arXiv research

Study shows Sasaki solitons with harmonic Weyl tensor are spheres.

New proof shows gradient Ricci solitons with harmonic Weyl tensor have at most three eigenvalues.

Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.

In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature

Study classifies gradient almost Ricci solitons with harmonic Weyl tensor.

The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric…

Study proves rigidity of certain gradient steady Ricci solitons with harmonic Weyl curvature.

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harm…

We investigate the local structure of four-dimensional Lorentzian quasi-Einstein manifolds under conditions on the Weyl tensor. We show that if the Weyl tensor is harmonic and the potential function preserves this harmonicity then, in the isotropic case, the manifold is necessarily a $pp$-wave. Using the quasi-Einstein…

Study rigidifies geometry of electrostatic systems with specific tensor properties.

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. …

In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a war…

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

One of the main aims of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ and with harmonic Weyl tensor, which improves the corresponding classification for complete locally conformally flat case, due to Miao and Tam [18…

Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = ρg$ with $ρ>0$ constant. We show that if $(M,g)$ satisfies $δ\mathcal W (\cdot, \cdot, \nabla f) = 0$, t…

We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmoni…

Study classifies Einstein spaces and warped products in weighted geometry.

We establish a $C^{1,α}$ compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the $C^{1,α}$ harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…

In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…

Among other results, a compact almost Kähler manifold is proved to be Kähler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact a…

Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.

New rigidity results for tensors on non-compact manifolds with curvature conditions.

We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$…

In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determi…

Study pseudo-harmonic maps on Weyl manifolds.

The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic …

The study finds conditions for almost-Kähler 4-manifolds to be Kähler.

The paper introduces comprehensive quasi-Einstein spacetimes and explores their properties.

Study pinched self-dual Weyl curvature in compact 4-manifolds.

We study conformal harmonic coordinates on Riemannian manifolds. These are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show their existence under general conditions. We find that conformal harmonic coordinates are a close conformal analogue of harmonic coordinates. We prove u…

We show that Weyl spaces provide a natural context for harmonic morphisms.

The paper classifies quasi-Einstein manifolds with harmonic Weyl curvature.

The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.

Refined asymptotics of scalar-flat ALE four-manifolds

Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu product. If the square of the Weyl tensor is nonzero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak's theorem, the expl…

New findings on compact manifolds with specific curvature properties.

In all dimensions and arbitrary signature, we demonstrate the existence of a new local potential -- a double (2,3)-form -- for the Weyl curvature tensor, and more generally for all tensors with the symmetry properties of the Weyl curvature tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -- a …

Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.

In this paper, we prove that the static triple with half harmonic Weyl curvature and positive scalar curvature must be the standard hemisphere.

Extends Weyl geometry from conformal to Weyl manifolds using ambient metrics.

We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends…

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

We study Kahler surfaces with harmonic anti-selfdual Weyl tensor. We provide an explicit local description, which we use to obtain the complete classification in the compact case. We give new examples of extremal Kahler metrics, including Kahler-Einstein metrics and conformally Einstein Kahler metrics. We also extend s…

In this paper, we will give a local version of the Hamilton-Ivey type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification on gradient shrinking solitons with vanishing Weyl tensor.

We prove that, in a space-time of dimension n>3 with a velocity field that is shear-free, vorticity-free and acceleration-free, the covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is zero. The other way, if the covariant divergence of the Weyl tensor is zero, then…