Modifications of the Ricci tensor and applications

In this paper, we focus on left-invariant m-quasi-Einstein metrics on simple Lie groups. First, we prove that X is a left-invariant Killing vector field if the metric on a compact simple Lie group is m-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial m-quasi-Einstein metrics except SU (3), E 8 and G 2 and most of them admit infinitely many metrics. Moreover, we prove that every compact simple Lie group admits non-trivial m-quasi-Einstein Lorentzian metrics and most of them admit infinitely many metrics. Finally, we prove that some non-compact simple Lie groups admit infinitely many non-trivial m-quasi-Einstein Lorentzian metrics.

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“…This tensor was extended, independently, by Barros and Ribeiro [4] and Limoncu [18] for an arbitrary vector field X on M n as follows:…”

Section: Introductionmentioning

confidence: 98%

Non-trivial m-quasi-Einstein metrics on simple Lie groups

Chen

Liang

Zhu

2015

Annali di Matematica

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“…[9,20,21]). Later on Barros-Ribeiro Jr [4] and Limoncu [16] generalized and studied the previous equation (1.1), independently, by considering a 1-form V ♭ instead of df , which is satisfied Ric + 1 2…”

Section: Introductionmentioning

confidence: 99%

Quasi-Einstein structures and almost cosymplectic manifolds

Chen

2020

RACSAM

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In this article, we study almost cosymplectic manifolds admitting quasi-Einstein structures (g, V, m, λ). First we prove that an almost cosymplectic (κ, µ)-manifold is locally isomorphic to a Lie group if (g, V, m, λ) is closed and on a compact almost (κ, µ)-cosymplectic manifold there do not exist quasi-Einstein structures (g, V, m, λ), in which the potential vector field V is collinear with the Reeb vector filed ξ. Next we consider an almost α-cosymplectic manifold admitting a quasi-Einstein structure and obtain some results. Finally, for a K-cosymplectic manifold with a closed, non-steady quasi-Einstein structure, we prove that it is η-Einstein. If (g, V, m, λ) is non-steady and V is a conformal vector field, we obtain the same conclusion.

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“…Instead of a gradient of a smooth function f , m-Bakry-Emery Ricci tensor was extended by Barros and Ribeiro Jr in [3] and Limoncu in [15] to an arbitrary vector field X on M n as follows:…”

Section: Introduction a Natural Extension Of The Ricci Tensor Is Thementioning

confidence: 99%

Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

Chen

Liang

2016

Arch. Math.

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We call a metric m-quasi-Einstein if Ric m X (a modification of the m-Bakry-Emery Ricci tensor in terms of a suitable vector field X) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contain Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field X such that the left-invariant Riemannian metric on a quadratic Lie group is m-quasi-Einstein is a Killing vector field. Then we construct infinitely many non-trivial m-quasi-Einstein metrics on solvable quadratic Lie groups G(n) for m finite.Mathematics Subject Classification. Primary 53C25, 53C20, 53C21; Secondary 17B30.

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Modifications of the Ricci tensor and applications

Abstract: By using two modified Ricci tensors, we prove some theorems which correspond to Myers's diameter estimate theorem and Bochner's vanishing theorem.

Cited by 48 publications

References 17 publications

Non-trivial m-quasi-Einstein metrics on simple Lie groups

Non-trivial m-quasi-Einstein metrics on simple Lie groups

Quasi-Einstein structures and almost cosymplectic manifolds

Non-trivial m-quasi-Einstein metrics on quadratic Lie groups

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