On a conformally flat Riemannian space with positive Ricci curvature

Tani, Makoto

doi:10.2748/tmj/1178243319

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Introduction2

Theorem 13 Let (M 4 G) Be a 4-dimensional Compact Orient1

In -1)1

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Cited by 37 publications

(29 citation statements)

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“…Introduction. M. Tani [3] proved that a compact and orientable Riemannian manifold admitting a conformally flat metric of positive Ricci curvature and constant scalar curvature is a space form, that is, it is a constant curvature space. It is our purpose to extend this result to complete Riemannian manifolds with Ricci curvature bounded from below.…”

mentioning

confidence: 99%

An application of Yau’s maximum principle to conformally flat spaces

Goldberg¹

1980

Proc. Amer. Math. Soc.

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mentioning

confidence: 99%

An application of Yau’s maximum principle to conformally flat spaces

Goldberg¹

1980

Proc. Amer. Math. Soc.

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“…Then, we can combine Theorem 1.3 with Tani's theorem [27] and Bonnet-Myers theorem to obtain the following result. This improves Corollary 3 in [7].…”

Section: Theorem 13 Let (M 4 G) Be a 4-dimensional Compact Orientmentioning

confidence: 99%

“…[21] and [24]), nonnegative or positive Ricci curvature (cf. [18] and [27]), nonnegative or positive scalar curvature (cf. [11], [12] and [16]), nonnegative or positive isotropic curvature (cf.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

Ribeiro

2016

Annali di Matematica

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ABSTRACT. The goal of this paper is to investigate the rigidity of 4-dimensional manifolds involving some pinching curvature conditions. To this end, we make use of the approach of biorthogonal curvature which is weaker than the sectional curvature. Here, we prove a rigidity result for 4-dimensional compact manifolds under a suitable lower bound condition on the minimum of the biorthogonal curvature. From this, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional manifolds.

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“…Summing up we have that M is a compact orientable conformally flat Riemannian manifold with constant scalar curvature and with positive Ricci curvature. Using Theorem A of Tani [7], we have that M is of constant sectional curvature o = K/(n -1) (in fact n(n -l)cr = t = nK).…”

Section: In -1)mentioning

confidence: 99%

On the minimal eigenvalue of the Laplacian operator for 𝑝-forms in conformally flat Riemannian manifolds

Perrone¹

1982

Proc. Amer. Math. Soc.

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ABSTRACT. Let (M, g) be a compact orientable conformally flat Riemannian manifold and pXi the minimal eigenvalue of the Laplacian operator for p-forms. We prove that if there exists a positive constant K such that p > Kg, where p is the Ricci tensor of M, then pXi > Kp(n -p + l)/(n -1) f°r eacn P, 1 < p < n/2, (n = dim M); moreover if the equality holds for some p then M is of constant curvature a = K/(n -1).

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On a conformally flat Riemannian space with positive Ricci curvature

Cited by 37 publications

References 1 publication

An application of Yau’s maximum principle to conformally flat spaces

An application of Yau’s maximum principle to conformally flat spaces

Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

On the minimal eigenvalue of the Laplacian operator for 𝑝-forms in conformally flat Riemannian manifolds

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