scholarly journals ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE

2007 ◽  
Vol 49 (2) ◽  
pp. 357-366 ◽  
Author(s):  
MARCOS SALVAI
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Isometry Group ◽  
Complex Structure ◽  
Riemannian Metrics ◽  
Constant Sectional Curvature ◽  
Almost Complex Structure ◽  
Time And Space ◽  
Identity Component ◽  
Almost Complex

AbstractLet H be the n-dimensional hyperbolic space of constant sectional curvature –1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space $\mathcal{G}_{n}$ of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of $ \mathcal{G}_{n}$ and H. Moreover, we show that $\mathcal{G}_{3}$ is Kähler and find an orthogonal almost complex structure on $\mathcal{G} _{7}$.

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

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

scholarly journals Compact Hermitian surfaces of pointwise constant holomorphic sectional curvature

1995 ◽  
Vol 37 (3) ◽  
pp. 343-349 ◽  
Author(s):  
Kouei Sekigawa ◽  
Takashi Koda
Keyword(s):  
Sectional Curvature ◽  
Differentiable Function ◽  
Tangent Bundle ◽  
Complex Structure ◽  
Hermitian Manifold ◽  
Holomorphic Sectional Curvature ◽  
Almost Complex Structure ◽  
Almost Hermitian Manifold ◽  
Almost Complex ◽  
Unit Tangent Bundle

Let M = (M, J, g) be an almost Hermitian manifold and U(M)the unit tangent bundle of M. Then the holomorphic sectional curvature H = H(x) can be regarded as a differentiable function on U(M). If the function H is constant along each fibre, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole U(M), then M is called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).

scholarly journals Positively curved 6-manifolds with simple symmetry groups

2002 ◽  
Vol 74 (4) ◽  
pp. 589-597 ◽  
Author(s):  
FUQUAN FANG
Keyword(s):  
Sectional Curvature ◽  
Isometry Group ◽  
Symmetry Groups ◽  
Positive Sectional Curvature ◽  
Identity Component ◽  
Simply Connected ◽  
Lie Subgroup ◽  
Positively Curved

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

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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