scholarly journals The Kähler-Einstein metrics on a $K3$ surface cannot be almost Kähler with respect to an opposite almost complex structure

1995 ◽  
Vol 18 (3) ◽  
pp. 506-514
Author(s):  
A. Bonome ◽  
R. Castro ◽  
E. García-Río ◽  
L. Hervella ◽  
Y. Matsushita
Keyword(s):  
Einstein Metrics ◽  
Complex Structure ◽  
K3 Surface ◽  
Almost Complex Structure ◽  
Almost Complex ◽  
Almost Kähler ◽  
Kähler Einstein Metrics

A NOTE ON THE GOLDBERG CONJECTURE OF WALKER MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 925-928 ◽  
Author(s):  
A. A. SALIMOV
Keyword(s):  
Compact Manifold ◽  
Complex Structure ◽  
Additional Restriction ◽  
Almost Complex Structure ◽  
Almost Complex ◽  
Walker Metric ◽  
Almost Kähler ◽  
Goldberg Conjecture ◽  
Walker Manifold

This paper is concerned with Goldberg conjecture. Using the ϕφ-operator we prove the following result. Let (M, φ, w g) be an almost Kähler–Walker–Einstein compact manifold with the proper almost complex structure φ. The proper almost complex structure φ on Walker manifold (M, w g) is integrable if ϕφgN+ = 0, where gN+ is the induced Norden–Walker metric on M. This resolves a conjecture of Goldberg under the additional restriction on Norden–Walker metric (gN+ ∈ Ker ϕφ).

scholarly journals On second non-HLC degree of closed symplecitc manifold

2021 ◽  
pp. 2150079
Author(s):  
Teng Huang
Keyword(s):  
Complex Structure ◽  
The Self ◽  
Almost Complex Structure ◽  
Harmonic Forms ◽  
Almost Complex ◽  
Almost Kähler Manifold ◽  
De Rham ◽  
Number Formula ◽  
Almost Kähler ◽  
Structure Formula

In this note, we show that for a closed almost-Kähler manifold [Formula: see text] with the almost complex structure [Formula: see text] satisfies [Formula: see text] the space of de Rham harmonic forms is contained in the space of symplectic-Bott–Chern harmonic forms. In particular, suppose that [Formula: see text] is four-dimensional, if the self-dual Betti number [Formula: see text], then we prove that the second non-HLC degree measures the gap between the de Rham and the symplectic-Bott–Chern harmonic forms.

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.

The almost complex structure on 𝕊6 and related Schrödinger flows

2018 ◽  
Vol 29 (14) ◽  
pp. 1850099 ◽  
Author(s):  
Qing Ding ◽  
Shiping Zhong
Keyword(s):  
Complex Structure ◽  
Type System ◽  
Text Structure ◽  
Almost Complex Structure ◽  
Nls Equation ◽  
Geometric Properties ◽  
Nonlinear Schrödinger ◽  
Almost Complex ◽  
Complex Functions

In this paper, by using the [Formula: see text]-structure on Im[Formula: see text] from the octonions [Formula: see text], the [Formula: see text]-binormal motion of curves [Formula: see text] in [Formula: see text] associated to the almost complex structure on [Formula: see text] is studied. The motion is proved to be equivalent to Schrödinger flows from [Formula: see text] to [Formula: see text], and also to a nonlinear Schrödinger-type system (NLSS) in three unknown complex functions that generalizes the famous correspondence between the binormal motion of curves in [Formula: see text] and the focusing nonlinear Schrödinger (NLS) equation. Some related geometric properties of the surface [Formula: see text] in Im[Formula: see text] swept by [Formula: see text] are determined.

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 Coclosed G2-structures inducing nilsolitons

2018 ◽  
Vol 30 (1) ◽  
pp. 109-128 ◽  
Author(s):  
Leonardo Bagaglini ◽  
Marisa Fernández ◽  
Anna Fino
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Nilpotent Lie Algebras ◽  
Almost Complex ◽  
Metric Structures ◽  
Trivial Center ◽  
Nilsoliton Metric

Abstract We show obstructions to the existence of a coclosed {\mathrm{G}_{2}} -structure on a Lie algebra {\mathfrak{g}} of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from {\mathfrak{g}} to a six-dimensional Lie algebra {\mathfrak{h}} , with the kernel contained in the center of {\mathfrak{g}} , then any coclosed {\mathrm{G}_{2}} -structure on {\mathfrak{g}} induces a closed and stable three form on {\mathfrak{h}} that defines an almost complex structure on {\mathfrak{h}} . As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed {\mathrm{G}_{2}} -structures. We also prove that each one of these Lie algebras has a coclosed {\mathrm{G}_{2}} -structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed {\mathrm{G}_{2}} -structures. The existence of contact metric structures is also studied.

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