On Riemannian g-natural Metrics of the Form a.gs + b.gh + c.gv on the Tangent Bundle of a Riemannian Manifold (M, g)

Abbassi, Mohamed Tahar Kadaoui; Sarih, Maâti

doi:10.1007/s00009-005-0028-8

Cited by 26 publications

(39 citation statements)

References 11 publications

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“…The h p,q all belong to the infinite-dimensional family of g-natural metrics on T M [1,2], but are much more tightly controlled, being constructed from a spherically symmetric family of metrics on R n via the Kaluza-Klein procedure. Then σ is said to be (p, q)-harmonic if σ is a harmonic section of T M with respect to h p,q (the metric g on M is fixed throughout).…”

Section: Introductionmentioning

confidence: 99%

Harmonic vector fields on space forms

Benyounes¹,

Loubeau²,

Wood

2014

Geom Dedicata

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Abstract. A vector field σ on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised CheegerGromoll metrics on T M with respect to which σ is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vector fields that are harmonic. In particular, the harmonic Killing fields and conformal gradient fields are classified, a loop of non-congruent harmonic conformal fields on the hyperbolic plane constructed, and the 2-dimensional classification achieved for conformal fields. A classification is then given of all harmonic quadratic gradient fields on spheres.

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Section: Introductionmentioning

confidence: 99%

Harmonic vector fields on space forms

Benyounes¹,

Loubeau²,

Wood

2014

Geom Dedicata

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“…We begin by computing the spectrum of the operator h = which can be derived from Gauss's formula for T r M and the expression for the LeviCivita connection of (T M, G) (see [2]). This formula yields Hence, using (6.20), we obtain τ = 4n(n + 1) K r 2 λ 2 + 4n(n − 1) − 2n…”

Section: Classification Resultsmentioning

confidence: 99%

“…https://doi.org/10.1017/S0004972709000252 (see [1,2] for the general theory of g-natural metrics on tangent bundles). Here g v denotes the vertical lift of g determined by…”

Section: Examples: Cr Geometry Of Tangent Sphere Bundlesmentioning

confidence: 99%

“…Using the Gauss equation, this formula can be derived from the fact thatR(X V , Y V )Z V = 0, which holds for the curvature of any g-natural metric on T M of type G = ag S + bg h + cg v (see [2]). Now the scalar curvature τ of (T r M, g η ) is related to the scalar curvatures τ M ,τ of (M, (λ 2 /4r 2 )g) and of the fibres of π by τ = τ M +τ − A 2 (6.20) where A is the O'Neill fundamental horizontal tensor field of the submersion π (see, for example, [16]).…”

Section: Examples: Cr Geometry Of Tangent Sphere Bundlesmentioning

confidence: 99%

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A Classification of Spherical Symmetric Cr Manifolds

Dileo¹,

Lotta²

2009

Bull. Aust. Math. Soc.

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In this paper we get different characterizations of the spherical strictly pseudoconvex CR manifolds admitting a CR-symmetric Webster metric by means of the Tanaka-Webster connection and of the Riemannian curvature tensor. As a consequence we obtain the classification of the simply connected, spherical symmetric pseudo-Hermitian manifolds.2000 Mathematics subject classification: primary 53C15, 53C25, 53C35; secondary 32V05.

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“…In particular, the scalar curvature of T (M ) can be constant also for a non-flat base manifold with constant sectional curvature. Then M.T.K.Abbassi & M.Sarih proved in [4] that the considered metrics by Oproiu form a particular subclass of the so-called g-natural metrics on the tangent bundle (see also [1,2,4,5,6,14]). …”

Section: Introductionmentioning

confidence: 99%

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

Munteanu

2008

MedJM

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In this paper we study a Riemanian metric on the tangent bundle T (M ) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M ) a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on T (M ). We found conditions under which T (M ) is almost Kählerian, locally conformal Kählerian or Kählerian or when T (M ) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T (M ). Moreover, we found that this map preserves also the natural almost contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively.Mathematics Subject Classifications (2000): 53B35, 53C07, 53C25, 53C55.

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On Riemannian g-natural Metrics of the Form a.gs + b.gh + c.gv on the Tangent Bundle of a Riemannian Manifold (M, g)

Cited by 26 publications

References 11 publications

Harmonic vector fields on space forms

Harmonic vector fields on space forms

A Classification of Spherical Symmetric Cr Manifolds

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

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