A note on critical point metrics of the total scalar curvature functional

Leandro, Benedito

doi:10.1016/j.jmaa.2014.11.040

Cited by 32 publications

(21 citation statements)

References 7 publications

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“…A function satisfying Equation (2.3) is said to have constant energy, following [15], where the author investigated properties of such functions. Equation (2.3) also appeared in the Critical Point Equation (CPE) conjecture (see [33]), where it is shown that the potential function of a CPE metric satisfies an equation similar to (2.3) if and only if it is an Einstein manifold (i.e., the CPE Conjecture holds).…”

Section: Resultsmentioning

confidence: 99%

Ricci almost solitons on semi‐Riemannian warped products

Borges

Tenenblat

2022

Mathematische Nachrichten

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It is shown that a gradient Ricci almost soliton on a warped product, true(Bn×hFm,g,f,λtrue)$\big (B^n\times _h F^m, g,f,\lambda \big )$ whose potential function f depends on the fiber, is either a Ricci soliton or λ is not constant and the warped product, the base and the fiber are Einstein manifolds, which admit conformal vector fields. Assuming completeness, a classification is provided for the gradient Ricci almost solitons on warped products, whose potential functions depend on the fiber. An important decomposition property of the potential function in terms of functions which depend either on the base or on the fiber is proven. In the case of a complete gradient Ricci soliton, the potential function depends only on the base.

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Section: Resultsmentioning

confidence: 99%

Ricci almost solitons on semi‐Riemannian warped products

Borges

Tenenblat

2022

Mathematische Nachrichten

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“…In [9], Hwang proved that the CPE conjecture is also true under certain conditions on the bounds of the potential function λ. Very recently, Neto [12] deduced a necessary and sufficient condition on the norm of the gradient of the potential function for a CPE metric to be Einstein. Similar kind of critical metric was studied by Wang and Wang in [17].…”

Section: Definitionmentioning

confidence: 99%

*- Critical point equation on N(k)-contact manifolds

Dey¹,

Majhi²

2020

MIF

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The object of the present paper is to characterize N (k)-contact metric manifolds satisfying the * -critical point equation. It is proved that, if (g, λ) is a non-constant solution of the * -critical point equation of a non-compact N (k)-contact metric manifold, then (1) the manifold M is locally isometric to the Riemannian product of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of positive curvature 4 for n > 1 and flat for n = 1, (2) the manifold is * -Ricci flat and (3) the function λ is harmonic. The result is also verified by an example.2000 Mathematics Subject Classification: 53C25, 53C15.

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“…Further, in [9], Hwang proved that the CPE conjecture is valid under certain condition on the bound of the potential function λ. Recently, Nato [10] obtained a necessary and sufficient condition on the norm of the gradient of the potential function for a CPE metric to be Einstein.…”

Section: Introductionmentioning

confidence: 99%

“…10) where we used (2.5) and(3.2). Making use of (3.9), equation(3.7) transforms into (ξλ − λ − 1){QX + 2nX} = 0, (3.11) for all vector field X on M .…”

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confidence: 99%

The critical point equation on Kenmotsu and almost Kenmotsu manifolds

Patra¹,

Ghosh²,

Bhattacharyya³

2020

Publ. Math. Debrecen

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In this paper, we study the critical point equation (shortly, CPE) within the framework of Kenmotsu and almost Kenmotsu manifolds. First, we prove that a complete Kenmotsu metric satisfying the CPE is Einstein and locally isometric to the hyperbolic space H 2n+1. In the case of Kenmotsu manifolds, it is possible to determine the potential function explicitly (locally). We also provide some examples of Kenmotsu and almost Kenmotsu manifolds that satisfy the CPE.

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A note on critical point metrics of the total scalar curvature functional

Cited by 32 publications

References 7 publications

Ricci almost solitons on semi‐Riemannian warped products

Ricci almost solitons on semi‐Riemannian warped products

*- Critical point equation on N(k)-contact manifolds

The critical point equation on Kenmotsu and almost Kenmotsu manifolds

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