On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

Abbassi, Mohamed Tahar Kadaoui; Sarih, Maâti

doi:10.1016/j.difgeo.2004.07.003

Cited by 108 publications

(141 citation statements)

References 18 publications

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“…The introduction of g-natural metrics moves from the classification of natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles [KSe], or equivalently, the description of all first order natural operators D : S 2 + T * (S 2 T * )T , transforming Riemannian metrics on manifolds into metrics on their tangent bundles [KoMSl] (see also [A]). Riemannian g-natural metrics have been completely described in [AS2]. They depend on six smooth functions from IR + to IR, special choices of which give all the well known examples of Riemannian metrics on T M as g s itself, the Cheeger-Gromoll metric g GC and the metrics investigated in [O] (cf.…”

Section: Introductionmentioning

confidence: 99%

HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

Abbassi

Calvaruso

Perrone

2010

The Quarterly Journal of Mathematics

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Let (M, g) be a Riemannian manifold. When M is compact and the tangent bundle T M is equipped with the Sasaki metric g s , the only vector fields which define harmonic maps from (M, g) to (T M, g s ), are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on T M , are particular examples of g-natural metrics. We equip T M with an arbitrary Riemannian g-natural metric G, and investigate the harmonicity of a vector field V of M , thought as a map from (M, g) to (T M, G). We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres.

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Section: Introductionmentioning

confidence: 99%

HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

Abbassi

Calvaruso

Perrone

2010

The Quarterly Journal of Mathematics

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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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“…These metrics depend on several smooth functions from ‫ޒ‬ C D OE0; C1/ to ‫ޒ‬ and as their name suggests, they arise from a very "natural" construction starting from a Riemannian metric g over M (see [Abbassi and Sarih 2005;Abbassi et al 2010a] and the references in [Abbassi 2008]). Given an arbitrary g-natural metric G on the tangent bundle TM of a Riemannian manifold .M; g/, there are six smooth functions˛i,ˇi W ‫ޒ‬ C !…”

Section: Natural Riemannian Metrics On T 1 Mmentioning

confidence: 99%

Instability of the geodesic flow for the energy functional

Perrone¹

2011

Pacific J. Math.

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Let .S n .r/; g 0 / be the canonical sphere of radius r. Denote by z G s the Sasaki metric on the unit tangent bundle T 1 S n .r/ induced from g 0 and by z G z s the Sasaki metric on T 1 T 1 S n .r/ induced from z G s . We resolve here, for n 7, a question raised by Boeckx, González-Dávila, and Vanhecke: namely, we prove that the geodesic flowis an unstable harmonic vector field for any r > 0 and n 7. In particular, in the case r D 1, is an unstable harmonic map. We show that these results are invariant under a four-parameter deformation of the Sasaki metric z G z s .

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On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

Cited by 108 publications

References 18 publications

HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

Harmonic vector fields on space forms

Instability of the geodesic flow for the energy functional

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