Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Integral formulas in Riemannian geometry

The aim of this note is to define almost Yamabe solitons as special conformal solutions of the Yamabe flow. Moreover, we shall obtain some rigidity results concerning Yamabe almost solitons. Finally, we shall give some characterizations for homogeneous gradient Yamabe almost solitons.

On conformal solutions of the Yamabe flow

We give a short Lie-derivative theoretic proof of the following recent result of Barros et al. “A compact non-trivial almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere”. Next, we obtain the result: a complete almost Ricci soliton whose metric $g$ is $K$-contact and flow vector field $X$ is contact, becomes a Ricci soliton with constant scalar curvature. In particular, for $X$ strict, $g$ becomes compact Sasakian Einstein.

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Almost ricci solitons and k-contact geometry

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form $\mathbb {R}^{4}, \mathbb {H}^{4}$ or $\mathbb {S}^{4}$. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that $M$ has divergence-free Bach tensor.

Bach-Flat Critical Metrics of the Volume Functional on 4-Dimensional Manifolds with Boundary

The purpose of this article is to investigate the geometry of static perfect fluid space-time metrics on compact manifolds with boundary. In the first part, we provide a boundary estimate for static perfect fluid space-time. In the second part, we establish a unified Bochner type formula for a large class of spaces that include the static perfect fluid space-time, critical metrics of the volume functional, static spaces and CPE metrics. Moreover, as a consequence of such a formula we obtain a gap result for a compact static perfect fluid space-time.

Static perfect fluid space-time on compact manifolds

The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial answer to Yau Conjecture on pinching theorem for 4-dimensional compact manifolds.

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Four-Dimensional Compact Manifolds with Nonnegative Biorthogonal Curvature

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in $1980$'s that every CPE metric must be Einstein. We prove that a $4$-dimensional CPE metric with harmonic tensor $W^+$ must be isometric to a round sphere $\Bbb{S}^4.$

Critical metrics of the total scalar curvature functional on 4-manifolds

We prove that a 4-dimensional compact manifold $M^4$ with harmonic Weyl tensor must be either locally conformally flat or isometric to a complex projective space $\mathbb {CP}^2,$ provided that the biorthogonal (sectional) curvature satisfies a suitable pinching condition. In particular, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional compact manifolds.

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Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

We prove that a critical metric of the volume functional on a 4-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $$\mathbb {R}^{4}$$
 , $$\mathbb {H}^{4}$$
 or $$\mathbb {S}^{4}$$
 Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics

Ernani Ribeiro

Papers

Static perfect fluid space-time on compact manifolds

Four-Dimensional Compact Manifolds with Nonnegative Biorthogonal Curvature

Critical metrics of the total scalar curvature functional on 4-manifolds

Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor

Volume Functional of Compact 4-Manifolds with a Prescribed Boundary Metric