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 aim of this note is to prove that any compact non-trivial almost Ricci soliton $$\big (M^n,\,g,\,X,\,\lambda \big )$$
 with constant scalar curvature is isometric to a Euclidean sphere $$\mathbb {S}^{n}$$
 . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $$X$$
 decomposes as the sum of a Killing vector field $$Y$$
 and the gradient of a suitable function.

Compact almost Ricci solitons with constant scalar curvature are gradient

The aim of this note is to prove that any compact non-trivial almost Ricci soliton $\big(M^n,\,g,\,X,\,\lambda\big)$ with constant scalar curvature is isometric to a Euclidean sphere $\Bbb{S}^{n}$. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $X$ decomposes as the sum of a Killing vector field $Y$ and the gradient of a suitable function.

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao–Tam critical metrics. We provide an estimate to the area of the boundary of Miao–Tam critical metrics on compact three-manifolds. In addition, we obtain a Bochner type formula which enables us to show that a Miao–Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in $\mathbb {S}^3$.

Critical Metrics of the Volume Functional on Compact Three-Manifolds with Smooth Boundary

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4-dimensional half conformally flat manifolds M4. In fact, we shall show that for a nontrivial must be isometric to a sphere and f is some height function on

Ernani Ribeiro

Papers

On conformal solutions of the Yamabe flow

Compact almost Ricci solitons with constant scalar curvature are gradient

Compact almost Ricci solitons with constant scalar curvature are gradient

Critical Metrics of the Volume Functional on Compact Three-Manifolds with Smooth Boundary

Critical point equation on four-dimensional compact manifolds