scholarly journals Extension of CR structures on three dimensional pseudoconvex CR manifolds

1998 ◽  
Vol 152 ◽  
pp. 97-129 ◽  
Author(s):  
Sanghyun Cho
Keyword(s):  
Finite Type ◽  
Complex Structure ◽  
Three Dimensional ◽  
Concave Side ◽  
Almost Complex Structure ◽  
Cr Manifolds ◽  
Almost Complex ◽  
Cr Structures ◽  
The Given

Abstract.Let be a smoothly bounded orientable pseudoconvex CR manifold of finite type and dimℝM = 3. Then we extend the given CR structure on M to an integrable almost complex structure on which is the concave side of M and M ⊂

scholarly journals THE STRUCTURE OF THE SPACE OF AFFINE KÄHLER CURVATURE TENSORS AS A COMPLEX MODULE

2011 ◽  
Vol 08 (08) ◽  
pp. 1849-1868 ◽  
Author(s):  
M. BROZOS-VÁZQUEZ ◽  
P. GILKEY ◽  
S. NIKČEVIĆ
Keyword(s):  
Complex Structure ◽  
Inner Product ◽  
Almost Complex Structure ◽  
The Real ◽  
Direct Sum ◽  
Irreducible Modules ◽  
Almost Complex ◽  
Affine Curvature ◽  
Curvature Tensors ◽  
The Given

In dimension m ≥ 4, results of Strichartz decompose the space 𝔄 of affine curvature tensors as a direct sum of 3 modules in the real setting and results of Bokan give a corresponding finer decomposition of 𝔄 in the Riemannian setting as the direct sum of 8 irreducible modules. In dimension m ≥ 8, results of Matzeu and Nikčević decompose the space 𝔎 of affine Kähler curvature tensors as the direct sum of 12 irreducible modules in the Hermitian setting (i.e. given an auxiliary inner product which is invariant under the given almost complex structure). In this paper, we decompose 𝔎 as a direct sum of six irreducible modules in the complex setting in dimension m ≥ 8. Corresponding decompositions into fewer modules are given in dimension m = 4 and m = 6.

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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