The almost complex structure on 𝕊6 and related Schrödinger flows

2018 â—˝  
Vol 29 (14) â—˝  
pp. 1850099 â—˝  
Author(s):  
Qing Ding â—˝  
Shiping Zhong
Keyword(s):  
Complex Structure â—˝  
Type System â—˝  
Text Structure â—˝  
Almost Complex Structure â—˝  
Nls Equation â—˝  
Geometric Properties â—˝  
Nonlinear Schrödinger â—˝  
Almost Complex â—˝  
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 â—˝  
Vol 17 (11) â—˝  
pp. 1429-1454 â—˝  
Author(s):  
FRANCESCO COSTANTINO
Keyword(s):  
Complex Structure â—˝  
Complex Structures â—˝  
Almost Complex Structure â—˝  
Homotopy Classes â—˝  
Almost Complex Structures â—˝  
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.

Integrability Condition on an Almost Complex Structure and Its Application

2005 â—˝  
Vol 21 (6) â—˝  
pp. 1459-1464 â—˝  
Author(s):  
Chia Kuei Peng â—˝  
Zi Zhou Tang
Keyword(s):  
Integrability Condition â—˝  
Complex Structure â—˝  
Almost Complex Structure â—˝  
Almost Complex

scholarly journals Coclosed G2-structures inducing nilsolitons

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

scholarly journals Holomorphic 2-Forms and Vanishing Theorems for Gromov–Witten Invariants

2009 â—˝  
Vol 52 (1) â—˝  
pp. 87-94 â—˝  
Author(s):  
Junho Lee
Keyword(s):  
Complex Structure â—˝  
Kähler Manifold â—˝  
Higher Dimensions â—˝  
Almost Complex Structure â—˝  
Vanishing Theorems â—˝  
Kahler Manifold â—˝  
Almost Complex â—˝  
Kähler Surfaces â—˝  
Compact Kähler Manifold

AbstractOn a compact Kähler manifold X with a holomorphic 2-form α, there is an almost complex structure associated with α. We show how this implies vanishing theorems for the Gromov–Witten invariants of X. This extends the approach used by Parker and the author for Kähler surfaces to higher dimensions.

Horizontal Lifts of some Geometric Objects to the Bundle of Pairs of Volume Forms

2013 â—˝  
Vol 59 (2) â—˝  
pp. 357-372
Author(s):  
Anna Gąsior
Keyword(s):  
Vector Fields â—˝  
Complex Structure â—˝  
Riemannian Metrics â—˝  
Almost Complex Structure â—˝  
Horizontal Lift â—˝  
Almost Complex â—˝  
Geometric Objects â—˝  
Volume Forms â—˝  
Infinitesimal Transformations

Abstract In this paper we present a bundle of pairs of volume forms V2. We describe horizontal lift of a tensor of type (1; 1) and we show that horizontal lift of an almost complex structure on a manifold M is an almost complex structure on the bundle V2. Next we give conditions under which the almost complex structure on V 2 is integrable. In the second part we find horizontal lift of vector fields, tensorfields of type (0; 2) and (2; 0), Riemannian metrics and we determine a family of a t-connections on the bundle of pairs of volume forms. At the end, we consider some properties of the horizontally lifted vector fields and certain infinitesimal transformations.

Normal Variations of Invariant Hypersurfaces of Framed Manifolds

1972 â—˝  
Vol 15 (4) â—˝  
pp. 513-521
Author(s):  
Samuel I. Goldberg
Keyword(s):  
Complex Manifold â—˝  
Contact Structure â—˝  
Complex Structure â—˝  
Almost Complex Structure â—˝  
Marked Contrast â—˝  
Almost Complex Manifold â—˝  
Almost Contact Structure â—˝  
Almost Complex â—˝  
Differentiable Vector â—˝  
Real Codimension

A hypersurface of a globally framed f-manifold (briefly, a framed manifold), does not in general possess a framed structure as one may see by considering the 4-sphere S4 in R5 or S5. For, a hypersurface so endowed carries an almost complex structure, or else, it admits a nonsingular differentiable vector field. Since an almost complex manifold may be considered as being globally framed, with no complementary frames, this situation is in marked contrast with the well known fact that a hypersurface (real codimension 1) of an almost complex manifold admits a framed structure, more specifically, an almost contact structure.

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