g-Natural Contact Metrics on Unit Tangent Sphere Bundles

Abbassi, Mohamed Tahar Kadaoui; Calvaruso, Giovanni

doi:10.1007/s00605-006-0421-9

Cited by 38 publications

(67 citation statements)

References 18 publications

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“…Remark 1 A Riemannian g-natural metric on T r M was originally defined as a metric induced by a Riemannian g-natural metric on TM [3][4][5][6]. In the article Abbassi and Kowalski [7], we were forced to re-define this concept in a more general form (see Important Remark in [7]).…”

Section: Basic Formulas On Tangent Bundlesmentioning

confidence: 98%

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On Einstein Riemannian g-natural metrics on unit tangent sphere bundles

Abbassi

Kowalski

2010

Ann Glob Anal Geom

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We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1-1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V 2 R m+1 = SO(m + 1)/SO(m − 1) and all g-natural metrics on T 1 S m (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V 2 R m+1 , m = 3, initially constructed by Kobayashi.

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Section: Basic Formulas On Tangent Bundlesmentioning

confidence: 98%

“…Let us start our section by the following result from Abbassi and Kowalski [7] Proposition 3 Every Riemannian SO(m + 1)-invariant metric on the Stiefel manifold In the article by Megan Kerr [15], the author uses parameters x 0 , x 1 , x 2 , x 3 …”

Section: Einstein Riemannian G-natural Metrics On T 1 Mmentioning

confidence: 99%

On Einstein Riemannian g-natural metrics on unit tangent sphere bundles

Abbassi

Kowalski

2010

Ann Glob Anal Geom

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“…Implicitly there is the metric connection ∇ for g E . One close look at the defining equations in [7] will prove the horizontal subspace is defined by (1), (2). Now we must abide to one more little detail, because for Scal M < 0 of course the definitions are not consistent.…”

Section: Metrics With G 2 Holonomymentioning

confidence: 99%

On vector bundle manifolds with spherically symmetric metrics

Albuquerque

2016

Ann Glob Anal Geom

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We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle E −→ M , over a Riemannian manifold M , when E is endowed with a metric connection. The tangent bundle of E admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of E; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type G 2 manifolds.

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“…The classification of all natural metrics on T M induced from g may be found in [1,2,3]. An analysis of the convexity properties has shown that the metrics correspond with six weight functions f 1 , .…”

Section: Natural Metrics On T Mmentioning

confidence: 99%

“…We review the known classification of g-natural metrics on T M by [1,2,3] and continue our study assuming metrics of the kind f 1 π * g ⊕f 2 π * g +f 3 µ⊗µ…”

Section: Introductionmentioning

confidence: 99%

Weighted Metrics on Tangent Sphere Bundles

Albuquerque¹

2011

The Quarterly Journal of Mathematics

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Natural metric structures on the tangent bundle and tangent sphere bundles S r M of a Riemannian manifold M with radius function r enclose many important unsolved problems. Admitting metric connections on M with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of T M to the almost Hermitian category. Our purpose is the study of the natural contact structure on S r M and the G 2 -twistor space of any oriented Riemannian 4-manifold.

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g-Natural Contact Metrics on Unit Tangent Sphere Bundles

Cited by 38 publications

References 18 publications

On Einstein Riemannian g-natural metrics on unit tangent sphere bundles

On Einstein Riemannian g-natural metrics on unit tangent sphere bundles

On vector bundle manifolds with spherically symmetric metrics

Weighted Metrics on Tangent Sphere Bundles

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