The Geometry of Generalised Cheeger-Gromoll Metrics

Benyounes, M.; Loubeau, E.; Wood, Chris M.

doi:10.3836/tjm/1264170234

Cited by 22 publications

(37 citation statements)

References 19 publications

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“…The interested reader may see properties of G in general in [1,3,11,12,16,18,21,22] and other references therein. One of the peculiar natural metrics is the Cheeger-Gromoll metric:…”

Section: Natural Metrics On T Mmentioning

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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“…The interested reader may see properties of G in general in [1,3,11,12,16,18,21,22] and other references therein. One of the peculiar natural metrics is the Cheeger-Gromoll metric:…”

Section: Natural Metrics On T Mmentioning

confidence: 99%

Weighted Metrics on Tangent Sphere Bundles

Albuquerque¹

2011

The Quarterly Journal of Mathematics

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“…The Riemannian metric h p,q,α is a generalization of the metric considered in [2,3] and is a special case of a metric considered in [10]. In particular, h 0,0,α (or h p,0,0 ) is the Sasaki metric [7,12], h 1,1,1 is the Cheeger-Gromoll metric [6,11,13].…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…It follows that in the case of horizontal conformality of a differential, and in consequence for a differential to be a harmonic morphism, it is sufficient to consider only h p,q metrics introduced in [3].…”

Section: Theoremmentioning

confidence: 98%

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Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

Kozłowski

Niedziałomski

2009

Ann Glob Anal Geom

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“…These metrics, known as generalized Cheeger-Gromoll metrics, has been introduced and studied in [6,5]. The original Cheeger-Gromoll metric [9,15] corresponds to p = q = 1 and h 0,0 is the Sasaki metric [17].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The geometry of generalized Cheeger–Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Boucetta

Essoufi

2019

Journal of Geometry and Physics

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Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics h p,q depending on two parameters with p ∈ R and q ≥ 0. This family has been introduced recently and possesses interesting geometric properties. If p = q = 0 we recover the Sasaki metric and when p = q = 1 we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with an Euclidean product on its total space. In this paper, we show that natural metrics can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger-Gromoll metrics on this new context. We show a rigidity result of this metrics which generalizes so far all rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if M is a Riemannian manifold of dimension n, then the Atiyah Lie algebroid associated to the O(n)principal bundle of orthonormal frames over M possesses a family depending on a parameter k > 0 of transitive Euclidean Lie algebroids structures say AO (M, k). When M is a space form of constant curvature c, we show that there exists two constants C n < 0 and K(n, c) > 0 such that (AO(M, k), h 1,1 ) is a Riemannian manifold with positive scalar curvature if and only if c > C n and 0 < k ≤ K(n, c).

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The Geometry of Generalised Cheeger-Gromoll Metrics

Cited by 22 publications

References 19 publications

Weighted Metrics on Tangent Sphere Bundles

Weighted Metrics on Tangent Sphere Bundles

Differential as a harmonic morphism with respect to Cheeger–Gromoll-type metrics

The geometry of generalized Cheeger–Gromoll metrics on the total space of transitive Euclidean Lie algebroids

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