scholarly journals A vertical Liouville subfoliation on the cotangent bundle of a Cartan space and some related structures

2014 ◽  
Vol 11 (06) ◽  
pp. 1450063 ◽  
Author(s):  
Cristian Ida ◽  
Adelina Manea
Keyword(s):  
Vector Field ◽  
Cotangent Bundle ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Linear Connections ◽  
Liouville Distribution ◽  
Almost Complex ◽  
Liouville Foliation

In this paper, we study some problems related to a vertical Liouville distribution (called vertical Liouville–Hamilton distribution) on the cotangent bundle of a Cartan space. We study the existence of some linear connections of Vrănceanu type on Cartan spaces related to some foliated structures. Also, we identify a certain (n, 2n-1)-codimensional subfoliation [Formula: see text] on T*M0given by vertical foliation [Formula: see text] and the line foliation [Formula: see text] spanned by the vertical Liouville–Hamilton vector field C* and we give a triplet of basic connections adapted to this subfoliation. Finally, using the vertical Liouville foliation [Formula: see text] and the natural almost complex structure on T*M0we study some aspects concerning the cohomology of c-indicatrix cotangent bundle.

XXII.—Tensor Fields and Connections on Cross-Sections in the tangent bundle of a differentiable manifold

1967 ◽  
Vol 67 (4) ◽  
pp. 277-288
Author(s):  
Kentaro Yano
Keyword(s):  
Vector Field ◽  
Cross Section ◽  
Tangent Bundle ◽  
Complex Structure ◽  
Linear Connection ◽  
Differentiable Manifold ◽  
Tensor Fields ◽  
Linear Connections ◽  
Almost Complex ◽  
The Cross

SynopsisTensor fields and linear connections in an n-dimensional differentiable manifold M can be extended, in a natural way, to the tangent bundle T(M) of M to give tensor fields of the same type and linear connections in T(M) respectively. We call such extensions complete lifts to T(M) of tensor fields and linear connections in M.On the other hand, when a vector field V is given in M, V determines a cross-section which is an n-dimensional submanifold in the 2n-dimensional tangent bundle T(M).We study first the behaviour of complete lifts of tensor fields on such a cross-section. The complete lift of an almost complex structure being again an almost complex structure, we study especially properties of the cross-section as a submanifold in an almost complex manifold.We also study properties of cross-sections with respect to the linear connection which is the complete lift of a linear connection in M and with respect to the linear connection induced by the latter on the cross-section. To quote a typical result: A necessary and sufficient condition for a cross-section to be totally geodesic is that the vector field V in M defining the cross-section in T(M) be an affine Killing vector field in M.

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.

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

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

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