scholarly journals On a Riemannian manifold admitting a certain vector field

1978 ◽  
Vol 29 (3) ◽  
pp. 250-260 ◽  
Author(s):  
Isuke Satō
Keyword(s):  
Riemannian Manifold ◽  
Vector Field

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.

Some results on harmonic and bi-harmonic maps

2017 ◽  
Vol 14 (07) ◽  
pp. 1750098 ◽  
Author(s):  
Ahmed Mohammed Cherif
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Convex Function ◽  
Open Problem ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Complete Riemannian Manifold ◽  
Strongly Convex Function ◽  
Strongly Convex ◽  
Homothetic Vector

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary [Formula: see text] to Riemannian manifold [Formula: see text] is necessarily constant with [Formula: see text] admitting a strongly convex function [Formula: see text] such that [Formula: see text] is a Jacobi-type vector field (or [Formula: see text] admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

Geometry of a vector field on a Riemannian manifold

1993 ◽  
Vol 54 (5) ◽  
pp. 1182-1183
Author(s):  
M. A. Cheshkova
Keyword(s):  
Riemannian Manifold ◽  
Vector Field

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

Geodesics on the unit tangent bundle

2003 ◽  
Vol 133 (6) ◽  
pp. 1209-1229 ◽  
Author(s):  
J. Berndt ◽  
E. Boeckx ◽  
P. T. Nagy ◽  
L. Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sphere Bundle ◽  
Sasaki Metric ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Unit Tangent Bundle ◽  
Unit Tangent Sphere Bundle

A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

scholarly journals On a class of exact locally conformal cosymlectic manifolds

1996 ◽  
Vol 19 (2) ◽  
pp. 267-278
Author(s):  
I. Mihai ◽  
L. Verstraelen ◽  
R. Rosca
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Conformal Transformation ◽  
Space Form ◽  
Conformal Vector Field ◽  
Cosymplectic Manifold ◽  
Almost Cosymplectic Manifold ◽  
Riemannian Connection ◽  
Locally Conformal ◽  
Hyperbolic Space Form

An almost cosymplectic manifoldMis a(2m+1)-dimensional oriented Riemannian manifold endowed with a 2-formΩof rank2m, a 1-formηsuch thatΩm Λ η≠0and a vector fieldξsatisfyingiξΩ=0andη(ξ)=1. Particular cases were considered in [3] and [6].Let(M,g)be an odd dimensional oriented Riemannian manifold carrying a globally defined vector fieldTsuch that the Riemannian connection is parallel with respect toT. It is shown that in this caseMis a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. MoreoverTdefines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal transformation of the curvature forms.The existence of a structure conformal vector fieldConMis proved and their properties are investigated. In the last section, we study the geometry of the tangent bundle of an exact locally conformal cosymplectic manifold.

scholarly journals A New Approach on Helices in Pseudo-Riemannian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Evren Zıplar ◽  
Yusuf Yaylı ◽  
İsmail Gök
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Frenet Frame ◽  
New Approach ◽  
Nonzero Constant ◽  
Slant Helix

A proper curveαin then-dimensional pseudo-Riemannian manifold(M,g)is called aVn-slant helix if the functiong(Vn,X)is a nonzero constant alongα, whereXis a  parallel vector field alongαandVnisnth Frenet frame. In this work, we study such curves and give important characterizations about them.

scholarly journals Pseudo-Reimannian manifolds endowed with an almost paraf-structure

1985 ◽  
Vol 8 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Vladislav V. Goldberg ◽  
Radu Rosca
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Mean Curvature ◽  
Integral Manifold ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Minimal Hypersurfaces ◽  
Reimannian Manifolds ◽  
The Mean ◽  
Orthogonal Distribution

LetM˜(U,Ω˜,η˜,ξ,g˜)be a pseudo-Riemannian manifold of signature(n+1,n). One defines onM˜an almost cosymplectic paraf-structure and proves that a manifoldM˜endowed with such a structure isξ-Ricci flat and is foliated by minimal hypersurfaces normal toξ, which are of Otsuki's type. Further one considers onM˜a2(n−1)-dimensional involutive distributionP⊥and a recurrent vector fieldV˜. It is proved that the maximal integral manifoldM⊥ofP⊥hasVas the mean curvature vector (up to1/2(n−1)). If the complimentary orthogonal distributionPofP⊥is also involutive, then the whole manifoldM˜is foliate. Different other properties regarding the vector fieldV˜are discussed.

scholarly journals Twistor Bundle of a Neutral Kähler Surface

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 43
Author(s):  
Włodzimierz Jelonek
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Complex Structure ◽  
Killing Vector ◽  
Kähler Manifold ◽  
Killing Vector Field ◽  
Kahler Manifold ◽  
Kähler Surface ◽  
Kähler Surfaces ◽  
Twistor Bundle

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

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