Indefinite symmetric spaces with $G_{2(2)}$-structure
Ines Kath
本文确定了完备性包含在$G_{2(2)}\subset SO(4,3)$中的所有签名为（4，3）的不可分解伪黎曼对称空间.
We determine all indecomposable pseudo-Riemannian symmetric spaces of signature (4,3) whose holonomy is contained in $G_{2(2)}\subset SO(4,3)$.
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Semisimplicity of indefinite extrinsic symmetric spaces and mean curvature
本文改进了Schienburg和Kim的一个结果，利用关于中曲率向量的形状算子给出了伪黎曼外对称空间半单性的一个判别准则.
Indefinite extrinsic symmetric spaces I
我们发现（可能退化）内积空间中的完全外对称空间与称为（弱）外对称三元组的某些代数对象之间存在一一对应关系。特别是，这产生了一个描述的任意外部对称空间在伪欧几里得空间由相应的无穷小对象。
Indefinite extrinsic symmetric spaces II
开发了一个描述不确定的外部对称空间相应的无穷小对象在前面的文件中，我们现在研究这些代数对象的分类问题。在大多数情况下，不定外对称空间的横射群不是半单的，这给分类带来了困难。我们使用最近发展起来的（h，K）-不变度量李代数的二次扩张方法来解决这个问题。我们得到了外对称空间的等距类与某个上同调集之间的一一对应。这允许系统地构造外部对称空间和明确的分类结果，例如，如果嵌入流形或周围空间的度量具有小的索引。我们将通过对所有洛伦兹外对称空间进行分类来说明这一点。
Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form
我们用$q\ge5$对符号为$(p,q)$的连通伪黎曼流形进行了分类，使得在$M$的每一点处，反对称曲率算子在类空2平面集上具有常秩2和常Jordan标准形，并且反对称曲率算子对$M$的至少一点不是幂零的.
The geometric characteristics of SGL submanifolds in an indefinite Sasakian statistical manifold equipped with a quarter symmetric metric connection
本文研究了在特定几何结构下SGL子流形的几何特性，分析了其分布的可积性和平行性，并通过示例展示了其应用。
Classification of indefinite hyper-Kaehler symmetric spaces
我们对不定单连通超Kaehler对称空间进行了分类。没有平坦因子的空间有交换和乐群和特征标（4 m，4 m）。在满足实性条件S = τS的n = 2 m复变数齐次四次多项式空间（S^4C^n）^τ上的四元数上，建立了8 m维单连通超Kaehler对称空间与一般线性群GL（m，H）的轨道之间的自然1-1对应，其中τ是由C^{2 m} = H^m的四元数结构导出的实结构.我们还定义和分类复超Kaehler对称空间。这种没有平坦因子的空间存在于任何可被4整除的（复）维中。
On the curvature symmetries in lightlike hypersurfaces of an indefinite Sasakian manifold
本文研究不定 Sasakian 流形中相切于结构向量场的类光超曲面，证明其不具备局部对称、半对称或半平行等曲率对称性。
On the Geometry of Connections with Totally Skew-Symmetric Torsion on Manifolds with Additional Tensor Structures and Indefinite Metrics
本文综述了作者在下列流形上具有全反对称挠的联络的几何学结果：具有Norden度量的殆复流形，具有B-度量的几乎切触流形，具有Hermitian和反Hermitian度量的几乎超复流形.
New examples of indefinite hyper-Kaehler symmetric spaces
根据数学中发展的伪黎曼对称空间的方法，DG/0408249给出了具有非交换完整性的不定超Kaehler对称空间的例子。此外，我们还对度量具有特征（4，4 n）的不可分解超Kaehler对称空间进行了分类.这样的空间存在当且仅当n= 0，1，3。
Curvature-homogeneous indefinite Einstein metrics in dimension four: the diagonalizable case
我们对所有度量签名中具有复可对角化曲率算子的曲率齐次爱因斯坦四维流形进行了分类。它们都是局部均匀的。更精确地说，任何这样的流形必须是局部对称的或局部等距于一个合适的具有左不变度量的李群。为了证明这一点，我们明确地确定了可能的局部等距类型的流形，具有上述性质，但不是局部对称的。
On compact manifolds admitting indefinite metrics with parallel Weyl tensor
具有平行Weyl张量但不是共形平坦或局部对称的紧致伪黎曼流形已知存在于大于4的无穷多个维度中。我们证明了这类流形的一些一般拓扑性质，即欧拉示性数和实庞特里亚金类的消失，以及基本群的无限性。我们还表明，在洛伦兹的情况下，他们中的每一个至少是5维的，并承认一个双重覆盖，这是一个在圆上的丛。
We study hyper-spheres, spheres and circles, with respect to an indefinite metric, in a tangent space on a 4-dimensional differentiable manifold. The manifold is equipped with a positive definite metric and an additional tensor structure of type (1, 1)11. The fourth power of the additional structure is minus identity and its components form a skew-circulant matrix in some local coordinate system. The both structures are compatible and they determine an associated indefinite metric on the manifold.
Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomomorphims containing the adjoint representation of $\mathfrak{g}$ is an infinitesimal isometry for $\langle\cdot,\cdot\rangle $. Among these Lie algebras are the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume. We prove a strong invariance property for nil-invariant symmetric bilinear forms, whic…
In this paper we build the structure equations and the integrable systems for a discrete centroaffine indefinite surface in $\R^3$. At the same time, some centroaffine invariants are obtained according to the structure equations. Using these centroaffine invariants, we study the Laplacian operator and the convexity of a discrete centroaffine indefinite surface. Furthermore, some interest examples are provided.
Lightlike hypersurfaces in indefinite $\mathcal{S}$-manifolds
在度量$g.f.f$ -流形中，我们研究了与特征向量场相切的类光超曲面$M$，由于$f$ -结构的存在，我们确定了$TM$和一个选定的屏蔽分布的一些分解，得到了两个关于该结构不变的分布.讨论了类光超曲面上$g.f.f$ -结构的存在性，在适当的假设下，得到了可积分布叶上的不定$\mathcal{S}$ -结构.讨论了不定$\mathcal{S}$ -空间型的全脐类光超曲面的存在性.最后，我们明确地描述了不定$\mathcal{S}$ -流形的类光超曲面。
We consider a $3$-dimensional differentiable manifold with two circulant structures -- a Riemannian metric and an additional structure, whose third power is the identity. The structure is compatible with the metric such that an isometry is induced in any tangent space of the manifold. Further, we consider an associated metric with the Riemannian metric, which is necessary indefinite. We find equations of a sphere and of a circle, which are given in terms of the associated metric, with respect to the Riemannian metric.
Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with isometry group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie group acting transitively on $M$. For metric index less than three, we find that the isometry group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decompo…
We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reversible. We present three examples of our framework: the Dirac operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an indefinite metric), the harmonic oscillator, and the construction via the Kasparov product of an indefinite spectral triple from a family of spectr…
Indefinite almost paracontact metric manifolds
本文引入了$(\varepsilon)$ -几乎仿切流形的概念，特别是$(\varepsilon)$ -仿Sasakian流形的概念.提出了几个例子。得到了$(\varepsilon)$ -paraSasakian流形的曲率张量和Ricci张量的一些典型恒等式.证明了如果半黎曼流形是平坦的、真循环的或真Ricci-循环的，则它不允许有$(\varepsilon)$ -paraSasakian结构。证明了对于$(\varepsilon)$ -paraSasakian流形，对称、半对称和常截面曲率的条件都是相同的.它表明，一个对称的类空（分别。类时）$(\varepsilon)$ -paraSasakian流形$M^{n}$局部等距于伪双曲空间$H_ν^{n}(1)$（分别为.伪球$S_ν^{n}(1)$）。最后证明了对于$(\varepsilon)$ -paraSasakian流形，Ricci-半对称、Ricci-对称和Einstein流形的条件是完全相同的。
We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch and Mesland on the self-adjointness and regularity of the sum of two weakly anticommuting operators, we show that we still have an equivalence between indefinite Kasparov modules and pairs of Kasparov modules. Importantly, the weakened version of indefinite Kasparov modules now inc…
Minimal Lagrangian submanifolds in indefinite complex space
考虑复线性空间C^n具有标准伪厄米特签名形式（2 p，2（n-p））。这在C^n上产生伪黎曼和辛结构。我们证明了那些子流形，这是拉格朗日和最小的关于这些结构的体积最小化在其拉格朗日同调类。我们还描述了几个家庭的最小拉格朗日子流形。特别地，我们刻画了C ^2中具有自然中立度量的极小Lagrange曲面和C^n中具有任意签名的等变极小拉格朗日子流形.
We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in [1], for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors' point of view, obtaining cleaner equations for the almost contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are $η$-umbilical.…
In this paper we provide a method to study critical points of strongly indefinite functionals on vector bundles. We focus mainly on energy functionals coupled with a fermionic part, that is with a Dirac-type operator. We consider the cases of the perturbed Dirac-Geodesics problem and the Yang-Mills-Dirac type equation in dimension two.
It is well known that if the dimension of the Sasaki cone is greater than one, then all Sasakian structures are either positive or indefinite. We discuss the phenomenon of type changing within a fixed Sasaki cone. Assuming henceforth that the dimension of the Sasaki cone is greater than one, there are three possibilities, either all elements are positive, all are indefinite, or both positive and indefinite Sasakian structures occur. We illustrate by examples how the type can change as we move in the Sasaki cone. If there exists a Sasakian structure in the cone whose total transverse scalar cur…
Finite dimensional imbeddings of harmonic spaces
在非紧调和流形$M$中，我们建立了由$\cosh r + c$形式的径向本征函数生成的本征空间$V_λ$的有限维性。因此，对于这样的调和流形，我们给出了$M$到$(V_λ,B)$的等距嵌入，其中$B$是$V_λ$上的非退化对称双线性不定形式（类似于实双曲空间$I\!\!\!H^{n}$到具有不定形式$Q(x,x) = -x_0^2 + \sum x_i^2$的$I\!\!\!R^{n+1}$的嵌入）。最后给出了$M$对称的条件。
An integral transform on a cylinder and the twistor correspondence
明确地建立了R^4上R不变不定自对偶共形结构的扭量对应。这些对应关系是用二维圆柱上的函数到平坦洛伦兹空间R^{1，2}上的函数的自然积分变换来表示的，它与波动方程和Radon变换有关。总结了不定自对偶4-空间和不定Einstein-Weyl 3-空间扭量构造的一般方法。
The Grassmannian of indefinite subspaces
本文研究研究较少的齐性空间 $\u039F_{m,n}(\u211D)/(\u039F_{p,q}(\u211D) \times \u039F_{m-p,n-q}(\u211D))$，即不定格拉斯曼流形，推导其性质并确定其几何与拓扑特征。
Submaximally symmetric CR-structures
维数为$(2n+1)$的流形上具有非退化Levi形式的超曲面型CR-结构具有最大对称维数$n^2+4n+3$ .我们证明了当$n>1$满足一定条件时，（局部）对称代数的下一个（次极大）可能维数对Levi-不定结构是$n^2+4$，对Levi-定结构是$n^2+3$.在CR维数$n=1$的例外情况下，次极大对称维数3由E. 1计算，嘉当
String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group ${\rm O}(n)$ to the indefinite-signature orthogonal group O(p, q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simp…
We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally symmetric about the starting point of the slower set, then if the surface is conformally equivalent to the Euclidean plane, the slower set remains in a bounded region, while if the surface is nonpositively curved and conformally equivalent to the hyperbolic plane, both sets may keep growing indefinitely.
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