On static manifolds and related critical spaces with zero radial Weyl curvature

“…There are several works about classification of Miao-Tam critical metrics involving the Weyl curvature tensor, see for instance, [3,4,5,7,9,17]. Hence, motivated by [12] and based in techniques developed in [1,2], we shall provide a classification of a such critical metrics considering a pointwise pinching condition.…”

Section: Introductionmentioning

confidence: 99%

Critical metrics of the volume functional with pinched curvature

Baltazar¹,

Queiroz²

2022

Preprint

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“…Qing and Yuan [28] classified complete Bach-flat vacuum static spaces with compact level sets of f ; in particular these spaces have harmonic curvature and have at most two distinct Ricci-eigen values at each point. Baltazar, Barros, Batista and Viana [3] and Ye [32] proved that n-dimensional, n ≥ 5, compact vacuum static spaces with positive scalar curvature and zero radial Weyl curvature are Bach-flat. Hwang and Yun [16] showed that n-dimensional, n ≥ 5, vacuum static spaces with vanishing of complete divergence of Bach tensor and Weyl tensor have harmonic curvature if it has compact level sets of f .…”

Section: Introductionmentioning

confidence: 99%

Four-dimensional static and related critical spaces with harmonic curvature

Kim

Shin

2018

Pacific J. Math.

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In this article we study any 4-dimensional Riemannian manifold (M, g) with harmonic curvature which admits a smooth nonzero solution f to the following equationwhere Rc is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. We show that a neighborhood of any point in some open dense subset of M is locally isometric to one of the following five types;and H 2 (k) are the two-dimensional Riemannian manifold with constant sectional curvature k > 0 and k < 0, respectively, (iii) the static spaces in Example 3 below, (iv) conformally flat static spaces described in Kobayashi's [18], and (v) a Ricci flat metric.We then get a number of Corollaries, including the classification of the following four dimensional spaces with harmonic curvature; static spaces, Miao-Tam critical metrics and V -static spaces.The proof is based on the argument from a preceding study of gradient Ricci solitons [17]. Some Codazzi-tensor properties of Ricci tensor, which come from the harmonicity of curvature, are effectively used.where Rc is the Ricci curvature of g, x is a constant and y(R) a function of R. There are several well-known classes of spaces which admit such solutions.

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On static manifolds and related critical spaces with zero radial Weyl curvature

Cited by 10 publications

References 33 publications

A short overview of Besse’s conjecture

A short overview of Besse’s conjecture

Critical metrics of the volume functional with pinched curvature

Four-dimensional static and related critical spaces with harmonic curvature

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