Algebraic nature of singular Riemannian foliations in spheres

非负sectional curvature中黎曼叶理的一个对偶定理

A duality theorem for Riemannian foliations in nonnegative sectional curvature

利用一类新的Jacobi场估计，我们将证明具有非负截面曲率的完备流形中奇异黎曼叶理的一个对偶定理。

黎曼叶理的上同调紧性

Cohomological tautness for Riemannian foliations

在本文中，我们提出了关于历史背景下Riemannian叶理紧度的一些新结果。本文的第一部分简要介绍了这一问题的历史。对于闭流形，黎曼叶理的紧性可以用上同调来刻画。我们将这一上同调特征推广到一类叶理，其中包括任何奇异黎曼叶理的一个闭流形的层理。

黎曼叶状结构上的对称紧张量

Symmetric tautness tensor on Riemannian foliations

通过研究与叶状结构平均曲率相关的对称2-张量，给出了紧流形上黎曼叶状结构紧性的条件，并提供了该张量在多种几何条件下的应用。

Transverse $\mathcal F^T$-entropy and transverse Ricci flow for Riemannian foliations

In this paper, we introduce an entropy functional on Riemannian foliation, inspired by the work of , which is monotonically along the transverse Ricci flow. We relate their gradient flow, via diffeomorphism preserving the foliated structure of the manifold with Riemannian foliation, to the transverse Ricci flow. Moreover, inspired by the work of Fuquan Fang and Yuhao Zhang, we give a necessary condition for codimension 4 Riemannian foliation admitting the transverse Einstein metric.

黎曼勒让德叶理的谱几何

Spectral geometry of Riemannian Legendre foliations

本文给出了具有常$ϕ$ -截面曲率的紧致Sasakian流形上的等谱极小Riemannian Legendre叶理的几何刻画。

非正曲流形上的奇异黎曼叶

Singular Riemannian Foliations on Nonpositively Curved Manifolds

本文证明了在非正曲率的紧致流形中不存在真奇异黎曼叶状容许截面。然后给出了Hadamard流形上的真奇异黎曼叶理的整体刻画。此外，通过使用拉紧浸入的理论，我们提供了一个简短的证明，这一结果在极性作用的特殊情况下。

平行焦点结构与奇异黎曼叶理

Parallel Focal Structure and Singular Riemannian Foliations

证明了具有平行焦点结构的子流形是等参等焦子流形的推广，它通过其平行焦点流形诱导出周围空间的奇异黎曼叶状结构.

Riemannian 叶状结构的上同调与符号差的同伦不变性

Homotopy invariance of cohomology and signature of a riemannian foliation

本文证明 Riemannian 叶状结构的基本符号差与 Alvarez 类均为叶状同伦不变量，并说明叶状同伦映射诱导基本 Lichnerowicz 上同调间的同构。

Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the…

On Riemannian foliations admitting transversal conformal fields

Let $(M,g_M,\mathcal F)$ be a closed, connected Riemannian manifold with a Riemannian foliation $\mathcal F$ of nonzero constant transversal scalar curvature. When $M$ admits a transversal nonisometric conformal field, we find some generalized conditions that $\mathcal F$ is transversally isometric to the sphere.

含有复叶的射影空间的黎曼叶解

Riemannian foliations of projective space admitting complex leaves

受Gray关于复射影空间中具有Fubini-Study度量的复子流形的管公式的工作的启发，研究了射影空间的黎曼叶理.证明了余维为1的$\mathbb{P}^n$的任何开子集都不存在复黎曼叶理。因此，没有黎曼面的射影平面的黎曼叶理，甚至局部。我们确定如何一个复杂的子流形可能会出现作为一个特殊的叶的非平凡奇异黎曼叶的最大尺寸。应用格雷管公式得到复二次曲面上某些全纯曲线的体积界。

Pseudo-Riemannian foliations and their graphs

We prove that a foliation $(M, F)$ of codimension $q$ on a $n$-dimen\-sio\-nal pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph $G = G(F)$ of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pse\-u\-do-Rieman\-ni\-an submersions and the fibres of different projections are orthogonal at common points. Relatively this metric the induced foliation $(G,\mathbb{F})$ on the graph is pseudo-Riemannian …

可缩流形上的黎曼叶理

Riemannian foliations on contractible manifolds

证明了完备可缩流形上的黎曼叶理有一个闭叶，且若一个闭叶有一个黎曼生成的基本群，则所有的叶都是闭的。在额外的拓扑或几何假设下，我们证明了叶理也是简单的。

奇异黎曼叶理叶空间中的闭测地线

On closed geodesics in the leaf spaces of singular Riemannian foliations

本文综述了近年来关于黎曼轨道折叠和奇异黎曼叶理的一些结果，并将它们结合起来，得到了几类奇异黎曼叶理（s.r.f.）的叶空间中闭测地线的存在性，即S.R.F.其允许截面或没有水平共轭点。我们还研究了黎曼叶理的缩短过程。

关于带有黎叶状结构的紧致爱因斯坦型流形

On a compact Einstein-type manifolds with Riemannian foliations

该研究探讨带有黎叶状结构的紧致爱因斯坦型流形，证明其成为全叶状且具分裂性质的必要条件，并给出紧致几乎里奇孤子与紧致爱因斯坦流形上全叶状黎叶状结构的性质。

具有平行或调和基本形式的黎曼叶理

Riemannian foliations with parallel or harmonic basic forms

本文考虑法丛具有平行或调和基本形式的黎曼叶理。我们估计的范数的O'Neill张量的曲率数据的整个流形。然后给出了一些例子。

Orthogonal foliations on riemannian manifolds

In this work, we find an equation that relates the Ricci curvature of a riemannian manifold $M$ and the second fundamental forms of two orthogonal foliations of complementary dimensions, $\mathcal{F}$ and $\mathcal{F}^{\bot}$, defined on $M$. Using this equation, we show a sufficient condition for the manifold M to be locally a riemannian product of the leaves of $\mathcal{F}$ and $\mathcal{F}^{\bot}$, if one of the foliations is totally umbilical. We also prove an integral formula for such foliations.

Kaehler流形上复叶黎曼叶理的刚性

Rigidity of Riemannian foliations with complex leaves on Kaehler manifolds

研究了Kaehler流形上具有复叶的黎曼叶理。张量T是叶理是全测地线的障碍，它被解释为某个向量丛的全纯截面。这使我们能够给出分类结果时，流形是紧凑的。

广义半黎曼浸没与叶状结构

Generalized Semi-Riemannian Submersions and Foliations

本文引入广义半黎曼浸没与叶状结构概念，建立正则叶状结构为广义半黎曼的充要条件，证明平稳横截叶状结构的诱导性质，得到时空上平稳类光叶状结构的刚性结果，并推导奥尼尔型截面曲率公式。

黎曼叶上横截椭圆算子的迹公式

The trace formula for transversally elliptic operators on Riemannian foliations

本文证明了紧致叶流形上横截椭圆算子的Duistermaat-Guillemin迹公式。

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part I

We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part~II.

奇异黎曼叶理的几何分解

Geometric resolution of singular Riemannian foliations

证明了李群在黎曼流形上的等距作用有一个保持横截几何的分解当且仅当该作用是无穷小极的。我们提供了几类等距行动，包括极性和变分完全的拓扑简单性的应用。所有结果都证明在更一般的情况下奇异黎曼叶。

奇异黎曼叶状结构与坍缩

Singular Riemannian foliations and collapse

本文综述了利用群作用实现流形坍缩并控制曲率的经典方法，及其在奇异黎曼叶状结构中的最新扩展。

几乎四元数Kaehler积流形的四元数CR-子流形上的黎曼叶理

Riemannian foliations on quaternion CR-submanifolds of an almost quaternion Kaehler product manifold

本文研究了几乎四元数Kaehler积流形的四元数CR-子流形的典型叶状结构。

由圆构成的伪黎曼测地线叶理

Pseudo-Riemannian geodesic foliations by circles

研究了在什么样的假设下，由$S^1$作用生成的可定向的伪黎曼测地线叶理。我们构造的例子表明，相反的黎曼的情况下，它并不总是正确的。然而，我们证明了这样的作用总是存在时，叶理不包含lightlike叶，即伪黎曼Wadsley定理。作为应用，我们证明了所有类空/类时测地线都是闭的Lorentzian曲面都被$S^1\times \R$覆盖.由此可见，每个洛伦兹曲面都包含一条非闭测地线。

Localization of Chern-Simons type invariants of Riemannian foliations

We prove an Atiyah-Bott-Berline-Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern-Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.

正曲率中奇异黎曼叶状结构的核心约化

Core reduction for singular Riemannian foliations in positive curvature

本文证明在正截面曲率条件下，若奇异黎曼叶状结构存在预截面，则叶空间有边界，并推论正曲率流形的极叶状结构叶空间非空边界。

Clifford代数与球面上新的奇异黎曼叶理

Clifford algebras and new singular Riemannian foliations in spheres

利用Clifford代数的表示，我们构造了圆球上的不可分解的奇异黎曼叶理，其中大多数是非齐次的。这推广了Ferus，Karcher和Munzner关于非齐次等参超曲面的构造.

Riemann叶状结构的横截LS-范畴

Transverse LS-Category for Riemannian Foliations

研究了紧致流形上黎曼叶理的横截Lusternik-Schnirelmann范畴。当横LS范畴是有限的时，我们得到了一个充要条件。我们还介绍了横LS范畴的概念，本质的横范畴的变化，并表明，这是有限的每一个黎曼叶理，并符合横范畴，如果后者是有限的。证明了本质横截范畴是M上基本可微函数临界叶闭包数的下界。

Bounded distance geodesic foliations in Riemannian planes

A conjecture of Burns and Knieper asks whether a 2-plane with a metric without conjugate points, and with a geodesic foliation whose lines are at bounded Hausdorff distance, is necessarily flat. We prove this conjecture in two cases: under the hypothesis that the plane admits total curvature, and under the hypothesis of visibility at some point. Along the way, we show that all geodesic line foliations on a Riemannian 2-plane must be homeomorphic to the standard one.

Direct sum for basic cohomology of codimension four taut Riemannian foliation

We discuss the decomposition of degree two basic cohomology for codimension four taut Riemannian foliation according to the holonomy invariant transversal almost complex structure J, and show that J is C pure and full. In addition, we obtain an estimate of the dimension of basic J-anti-invariant subgroup. These are the foliated version for the corresponding results of T. Draghici et al.

加权黎曼叶状结构上的调和映射

Harmonic maps on weighted Riemannian foliations

本文研究加权叶状结构上的横截f-调和映射与$(F,F')_f$-调和映射，推广了极小叶状结构中两类调和映射的等价关系。

黎曼叶状结构与拟复形

Riemannian foliations and quasifolds

本文证明了在温和完备性条件下，Killing黎曼叶状结构的叶空间是微分拟流形，且其完整群胚在局部上与可数群的作用群胚Morita等价。

平坦退化度量与黎曼叶状结构

Flat degenerate metrics and Riemannian foliations

本文针对Bandyopadhyay等人关于具有常数秩的平坦非负定度量闭流形的猜想提出反例，并探讨了该问题与横截平坦黎曼叶状结构之间的联系。

Totally Geodesic Riemannian Foliations on Compact Lie Groups

In 86, Ranjan questioned whether a submersion $π: G\to B$ from a compact simple Lie group with bi-invariant metric is a coset foliation or not, provided the submersion is Riemannian with totally geodesic fibers. Here we answer this question affirmatively, even when the submersion is defined only in an open subset of $G$ (assuming suitable hypothesis).

Mean curvature flow of singular Riemannian foliations

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. These results generalize previous results of Liu and Terng, Pacini and Koike. In particular our results can be applied to partitions of Riemannian manifolds into orbits of actions of compact groups of isometries.

黎曼叶理上热算子的迹

Traces of heat operators on Riemannian foliations

我们考虑的基本热运营商的功能上的黎曼叶状的紧凑，黎曼流形，我们表明，这个运营商的轨迹有一个特别短的时间渐近展开。在这个扩展中的系数是从局部横向几何不变量-函数可计算的分析流形在一个任意小的邻域的叶封闭。利用这种展开式，我们证明了一些关于基本拉普拉斯算子谱的结果，例如类似于Weyl渐近公式。此外，我们明确地计算了前两个非平凡系数的特殊情况下，如余维两个叶理和叶理与定期关闭的扩展。

On the topology of leaves of singular Riemannian foliations

In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation $(M,\fol)$. If $M$ is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If $M$ has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well.

黎曼叶理的修正微分和基本上同调

Modified differentials and basic cohomology for Riemannian foliations

我们定义了一个新的版本的外导数的基本形式的黎曼叶理，以获得一个新的形式的基本上同调，满足庞加莱对偶的横定向的情况下。我们使用这种扭曲的基本上同调，以显示曲率之间的关系，紧，和消失的基本欧拉示性数和基本签名。

Conformal Minimal Foliations on Semi-Riemannian Lie Groups

We study left-invariant foliations ${\mathcal F}$ on semi-Riemannian Lie groups $G$ generated by a subgroup $K$. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations ${\mathcal F}$ when the subgroup $K$ is one of the important $\text{SU}(2)$, $\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SU}(2)$, $\text{SU}(2)\times\text{SL}_{2}(\mathbb R)$, $\text{SU}(2)\times\text{SO}(2)$, $\text{SL}_{2}(\mathbb R)\times\text{SU}(2)$. This way we construct new multi-dimensional families of Lie groups $G$ carrying such foliatio…

奇异黎曼叶：例外叶; Tautness

Singular Riemannian Foliations: Exceptional Leaves; Tautness

对于一个奇异的黎曼叶理，其叶子是正确嵌入的，我们在本文的第一部分中显示的存在的全球性管状邻域，我们开发了一个全球性的描述的叶理分层类型的叶子。第二部分讨论Lytchak和Thorbergsson引入的对无水平共轭点的叶理的进一步限制，在等距群作用的特殊情况下，它等于变分完备的概念。由此，我们推出了一个整体的几何描述-叶的焦点正是奇点-以及拓扑之一：紧叶。

坍缩具有平坦叶的正则黎曼流形

Collapsing regular Riemannian foliations with flat leaves

本文研究了如何保持截面曲率有界的情况下，对具有平坦叶的正则黎曼流形进行坍缩，并证明在特定条件下流形由环面作用给出。

奇异黎曼叶理的等域性

Equifocality of a singular riemannian foliation

完备黎曼流形M上的奇异叶理称之为黎曼的，如果在一点上垂直于叶的测地线保持垂直于它所遇到的每一叶。我们证明了正则叶是等焦的，即，正规叶状向量场的端点映射具有常秩。这意味着我们可以通过取正则叶的所有平行子流形的平凡完整性来重建奇异叶理。此外，叶上具有平凡完整性的法向叶状向量场的端点映射是覆叠映射。这些结果推广了作者以前关于带截面的奇异黎曼叶理的结果。

正则黎曼流的基本阿贝尔映射

Basic Albanese maps of regular Riemannian foliations

本文提出正则黎曼流的基本阿贝尔映射概念，通过基本1形式定义，并将其与环境流形的经典阿贝尔映射联系起来，同时展示了多个重要例子。

Lie groupoids and semi-local models of Singular Riemannian foliations

We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid.

近似Kaehler几何与黎曼叶理

Nearly Kaehler geometry and Riemannian foliations

考虑了具有正则Hermite联络的严格完备的近Kaehler流形。研究了正则厄米联络的完整表示。证明了严格完备的近似Kaehler是齐次近似Kaehler空间、四元数Kaehler流形上的扭量空间和6维近似Kaehler流形的局部黎曼积.作为应用，我们得到了容许相容Kaehler结构的全测地黎曼叶理的结构结果。最后，我们得到了一个分类结果的同质的情况下，减少了一个猜想的Wolf和Gray的6维形式。

度量与奇异黎曼叶的迈尔斯-斯蒂罗德定理

Myers-Steenrod Theorems for Metric and Singular Riemannian Foliations

本文研究了度量与奇异黎曼叶的等距群结构，证明了其维数上界，并指出该上界由特定纤维丛诱导。

具有幂零结构李代数的Riemann叶层的Alvarez类的刚性

Rigidity of the Álvarez classes of Riemannian foliations with nilpotent structure Lie algebras

证明了如果闭流形M上的黎曼叶状结构F的结构代数是幂零的，则（M，F）的Alvarez类沿每条闭路的积分是一个代数数的指数.利用这一结果和arXiv：1009.1098v2中的变形下Álvarez类的连续性，我们证明了闭流形M上的Álvarez类和黎曼叶理的几何紧性在变形下是不变的，如果M的基本群具有多项式增长.

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part II

Using the curvature-dimension inequality proved in Part~I, we look at consequences of this inequality in terms of the interaction between the sub-Riemannian geometry and the heat semigroup $P_t$ corresponding to the sub-Laplacian. We give bounds for the gradient, entropy, a Poincaré inequality and a Li-Yau type inequality. These results require that the gradient of $P_t f$ remains uniformly bounded whenever the gradient of $f$ is bounded and we give several sufficient conditions for this to hold.

Differentiable classification of 4-manifolds with singular Riemannian foliations

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, or $\mathbb{CP}^2\#\pm \mathbb{CP}^2$. As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard $\mathbb{S}^4$, $\pm\mathbb{CP}^2$ and $\mathbb{S}^2\times\mathbb{S}^2$. A classification of singular R…

黎曼叶上基本Dirac算子的指标理论

Index theory for basic Dirac operators on Riemannian foliations

本文证明了一个关于黎曼叶状算子上的基本Dirac-type算子的解析指数的公式，解决了一个多年来一直悬而未决的问题。我们还考虑了更一般的指数扭曲的基本狄拉克算子的表示的正交群。该公式是一个积分的总和爆破的地层的叶理，也涉及相关的椭圆算子的eta不变量。作为特例，利用两种独立的证明方法得到了基本欧拉示性数的Gauss-Bonnet公式。

A note on transverse divergence and taut Riemannian foliations

In this note we give a characterization of taut Riemannian foliations using the transverse divergence. This result turns out to be a convenient tool in the case of some standard examples. Furthermore, we show that a classical tautness result of Haefliger can be obtained in our particular setting as a straightforward consequence. In the final part of the paper we obtain a tautness characterization for transversally oriented foliations with dense leaves and investigate the case of a submanifold of lower dimension which is closed and orthogonal to the leaves.

Semi-Riemannian hypersurfaces in $\mathbb{L}^{n+1}$ with a totally geodesic foliation of codimension one

We classify hypersurfaces of the Minkowski space $Ł^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.

线性叶的闭包

The closure of linear foliations

本文通过简化几何证明方法，证明了线性叶的闭包本身是光滑的奇异黎曼流形。方法避免了传统证明中的技术工具，直接展示了线性半局部模型的光滑性。