scholarly journals The projective curvature of the tangent bundle with natural diagonal metric

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 401-410 ◽  
Author(s):  
Cornelia-Livia Bejan ◽  
Simona-Luiza Druţă-Romaniuc
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Total Space ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Necessary And Sufficient

Our study is mainly devoted to a natural diagonal metric G on the total space TMof the tangent bundle of a Riemannian manifold (M, 1). We provide the necessary and sufficient conditions under which (TM,G) is a space form, or equivalently (TM,G) is projectively Euclidean. Moreover, we classify the natural diagonal metrics G for which (TM,G) is horizontally projectively flat (resp. vertically projectively flat).

scholarly journals Some Curvature Properties on Lorentzian Generalized Sasakian-Space-Forms

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Rongsheng Ma ◽  
Donghe Pei
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Conformally Flat ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Projectively Flat ◽  
Generalized Sasakian Space Form ◽  
Necessary And Sufficient

In this paper, we investigate the Lorentzian generalized Sasakian-space-form. We give the necessary and sufficient conditions for the Lorentzian generalized Sasakian-space-form to be projectively flat, conformally flat, conharmonically flat, and Ricci semisymmetric and their relationship between each other. As the application of our theorems, we study the Ricci almost soliton on conformally flat Lorentzian generalized Sasakian-space-form.

scholarly journals Magnetic vector fields: New examples

2018 ◽  
Vol 103 (117) ◽  
pp. 91-102
Author(s):  
Jun-Ichi Inoguchi ◽  
Marian Munteanu
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Magnetic Vector ◽  
Necessary And Sufficient

In a previous paper, we introduced the notion of magnetic vector fields. More precisely, we consider a vector field ? as a map from a Riemannian manifold into its tangent bundle endowed with the usual almost K?hlerian structure and we find necessary and sufficient conditions for ? to be a magnetic map with respect to ? itself and the K?hler 2-form. In this paper we give new examples of magnetic vector fields.

scholarly journals The scalar curvature of the tangent bundle of a Finsler manifold

2011 ◽  
Vol 89 (103) ◽  
pp. 57-68
Author(s):  
Aurel Bejancu ◽  
Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Degree Zero ◽  
Necessary And Sufficient

Let Fm = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on the slit tangent bundle TM0 = TM \{0} of M. We express the scalar curvature ?~ of the Riemannian manifold (TM0,G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ?~ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of TM0. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose ?~ satisfies the above condition.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

scholarly journals The geometry of modified Riemannian extensions

2009 ◽  
Vol 465 (2107) ◽  
pp. 2023-2040 ◽  
Author(s):  
E. Calviño-Louzao ◽  
E. García-Río ◽  
P. Gilkey ◽  
R. Vázquez-Lorenzo
Keyword(s):  
Normal Form ◽  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jordan Normal Form ◽  
Jacobi Operators ◽  
Necessary And Sufficient

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and gives necessary and sufficient conditions for a modified Riemannian extension to be Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3, 3), whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four-dimensional results in Osserman geometry.

scholarly journals On interpolating sesqui-harmonic Legendre curves in Sasakian space forms

2019 ◽  
Vol 17 (01) ◽  
pp. 2050005 ◽  
Author(s):  
Fatma Karaca ◽  
Cihan Özgür ◽  
Uday Chand De
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Legendre Curve ◽  
Sasakian Space Forms ◽  
Legendre Curves ◽  
Necessary And Sufficient

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain a proper example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

Scalar curvature of Lagrangian Riemannian submersions and their harmonicity

2017 ◽  
Vol 14 (12) ◽  
pp. 1750171 ◽  
Author(s):  
Şemsi Eken Meri̇ç ◽  
Erol Kiliç ◽  
Yasemi̇n Sağiroğlu
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Hermitian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Necessary And Sufficient

In this paper, we consider a Lagrangian Riemannian submersion from a Hermitian manifold to a Riemannian manifold and establish some basic inequalities to obtain relationships between the intrinsic and extrinsic invariants for such a submersion. Indeed, using these inequalities, we provide necessary and sufficient conditions for which a Lagrangian Riemannian submersion [Formula: see text] has totally geodesic or totally umbilical fibers. Moreover, we study the harmonicity of Lagrangian Riemannian submersions and obtain a characterization for such submersions to be harmonic.

scholarly journals Vector bundles over (8k + 5)-dimensional manifolds

1989 ◽  
Vol 111 (1-2) ◽  
pp. 183-197 ◽  
Author(s):  
Tze-Beng Ng
Keyword(s):  
Tangent Bundle ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Suppose that M is a closed, connected and smooth manifold of dimension n = 8k + 5, with k ≧1. Let η be an n-plane bundle over M. Under suitable conditions on M, we derive necessary and sufficient conditions for the span of η to be ≧j, j = 5 or 6. We then apply the results to the tangent bundle of M. In particular, we prove a conjecture of E. Thomas, namely, if M is 3-connected mod 2, then span M ≧ 5 if, and only if, χ2(M) = 0. We prove that if also w8k(M) = 0, then span M≧6. We also derive some immersion theorems for M.

Cockcroft properties of graphs of 2-complexes

1994 ◽  
Vol 124 (2) ◽  
pp. 363-369 ◽  
Author(s):  
N.D. Gilbert ◽  
James Howie
Keyword(s):  
Fundamental Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Total Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for the 2-skeleton of the total space of a graph of 2-complexes to be Cockcroft, or L-Cockcroft for some subgroup L of the fundamental group. These conditions are used to construct new examples of Cockcroft and absolutely Cockcroft 2-complexes.

scholarly journals On the Class of (q,α,β)-Metrics

ISRN Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ahmad Alimohammadi
Keyword(s):  
Sufficient Conditions ◽  
Riemannian Metric ◽  
Necessary And Sufficient Conditions ◽  
Finsler Metrics ◽  
Projectively Flat ◽  
Necessary And Sufficient

We study a class of Finsler metrics in the form F=α+βq/αq-1, where α is a Riemannian metric, β is a 1-form, and 1<q<2.  F is called (q,α,β)-metrics. We find the necessary and sufficient conditions under which the class of (q,α,β)-metrics is locally projectively flat and Douglas metrics, respectively.

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