We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $\mathbb{S}^{n}$ with the round metric, $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with the product metric or $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with a rotationally symmetric Derdziński metric.
We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformal metrics holonomic with respect to the same coordinate system.
This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it follows from the classification that any closed G2-structure inducing a metric that is both conformally flat and complete must be equivalent to the flat G2-structure on $\mathbb{R}^7.$
In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.
The object of the present paper is to study 3-dimensional conformally flat quasi-Para-Sasakian manifolds. First, the necessary and sufficient conditions are provided for 3-dimensional quasi-Para-Sasakian manifolds to be conformally flat. Next, a characterization of 3-dimensional conformally flat quasi-Para-Sasakian manifold with \b{eta}=const. is given.
We give global restrictions on the possible boundaries of compact, orientable, locally conformally flat manifolds of dimension $4k$ in terms of integrality of eta invariants.
Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent, $(GR)_n$ structure are completely classified in this short report. A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is shown to be either a de Sitter space or an anti-de Sitter space. In particular, a conformally flat generalized Ricci recurrent spacetime must be either a de Sitter spacetime or an anti-de Sitter spacetime.
We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.
Inspired by the work of F. Hang and X. Wang and partial results by S. Raulot, we prove a scalar curvature rigitidy result for locally conformally flat manifolds with boundary in the spirit of the well-known Min-Oo conjecture.
In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.
In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$ and also the rigidity result when certain renormalized volume is zero.
In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension $n\geq 3$ is locally a warped product with $(n-1)$-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.
In this paper we study the invariant Walker structures over the conformally flat four-dimensional homogeneous manifolds according to the Seger types of the Ricci operator.
We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler metrics on Hopf manifolds, explicitly characterized by Vaisman. Finally, we review the properties of zero projectively flat metrics. As applications, we refine a list of possible projectively flat metrics by Li, Yau, and Zheng; moreover we prove that projectively flat astheno-Kähler…
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of th…
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$.
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
The local classification of conformally flat Lorentzian manifolds with special holonomy groups is obtained. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.
We show that under some non-degeneracy assumption the only submersive harmonic morphism on a conformally flat $3-$sphere is the Hopf fibration. The proof involves an appropriate use the Chern-Simons functional.
Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}δ_{ij}$ on the Euclidean space $\mathbb{R}^{m+1}$ so that a minimal hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, δ_{ij})$ in a Euclidean space becomes a biharmonic hypersurface $M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}δ_{ij})$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in [Ou1] and [OT] as special cases.
In this paper, we study conformally flat hypersurfaces of dimension $n(\geq 4)$ in $\mathbb{S}^{n+1}$ using the framework of Möbius geometry. First, we classify and explicitly express the conformally flat hypersurfaces of dimension $n(\geq 4)$ with constant Möbius scalar curvature under the Möbius transformation group of $\mathbb{S}^{n+1}$. Second, we prove that if the conformally flat hypersurface with constant Möbius scalar curvature $R$ is compact, then $$R=(n-1)(n-2)r^2, ~~0<r<1,$$ and the compact conformally flat hypersurface is Möbius equivalent to the torus $$\mathbb{ S}^1(\sqrt{1-r^2})…
In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$ using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed Möbius form under the Möbius transformation group of $\mathbb{R}^4$. Such examples come from cones, cylinders, or rotational hypersurfaces over the surfaces with constant Gaussian curvature in $3$-spheres, Euclidean $3$-spaces, or hyperbolic $3$-spaces, respectively. Second, we investigate the global behavior of the generic conformally flat hypersurface and give some int…
We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works by Q.-M. Cheng, M.H. Noronha, B.-L. Chen and X.-P. Zhu, and S. Zhu.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless $κ$-Einstein soliton and is realized on the cotangent bundle of an affine surface.
We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature $(2,2)$. This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat 3-metric with the Guichard condition in the interior of the space, an evolution of orthogonal (local-) Riemannian $2$-metrics with constant Gauss curvature $-1$ is determined; for a $2$-metric belonging to a certain class of orthogonal analytic $2$-metrics with constant Gauss cur…
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.
The problem of conformal transformation and conformal flatness of Finsler spaces has been studied by so many researchers $\left[ 6,16,17,20,21\right] .$ Recently, Prasad et. al $\left[ 19\right] $ have studied three dimensional conformally flat Landsberg and Berwald spaces and have given some important results. The purpose of the present paper is to extend the idea of conformal change to four dimensional Finsler spaces and find the suitable conditions under which a four dimensional conformally flat Landsberg space becomes a Berwald space.
The local structure of 4-dimensional, conformally flat, almost $ε$-Kählerian (i.e., almost pseudo-Kählerian and almost para-Kählerian) manifolds is characterized with the help of left-regular and right-regular paraquaternionic functions. Examples of such structures are discussed.
Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifolds. As applications of these results we exhibit explicit examples of metrics g, conformal to g they are solutions for this problem.
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all radially symmetric gradient Yamabe solitons. We also show that any eternal solutions to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space-time maximum must be a steady Yamabe soliton.
It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson-Walker warped product, if the gradient of the potential function is non null, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.
The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition. Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition. In this paper, w…
Peng Wu recently announced a beautiful characterization of conformally Kaehler, Einstein metrics of positive scalar curvature on compact oriented 4-manifolds via the condition det (W^+) > 0. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences of our method, which builds on techniques previously developed in (LeBrun 2015).
In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.
First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian manifold satisfying weakly (and doubly weakly) conformally flat and quasi-Einstein ($η$-Einstein) conditions. Finally, we provide local classification and characterization of a semi-Riemannian (including the 4-dimensional spacetime) with harmonic Weyl tensor and a non-homothetic conformal (including closed) vector field, in terms of Petrov types and Bach tensor.
In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence of this identity, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.
We show that a statistical manifold manifold of a constant non-zero curvature can be realised as a level line of Hessian potential on a Hessian cone. We construct a Sasakian structure on $TM\times\R$ by a statistical manifold manifold of a constant non-zero curvature on $M$. By a statistical Lie algebra of a constant non-zero Lie algebra we construct a l.c.K Lie algebra.
We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional curvature and we find some expressions for the natural lifted metric $G$, such that the tangent bundle $(TM,G)$ become conformally flat.
In this paper, we present some new properties for p-biharmonic hypersurfaces in Riemannian manifold. We also characterize the p-biharmonic submanifolds in an Einstein space. We construct a new example of proper p-biharmonic hypersurfaces. We present some open problems.
We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the sphere and extend a result due to Shiohama and Xu \cite{SX} for compact hypersurfaces in any space form.
The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and …
In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points (i.e. admissible GCF-manifolds), * frontals as hypersurfaces in the lightcone Q^{n+1}_+, * frontals as hypersurfaces in the hyperbolic space H^{n+1}, * sapacelike frontals as hypersurfaces in the de Sitter space S^{n+1}_1. Recently, the duality of conformally flat Riemannian mani…
On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian $Δ^k$, $k\in…
In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectivel…
