Sergiu Moroianu
Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.
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, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl t…
We consider a priori estimates of Weyl's embedding problem of $(\mathbb{S}^2, g)$ in general $3$-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang, we obtain an isometric embedding of $(\mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl's embedding theorem in space form under the condition that $g\in C^2$ with $D^2g$ Dini continuous.
We show any Weyl curvature model can be geometrically realized by a Weyl manifold
We determine the space of algebraic pseudo-Hermitian Kähler-Weyl curvature tensors and the space of para-Hermitian Kähler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian Weyl manifolds in arbitrary dimension.
We introduce and study the notion of Sasaki--Weyl manifold, which is a natural generalization of the notion of Sasaki manifold. We construct a reduction of Sasaki--Weyl manifolds and we show that it commutes with several reductions already existing in the literature.
We show that the Weyl structure of an almost-Hermitian Weyl manifold of dimension at least 6 is trivial if the associated curvature operator satisfies the Kaehler identity. Similarly if the curvature of an almost para-Hermitian Weyl manifold of dimension at least 6 satisfies the para-Kaehler identity, then the Weyl structure is trivial as well.
In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature
Compact four-manifolds with pinched self-dual Weyl curvature
本文研究了具有调和自双曲Weyl曲率和挤压条件的紧致四流形的性质。
In this article, we study Einstein-Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic $(κ,μ)$-manifold is Einstein or cosymplectic if it admits a closed Einstein-Weyl structure or two Einstein-Weyl structures. Next for a three dimensional compact almost $α$-cosymplectic manifold admitting closed Einstein-Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost $α$-cosymplectic admitting two Einstein-Weyl structures is either Einstein or $α$-cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compac…
We give a classification of compact conformally Kahler Einstein-Weyl manifolds whose Ricci tensor is hermitian.
This is a survey on quaternion Hermitian Weyl (locally conformally quaternion Kähler) and hyperhermitian Weyl (locally conformally hyperkähler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a study is two-fold: it comes from the constantly growing interest in Weyl (and Einstein-Weyl) geometry and, on the other hand, from the necessity of understanding the existing classes of quaternion Hermitian manifolds.
A Riemannian manifold is called Weyl homogeneous, if its Weyl tensors at any two points are "the same", up to a positive multiple. A Weyl homogeneous manifold is modeled on a homogeneous space $M_0$, if its Weyl tensor at every point is "the same" as the Weyl tensor of $M_0$, up to a positive multiple. We prove that a Weyl homogeneous manifold $M^n, n \ge 4$, modeled on an irreducible symmetric space $M_0$ of types II or IV (compact simple Lie group with a bi-invariant metric or its noncompact dual) is conformally equivalent to $M_0$.
We find geometric conditions on a Hermitian-Weyl manifold under which the complex structure is a pseudo-harmonic map in the sense of G. Kokarev \cite{K09} from the manifold into its twistor space. This is done under the assumption that the dimension of the manifold is four or the Hermitian-Weyl structure is locally conformally Kähler.
Weyl group of the group of holomorphic isometries of a Kähler toric manifold
本文计算了具有实解析凯勒度量的凯勒环流形全纯等距群的Weyl群（Segal意义下）。
The infimum of the Weyl functional is shown to be surprisingly small on many compact 4-manifolds that admit positive-scalar-curvature metrics. Results are also proved that systematically compare the scalar and self-dual Weyl curvatures of certain almost-Kaehler 4-manifolds.
Starting from a real analytic surface $\mathcal{M}$ with a real analytic conformal Cartan connection A. Borówka constructed a minitwistor space of an asymptotically hyperbolic Einstein-Weyl manifold with $\mathcal{M}$ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on $\mathcal{M}$ can be extended to symmetries of the obtained Einstein-Weyl manifold.
Pinched self-dual Weyl curvature on Einstein four-manifolds
研究Einstein四维流形中挤压自双曲Weyl曲率的性质，证明其在特定条件下具有反自双曲结构。
We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.
Conformal-projective transformations on statistical and semi-Weyl manifolds with torsion
该研究探讨带挠率的统计流形与半Weyl流形上的共形射影变换，证明相关结构的不变性、子流形诱导结构的等价性、脐点的保性，分析类光与非退化超曲面的异同，并给出半Weyl流形的仿射分布实现方式。
Almost-Kähler four manifolds with harmonic self-dual Weyl curvature
本文研究具有调和自对偶Weyl曲率的紧致近Kähler四维流形，证明了在常数量曲率且$c_{1}\cdotω\geq 0$条件下该流形为Kähler流形，并建立了一个积分曲率不等式。
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, which are generalizations of identities in dimension four.
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 warped product with $(n-1)$--dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.
The aim of this paper is to study complete (noncompact) steady $m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady $m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional manifold $(M^{n},g)$, $(n\geq4),$ with nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with $(n-1)-$dimensional Einstein fiber, provided that $M$ has fourth order divergence-free Weyl tensor (i.e., ${\rm div}^{4}W=0$).
In this paper we consider the geometric behavior near infinity of some Einstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$ space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential set and has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n, g)$ must be asymptotically locally hyperbolic. One interesting application of this theorem is to show a rigidity result for the hyperbolic space under an integral condition for the curvature.
We show that a compact K-contact manifold $(M,g,ξ)$ has a closed Weyl-Einstein connection compatible with the conformal structure $[g]$ if and only if it is Sasaki-Einstein.
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 to the covariant derivative level the classical one for the Weyl tensor obtained by Derdzinski in 1983. As a consequence, we deduce some integral identities involving the Weyl tensor and its derivatives on a compact four dimensional Einstein manifold.
Motivated by the study of Weyl structures on conformal manifolds admitting parallel weightless forms, we define the notion of conformal product of conformal structures and study its basic properties. We obtain a classification of Weyl manifolds carrying parallel forms, and we use it to investigate the holonomy of the adapted Weyl connection on conformal products. As an application we describe a new class of Einstein-Weyl manifolds of dimension 4.
The object of this paper is to obtain the concircular curvature tensor of the semi symmetric non-metric connection on the Weyl manifold and to give a necessary and sufficient condition for a semi symmetric non-metric connection to be S-concircular.
For the root system of type $B_l$ and $C_l$, we generalize the result of \cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \cite{DZ1998}.
We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $σ_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $σ_2$ is isometric to a quotient of the round $\mathbb{S}^4$.
We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also relate the Cotton-York tensor of an umbilic hypersurface to the Weyl tensor of the ambient. As a consequence, a conformal 3-manifold or a self-dual 4-manifold admitting a rational curve as a null-geodesic is conformally flat. We show that the projective structur…
A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in $\mathbb{R}^16$. As a consequence, we show that a Riemannian manifold having the same Weyl tensor a…
In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional Einstein fibers, provided that the potential function is proper. As a consequence, we prove a result about the nonexistence of multiple black holes in static spacetimes.
Fedosov manifolds with parallel symplectic Weyl curvature
该论文将Roter关于平行Weyl曲率流形的经典定理推广到辛范畴，证明具有平行辛Weyl曲率的Fedosov流形必为局部对称或Weyl平坦。
We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.
Assume that $M$ is a smooth manifold with a symplectic structure $ω$. Then Weyl manifolds on the symplectic manifold $M$ are Weyl algebra bundles endowed with suitable transition functions. From the geometrical point of view, Weyl manifolds can be regarded as geometrizations of star products attached to $(M,ω)$. In the present paper, we are concerned with the automorphisms of the Weyl manifold corresponding to Poincaré-Cartan class ($c_0$ is a $\check{\rm C}$ech cocycle corresponding to the symplectic structure $ω$.) $[c_0+\sum_{\ell=1}^\infty c_{\ell} ν^{2\ell}]\in \check{H}^2 (M)[[ν^2]]$. We…
Quasi-Einstein manifolds with Harmonic Weyl curvature
本文研究了具有调和韦伊尔曲率的准爱因斯坦流形，扩展了四维空间中的相关工作，并给出了新的非局部共形扁且非D-平坦的准爱因斯坦流形例子。
The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of $h_4$ is that it is nonnegative for Einstein manifolds, hence it provides a geometric obstruction to the existence of Einstein metrics in dimensions $\geq 4$, independently from the sign of the Einstein constant. This motivates our study of the positivity of this invariant. Here in this paper, we prove many constructions of metrics with positiv…
Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl's asymptotic formula.
We prove that a $4$-dimensional simply connected, compact critical metric of the volume functional with harmonic anti-self dual Weyl tensor and boundary isometric to a standard sphere $\mathbb{S}^{3}$ is isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$
Manifolds with harmonic Weyl curvature and curvature operator of the second kind
该研究证明了高维流形在特定曲率条件下的分类，并推广了相关领域的已有成果。
Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups I
本文研究了仿射Weyl群轨道空间上的广义Frobenius流形结构构造及其单值群的性质。
Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle.
The classical Weyl law for Schrödinger operators on complete Riemannian manifolds
本文提出一个判断Schrödinger算子在完备黎曼流形上经典Weyl定律成立的准则，不依赖传统几何或分析假设，引入几何-分析不变量量化几何、势函数增长与振荡尺度的平衡。
Weyl energy and connected sums of four-manifolds
针对非局部共形平坦且非同时自对偶或反自对偶的闭定向黎曼四维流形，证明其连通和上存在度量使Weyl能量严格小于两流形Weyl能量之和。
We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for some "generic" low-dimensional instances, where explicit examples of non-closed Clifford-Weyl structures can be constructed.
It is shown that in every dimension n=3j+2, j=1,2,3,..., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.
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, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of Andrei Moroianu in [9] is generalized in the following sense. The only non-closed Weyl manifolds of dimension greater than 3 that admit solutions of the real Killing equation are 4-dimensional and non-compact. Any Weyl manifold of these dimensions admitting a real Killing spinor h…
New examples of compact Weyl-parallel manifolds
本文构造了所有维数 n≥5 及任意不定符号差下、具有平行 Weyl 张量但非共形平坦且非局部对称的紧致伪 Riemann 流形新例子。
We extend the Siu--Beauville theorem to a certain class of compact Kaehler--Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As applications we obtain restrictions on the fundamental groups of such Kaehler--Weyl manifolds, and show that in certain cases they are in fact Kaehler.
We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that every such manifold has first Betti number $b_1 =1$ and Hodge numbers $h^{p,0} =0$ for $p>0$, $h^{0,1} =1$, $h^{0,q} =0$ for $q>1$.
We classify the possible local holonomy groups of Weyl connections. The Berger-Simons theorem and the Merkulov-Schwachhöfer classification of holonomy groups of irreducible torsion-free connections leaves us with the remaining case, where the Weyl connection $D$ is reducible and non-closed. In this case, it was shown by F. Belgun and A. Moroianu that the Weyl structure is an adapted Weyl structure of a non-closed conformal product. Furthermore we prove that non-closed Einstein-Weyl product structures only exist in dimension $4$.
It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$\int_M|W^+_g|^2dμ_{g}\geq 2π^2(2χ(M)+3τ(M))-\frac{8π^2}{|π_1(M)|},$$ where $W^+_g$, $χ(M)$ and $τ(M)$ respectively denote the self-dual Weyl tensor of $g$, the Euler characteristic and the signature of $M$. This generalizes Gursky's inequality \cite{gur} for the case of $b_1(M)>0$ in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gursky's inequalities for the case of $b_2^+(M)>0$ or $δ_gW^+_g=0$, and obtain topological obstructions …
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]. In particular, we prove that a critical metric with harmonic Weyl tensor on a simply connected compact manifold with boundary isometric to a standard sphere $\mathbb{S}^{n-1}$ must be isometric to a geodesic ball in a simply connected space form $\Bbb{R}^n,$ $\Bbb{H}^n$ and $\…
Compact locally homogeneous manifolds with parallel Weyl tensor
本文构造了新的紧致ECS流形例子，它们测地不完备、秩2且局部齐次，覆盖从5开始的所有奇数维度，区别于以往秩1、测地完备且非局部齐次的例子。
This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local structure of ECS manifolds, and outline an argument showing that compact ECS manifolds exist in infinitely many dimensions greater than 4. We also discuss some properties of compact manifolds that admit ECS metrics, and provide a list of open questions about compact ECS manifolds.
We give an elementary proof of the fact that any 4-dimensional para-Hermitian manifold admits a unique para-Kaehler--Weyl structure. We then use analytic continuation to pass from the para-complex to the complex setting and thereby show any 4-dimensional pseudo-Hermitian manifold also admits a unique Kaehler--Weyl structure.
A recent result of M. Kourganoff states that if $D$ is a closed, reducible, non-flat, Weyl connection on a compact conformal manifold $M$, then the universal covering of $M$, endowed with the metric whose Levi-Civita covariant derivative is the pull-back of $D$, is isometric to $\mathbb{R}^q\times N$ for some irreducible, incomplete Riemannian manifold $N$. Moreover, he characterized the case where the dimension of $N$ is $2$ by showing that $M$ is then a mapping torus of some Anosov diffeomorphism of $T^{q+1}$. We show that in this case one necessarily has $q=1$ or $q=2$.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。