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact [Formula: see text]-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text]. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover, we prove Ricci flatness of the canonical connection for the almost Kähler structure. Regarding the other two manifolds we compute the Kodaira dimension for certain almost complex structures. Finally, we construct a natural hypercomplex structure providing a twistorial description.

Remarks about the Kaluza-Klein metric on tangent bundle

2019 ◽  
Vol 16 (03) ◽  
pp. 1950040
Author(s):  
Murat Altunbas ◽  
Lokman Bilen ◽  
Aydin Gezer
Keyword(s):  
Tangent Bundle ◽  
Complex Structure ◽  
Sufficient Conditions ◽  
Complex Structures ◽  
Almost Complex ◽  
Compatible Almost Complex Structure ◽  
Almost Kähler ◽  
Necessary And Sufficient ◽  
Curvature Tensors ◽  
Kaluza Klein

The paper is concerned with the Kaluza–Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.

scholarly journals Berezin–Toeplitz Quantization and the Least Unsharpness Principle

2020 ◽  
Author(s):  
Louis Ioos ◽  
David Kazhdan ◽  
Leonid Polterovich
Keyword(s):  
Symplectic Manifolds ◽  
Complex Structures ◽  
Almost Complex Structures ◽  
Almost Complex

Abstract We show that compatible almost-complex structures on symplectic manifolds correspond to optimal positive quantizations.

XX.—Differential Geometry on Hypersurfaces in a Cayley Space

1964 ◽  
Vol 66 (4) ◽  
pp. 216-231 ◽  
Author(s):  
K. Yano ◽  
T. Sumitomo
Keyword(s):  
Complex Structure ◽  
Hermitian Structure ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Almost Complex Structure ◽  
Principal Curvatures ◽  
Totally Umbilical ◽  
Almost Complex ◽  
Necessary And Sufficient

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

Sign in / Sign up

Export Citation Format

Share Document