Conformal Killing-Yano Tensors on Manifolds with Mixed 3-Structures

Integrability Conditions for Killing-Yano Tensors and Maximally Symmetric Spaces in the Presence of Torsion

The integrability conditions for the existence of Killing-Yano tensors or, equivalently, covariantly closed conformal Killing-Yano tensors, in the presence of torsion are worked out. As an application, all metrics and torsions compatible with the existence of a Killing-Yano tensor of order n-1 are obtained. Finally, the issue of defining a maximally symmetric space with respect to connections with torsion is addressed.

Twistor spinors and extended conformal superalgebras

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twis…

Killing-Yano Tensors of Order n-1

The properties of a Killing-Yano tensor of order n-1 in an n-dimensional manifold are investigated. The integrability conditions are worked out and all metrics admitting a Killing-Yano tensor of order n-1 are found. It is pointed out a connection between such tensors and a generalization of the concept of angular momentum. A theorem on how to generate closed conformal Killing vectors using the symmetries of a manifold is proved and used to find all Killing-Yano tensors of order n-1 of a maximally symmetric space.

Killing-Yano tensors and multi-hermitian structures

We show that the Euclidean Kerr-NUT-(A)dS metric in $2m$ dimensions locally admits $2^m$ hermitian complex structures. These are derived from the existence of a non-degenerate closed conformal Killing-Yano tensor with distinct eigenvalues. More generally, a conformal Killing-Yano tensor, provided its exterior derivative satisfies a certain condition, algebraically determines $2^m$ almost complex structures that turn out to be integrable as a consequence of the conformal Killing-Yano equations. In the complexification, these lead to $2^m$ maximal isotropic foliations of the manifold and, in Lor…

New currents with Killing-Yano tensors

New relations involving the Riemann, Ricci and Einstein tensors that have to hold for a given geometry to admit Killing-Yano tensors are described. These relations are then used to introduce novel conserved "currents" involving such Killing-Yano tensors. For a particular current based on the Einstein tensor, we discuss the issue of conserved charges and consider implications for matter coupled to gravity. The condition on the background geometry to allow asymptotic conserved charges for a current introduced by Kastor and Traschen is found and a number of other new aspects of this current are c…

Invariant Conformal Killing-Yano $2$-forms on five dimensional Lie Groups

We study left invariant conformal Killing-Yano (CKY) $2$-forms on Lie groups endowed with a left invariant metric. We classify all $5$-dimensional metric Lie algebras carrying CKY tensors that are obtained as a one-dimensional central extension of 4-dimensional metric Lie algebras endowed with a invertible parallel skew-symmetric tensor. On the other hand, we also classify $5$-dimensional metric Lie algebras with center of dimension greater than one admitting strict CKY tensors. In addition, we determine all possible CKY tensors on these metric Lie algebras. In particular, we exhibit the first…

Generalized symmetry superalgebras

We generalize the symmetry superalgebras of isometries and geometric Killing spinors on a manifold to include all the hidden symmetries of the manifold generated by Killing spinors in all dimensions. We show that bilinears of geometric Killing spinors produce special Killing-Yano and special conformal Killing-Yano forms. After defining the Lie algebra structure of hidden symmetries generated by Killing spinors, we construct the symmetry operators as the generalizations of the Lie derivative on spinor fields. All these constructions together constitute the structure of generalized symmetry supe…

Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime

The higher-dimensional Kerr-NUT-de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing-Yano (CKY) tensor as a ``hidden'' symmetry which provides the separation of variables for the geodesic equations and Klein-Gordon equations. We present a classification of higher-dimensional spacetimes admitting a rank-2 closed CKY tensor. This provides a generalization of the Kerr-NUT-de Sitter spacetime. In particular, we show that the Kerr-NUT-de Sitter spacetime is the only s…

Basic Gravitational Currents and Killing-Yano Forms

It has been shown that for each Killing-Yano (KY)-form accepted by an $n$-dimensional (pseudo)Riemannian manifold of arbitrary signature, two basic gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general geometrical facts implied by these conservation laws are also elucidated. In particular, the conservation of the one-form currents implies that the scalar curvature of the manifold is a flow invariant for all of its Killing vector fields. It also directly follows that, while all KY…

曲率挤压条件下共形 Killing 向量场的 Bochner-Yano 型定理

Bochner-Yano Type Theorems for Conformal Killing Vector Fields under Curvature Pinching Conditions

研究曲率挤压条件下闭黎曼流形上共形 Killing 向量场，通过建立新 Bochner 型恒等式和 Moser 迭代得梯度估计，证明 Ricci 曲率有界时非平凡共形 Killing 向量场处处非零，推广了相关经典定理。

Gauged twistor spinors and symmetry operators

We consider gauged twistor spinors which are supersymmetry generators of supersymmetric and superconformal field theories in curved backgrounds. We show that the spinor bilinears of gauged twistor spinors satify the gauged conformal Killing-Yano equation. We prove that the symmetry operators of the gauged twistor spinor equation can be constructed from ordinary conformal Killing-Yano forms in constant curvature backgrounds. This provides a way to obtain gauged twistor spinors from ordinary twistor spinors.

非对角化Killing张量的Stäckel问题：Yano-Patterson提升、强对称的代数和动量二次积分

Stäckel problem for non-diagonal Killing tensors: Yano-Patterson lifts, algebra of strong symmetries and quadratic in momenta integrals

本文提出了一种非对角化Killing张量的Stäckel问题构造方法，通过Yano-Patterson提升和Nijenhuis几何，生成可积的哈密顿系统，并应用于流体类型PDE的可积性证明。

共形Killing--Yano Ricci孤立子：结构、相容性与刚性

Conformal Killing--Yano Ricci solitons: Structure, compatibility, and rigidity

本文引入共形Killing--Yano Ricci孤立子结构，建立相容性恒等式，证明在4维Lorentzian下其强制局部Kerr--NUT--(A)dS几何，并给出谱阻碍与显式构造，为研究隐藏对称性的共形Ricci孤立子提供几何框架。

The conformal Killing equation on forms -- prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on $k$-forms to a twisting of the conformal Killing equation on (k - l)-forms for various inte…

齐性平面波上的不可约 Killing 张量与共形 Killing 张量

Irreducible Killing and conformal Killing tensors on homogeneous plane waves

该研究借助 BGG 算子分类齐性平面波上的不可约 Killing 与共形 Killing 2-张量，并给出显式公式及存在条件。

First BGG operators on homogeneous conformal geometries

We study first BGG operators and their solutions on homogeneous conformal geometries. We focus on conformal Killing tensors, conformal Killing--Yano forms and twistor spinors in particular. We develop an invariant calculus that allows us to find solutions explicitly using only algebraic computations. We also discuss applications to holonomy reductions and conserved quantities of conformal circles. We demonstrate our result on examples of homogeneous conformal geometries coming mostly from general relativity.

FLRW时空的统计共形Killing向量场

Statistical conformal Killing Vector Fields for FLRW Space-Time

本文从新视角研究FLRW时空的共形Killing向量场，证明存在九个共形向量场（六个Killing），并分类统计结构及联络条件。

On Spaces with the Maximal Number of Conformal Killing Vectors

It is natural to expect and simple to prove that every conformally flat space possess the maximal number of conformal Killing vector fields (CKVs). On the other hand, it is interesting to ask whether the converse is true. Is conformal flatness a necessary condition for the existence of the maximal number of CKVs? In this review article it is proven that the answer is yes, a space admits the maximal number of CKVs if, and only if, it is conformally flat.

Conformal Killing forms on Riemannian manifolds

Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly Kaehler and weak G_2-manifolds. Moreover, we give a complete description of special conformal Killing forms. A further result is a sharp upper bound on the dimension of the space of conformal Killing forms.

Killing and Conformal Killing tensors

We introduce an appropriate formalism in order to study conformal Killing (symmetric) tensors on Riemannian manifolds. We reprove in a simple way some known results in the field and obtain several new results, like the classification of conformal Killing $2$-tensors on Riemannian products of compact manifolds, Weitzenböck formulas leading to non-existence results, and construct various examples of manifolds with conformal Killing tensors.

Conformal Killing forms on nearly Kähler manifolds

We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of $d ω$ and its Hodge dual $* dω$ where $ω$ is the fundamental 2-form of the nearly Kähler structure. The proof is based on a fundamental integrability condition for conformal Killing forms. We have partial results in the case of conformal Killing 2-forms. In particular we show the non-existence of J-anti-invariant Killing 2-forms.

共形Killing张量及其Killing尺度

Conformal Killing tensors and their Killing scales

本文通过共形不变的tractor演算和微分延拓，刻画了共形Killing张量成为Killing张量无迹部分的条件，给出了Einstein Killing尺度的代数刻画，并研究了共形Killing向量的类似问题。

A characterization of quaternionic projective space by the conformal-Killing equation

We prove that a compact quaternionic-Kähler manifold of dimension $4n\geq 8$ admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionic-Kähler structure.

Conformal Killing $L^{2}-$forms on complete Riemannian manifolds with nonpositive curvature operator

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemannian manifolds.

Conformal Killing graphs with prescribed mean curvature

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.

几乎交换李群上的不变共形Killing形式

Invariant conformal Killing forms on almost abelian Lie groups

本文完全描述了几乎交换度量李代数上的共形Killing-Yano $p$-形式，证明了非平行形式仅存在于$p=1$或$p=n-1$时，并完成了至多5维的分类及格存在性区分。

Normal Conformal Killing Forms

We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal twistor equations. The case of decomposable normal conformal holonomy representations is discussed. A typical example with an irreducible holonomy representation are the so-called Fefferman spaces. We also apply our results to describe the geometry of solutions with conformal Kill…

L2-transverse conformal Killing forms on complete foliated manifolds

In this article, we study the L2-transverse conformal Killing forms on complete foliated Riemannian manifolds and prove some vanishing theorems. Also, we study the same problems on Kahler foliations with a complete bundle-like metric.

Kähler几何中的共形Killing形式

Conformal Killing forms in Kaehler geometry

本文在Kähler几何中显式求解中间度数的共形Killing形式方程，完成紧致情形的分类，并首次给出非Hamiltonian 2-形式来源的实例，基于Calabi型流形，同时提供Hermitian Killing形式的结构与例子。

Transverse conformal Killing forms on Kahler foliations

On a closed, connected Riemannian manifold with a Kähler foliation of codimension $q=2m$, any transverse Killing $r\ (\geq 2)$-form is parallel (S. D. Jung and M. J. Jung [\ref{JJ2}], Bull. Korean Math. Soc. 49 (2012)). In this paper, we study transverse conformal Killing forms on Kähler foliations and prove that if the foliation is minimal, then for any transversal conformal Killing $r$-form $ϕ$ $(2\leq r \leq q-2)$, $Jϕ$ is parallel. Here $J$ is defined in Section 4.

The conformal-Killing equation on G2 and Spin7 structures

Let M be a 7-manifold with a G2-structure defined by ϕ\inΩ^{3}_{+}(M). We prove that ϕ is conformal-Killing with respect to the associated metric g(ϕ) if and only if the G2-structure is nearly parallel. Let M be an 8-manifold with a Spin7-structure defined by ψ\inΩ^{4}_{+}(M). We prove that ψ is conformal-Killing with respect to the associated metric g(ψ) if and only if the Spin7-structure is parallel.

Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations

We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with $C$-positive normal curvature, if there is a closed basic 1-form $ϕ$ such that $Δ_Bϕ=qCϕ$, then the foliation is transversally isometric to the quotient of a $q$-sphere.

带Killing场的洛伦兹曲面的共形类

Conformal class of Lorentzian surfaces with Killing fields

该研究针对带非零Killing场的二维洛伦兹环面共形类问题，定义了对应圆周向量场（差标量因子）的有限维纤维非单射映射，借此刻画了满足L. Mehidi共轭点存在相关条件的这类环面的共形类。

关于李群上不变共形Killing形式的综述

A survey on invariant conformal Killing forms on Lie groups

本文综述了李群上左不变共形Killing p-形式的最新研究进展，并指出了未来研究中的有趣开放问题。

A prolongation of the conformal-Killing operator on quaternionic-Kahler manifolds

A 2-form on a quaternionic-Kahler manifold (M, g) is called compatible (with the quaternionic structure) if it is a section of the direct sum bundle S^2(H) \oplus S^2(E). We construct a connection D on S^2(H) \oplus S^2(E)\oplus TM, which is a prolongation of the conformal-Killing operator acting on compatible 2-forms. We show that D is flat if and only if the quaternionic-Weyl tensor of (M,g) is zero. Consequences of this result are developed. We construct a skew-symmetric multiplication on the space of conformal-Killing 2-forms on (M,g) and we study its properties in connection with the subs…

Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds - Characterization and Killing-Field Decomposition

Given a maximally non-integrable 2-distribution ${\mathcal D}$ on a 5-manifold $M$, it was discovered by P. Nurowski that one can naturally associate a conformal structure $[g]_{\mathcal D}$ of signature (2,3) on $M$. We show that those conformal structures $[g]_{\mathcal D}$ which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of $[g]_{\mathcal D}$ can be decomposed into a symmetry of ${\mathcal D}$ and an almost Einste…

On the numerical computation of Killing and conformally Killing vector fields on compact Riemannian manifolds

The defining equations for Killing vector fields and conformal Killing vector fields are overdetermined systems of PDE. This makes it difficult to solve the systems numerically. We propose an approach which reduces the computation to the solution of a symmetric eigenvalue problem. The eigenvalue problem is then solved by finite element techniques. The formulation itself is valid in any dimension and for arbitrary compact Riemannian manifolds. The numerical results which validate the method are given in two dimensional case.

Conformal Killing spinors and special geometric structures in Lorentzian geometry - a survey

This paper is a survey on special geometric structures that admit conformal Killing spinors based on lectures, given at the ``Workshop on Special Geometric Structures in String Theory'', Bonn, September 2001 and at ESI, Wien, November 2001. We discuss the case of Lorentzian signature and explain which geometries occur up to dimension 6.

李群上的共形Killing对称张量

Conformal Killing symmetric tensors on Lie groups

该研究提出Killing型度量李代数的概念，证明满足两步幂零、低维等多种条件的李代数关于任意正定度量均为Killing型，丰富了李群上共形Killing对称张量的相关理论。

Complex surfaces and null conformal Killing vector fields

We study the relation between the existence of null conformal Killing vector fields and existence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with metric of signature (2,2). We establish first the topological types of pseudo-Hermitian surfaces admitting a nowhere vanishing null vector field. Then we show that a pair of orthogonal, pointwise linearly independent, null, conformal Killing vector fields defines a para-hyperhermitian structure and use this fact for a classification of the smooth compact four-manifolds admitting such a pair of vector fields…