Ingemar Eriksson, José M. M. Senovilla
We provide a simple proof that conformally semi-symmetric spacetimes are actually semi-symmetric. We also present a complete refined classification of the semi-symmetric spacetimes.
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We find all Ricci semi-symmetric as well as all conformally semi-symmetric spacetimes. Neither of these properties implies the other. We verify that only conformally flat spacetimes can be Ricci semi-symmetric without being conformally semi-symmetric and show that only vacuum spacetimes and spacetimes with just a $Λ$-term can be Ricci semi-symmetric without being conformally semi-symmetric.
This paper has been withdrawn by the authors because it has been merged with paper arXiv:0903.3501v1 [math.DG]
In this review, we collect several results for conformally standard stationary spacetimes (SxR,g) obtained in terms of a Finsler metric of Randers type on the orbit manifold S that we call Fermat metric. This metric is obtained by applying the relativistic Fermat principle and it turns out that it encodes all the causal aspects of the spacetime.
In this paper we study the r-stability of closed spacelike hypersurfaces with constant $r$-th mean curvature in conformally stationary spacetimes of constant sectional curvature. In this setting, we obtain a characterization of $r-$stability through the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature. As an application, we treat the case in which the spacetime is the de Sitter space.
In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $η$-pinched.
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.
Baryogenesis in Conformally Flat Spacetimes
本文基于因果费米子系统的重子生成机制，在共形平坦时空中推导出依赖于质量、共形因子和正则化向量场的重子生成率公式，并覆盖FLRW、Milne等时空，为定量预测奠定基础。
Enveloping space of a globally hyperbolic conformally flat spacetime
本文通过构造包含所有共形平坦柯西扩张的包络空间，重新证明全局双曲共形平坦时空 C 0-极大扩张的存在唯一性，并证明其保持包含关系。
It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.
In this paper we generalize the basic Lichnerowicz cohomology on transversally locally conformally Kählerian foliations and we study its relation with basic Bott-Chern cohomology and $0$--th basic Dolbeault cohomology with values in the associated foliated weight bundle.
Causal completion of a globally hyperbolic conformally flat spacetime
本文研究全局双曲时空的因果完备化，证明在单连通且共形平坦条件下其为同胚于S×[0,1]的带边拓扑流形，并刻画三类时空族的完备化。
Using a combination of techniques from conformal and complex geometry, we show the potentialization of 4-dimensional closed Einstein-Weyl structures which are half-algebraically special and admit a "half-integrable" almost-complex structure. That is, we reduce the Einstein-Weyl equations to a single, conformally invariant, non-linear scalar equation, that we call the "conformal HH equation", and we reconstruct the conformal structure (curvature and metric) from a solution to this equation. We show that the conformal metric is composed of: a conformally flat part, a conformally half-flat part r…
How to detect the spacetime curvature without rulers and clocks. II. Three-dimensional spacetime
本文将无尺无钟的曲率探测方法推广至三维时空，证明“良好缝合”结构等价于共形平坦，从而仅凭事件间因果关系即可识别非共形平坦时空的曲率。
Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $(M,J,θ)$ a $d$(bounded) locally conformally Kähler manifold if the lifted Lee form $\tildeθ$ on the universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies the inequality $(-1)^{n}χ(M^{2n})\geq$ (resp. $>$) $0$.
A note on locally conformally Kaehler structures and small deformations of Hopf manifolds
本文通过构造对角型Hopf流形附近的复解析族，结合lcK势在小参数复解析族中保持的已知结论，证明了所有Hopf流形都容许局部共形Kaehler结构。
Maximality of the futures of points in globally hyperbolic maximal conformally flat spacetimes
本文证明了 globally 超曲率最大 conformally 平面时空中的点的未来集在特定条件下具有极大性，并提供了新证明。
Anosov representations were introduced by F. Labourie [18] for fundamental groups of closed negatively curved surfaces, and generalized by O. Guichard and A. Wienhard [19] to representations of arbitrary Gromov hyperbolic groups into real semisimple Lie groups. In this paper, we focus on Anosov representations into the identity component O0(2, n) of O(2, n) for n $\ge$ 2. Our main result is that any Anosov representation with negative limit set as defined in [8] is the holonomy group of a spatially compact, globally hyperbolic maximal (abbrev. CGHM) conformally flat spacetime. The proof of the…
In this note we provide a direct proof of the complete classification of conformally flat isoparametric submanifolds of Euclidean space.
In this note we prove that a special family of Killing potentials on certain Hirzebruch complex surfaces, found by Futaki and Ono, gives rise to new conformally Kähler, Einstein-Maxwell metrics. The correspondent Kähler metrics are ambitoric but they are not given by the Calabi ansatz. This answers in positive questions raised by Futaki-Ono.
Conformally symplectic structures and the Lefschetz condition
本文证明紧辛流形在次数 1 满足硬 Lefschetz 条件且具有相容近复结构时，局部共形辛结构必为全局共形辛结构，给出 Vaisman 定理的辛类比。
In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regularity assumption) they are obtained as mapping tori of isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular, we obtain examples of inhomogeneous locally (non-globally) conformal nearly Kähler compact manifolds.
In this largely expository note, we discuss the mapping properties of the Laplacian (and other geometric elliptic operators) in spaces with an isolated conical singularity following the approach developed by B.-W. Schulze and collaborators. Our presentation aims at illustrating the versatility of these results by describing how certain representative (and seemingly disparate) applications in Geometric Analysis follow from a common setup.
In this note we revisit the notion of conformal barycenter of a measure on $\SS^n$ as defined by Douady and Earle in Acta Math. Vol 157, 1986. The aim is to extend rational maps from the Riemann sphere $\Cbar\isom\SS^2$ to the (hyperbolic) three ball $\BB^3$ and thus to $\SS^3$ by reflection. The construction which was pioneered by Douady and Earle in the case of homeomorphisms actually gives extensions for more general maps such as entire transcendental maps on $\C\subset\Cbar$. And it works for maps in any dimension.
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