The trace formula for transversally elliptic operators on Riemannian foliations

横截凯勒叶状结构的Aubin-Yau定理

An Aubin-Yau theorem for transversally Kähler foliations

本文在横截凯勒叶状结构中推导了Aubin-Yau定理，通过同调定向条件，简化了Vaisman Aubin-Yau定理的证明。

横截负曲率叶状结构的拓扑与底部谱

Topology and bottom spectrum of transversally negatively curved foliations

该研究针对叶空间单连通且负曲率的黎曼叶状结构，证明叶片法向指数映射是微分同胚，还给出叶片为极小子流形时流形及水平拉普拉斯算子谱底部的精确估计。

关于 Riemannian foliations 之间水平能量泛函的临界映射

On critical maps of the horizontal energy functional between Riemannian foliations

本文研究 Riemannian foliations 间光滑映射的水平能量泛函临界点，利用应力-能量张量建立单调性公式，并给出无穷远渐近条件下的 Jin 型定理。

Transversal infinitesimal automorphisms on Kähler foliations

Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field

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.

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.

The transversal wave equation and the noncommutative geodesic flow in Riemannian foliations

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and the corresponding transversal bicharacteristic flow and give a description of the noncommutative geodesic flow in transversal geometry of Riemannian foliations.

Modified differentials and basic cohomology for Riemannian foliations

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincaré duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.

Chern characters in equivariant basic cohomology

From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversally oriented Riemannien foliation, we can associate a manifold, called basique manifold, which is équiped with an action of orthogonal group, El Kacimi-Alaoui has shown how to associate a transversally elliptic basic differential operator an operator on a vector bundle, called usef…

The dual foliation of some singular Riemannian foliations

In this paper, we use the methods of subriemannian geometry to study the dual foliation of the singular Riemannian foliation induced by isometric Lie group actions on a complete Riemannian manifold M. We show that under some conditions, the dual foliation has only one leaf.

Low dimensional Singular Riemannian Foliations in spheres

Singular Riemannian Foliations are particular types of foliations on Riemannian manifolds, in which leaves locally stay at a constant distance from each other. Singular Riemannian Foliations in round spheres play a special role, since they provide "infinitesimal information" about general Singular Riemannian Foliations. In this paper we show that Singular Riemannian Foliations in spheres, of dimension at most 3, are orbits of an isometric group action.

Riemannian foliations of spheres

We show that a Riemannian foliation on a topological $n$-sphere has leaf dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.

Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow

The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

On Riemannian Foliations over Positively Curved Manifolds

We prove that, under reasonable conditions, odd co-dimension Riemannian foliations cannot occur in positively curved manifolds.

Algebraic nature of singular Riemannian foliations in spheres

We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations.

Singular Riemannian foliations and applications to positive and nonnegative curvature

We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call A-foliations and B-foliations, respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian 4- and 5-manifolds with positive or nonnegative sectional curvature.

Lightlike foliations of semi-Riemannian manifolds

Using screen distributions and lightlike transversal vector bundles we develop a theory of degenerate foliations of semi-Riemannian manifolds.

Monge-Ampère operators, energy functionals, and uniqueness of Sasaki-extremal metrics

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics minimizing it. More generally, a relative version of the K-energy is convex, and bounded below if there exists a Sasaki-extremal metric, providing an important necessary condition for Sasaki-extremal metrics. Another application is a proof of the uniqueness of Sasaki-extremal metr…

A duality theorem for Riemannian foliations in nonnegative sectional curvature

Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.

Cohomological tautness for Riemannian foliations

In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold.

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.

Spectral geometry of Riemannian Legendre foliations

We obtain geometric characterizations of isospectral minimal Riemannian Legendre foliations on compact Sasakian manifolds of constant $ϕ$-sectional curvature.

Singular Riemannian Foliations on Nonpositively Curved Manifolds

We prove the nonexistence of a proper singular Riemannian foliation admitting section in compact manifolds of nonpositive curvature. Then we give a global description of proper singular Riemannian foliations admitting sections on Hadamard manifolds. In addition by using the theory of taut immersions we provide a short proof of this result in the special case of a polar action.

Parallel Focal Structure and Singular Riemannian Foliations

We prove that a submanifold with parallel focal structure, which is a generalization of isoparametric and equifocal submanifolds, induces a singular Riemannian foliation of the ambient space by its parallel and focal manifolds.

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…

Riemannian foliations of projective space admitting complex leaves

Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any open subset of $\mathbb{P}^n$ of codimension one. As a consequence there is no Riemannian foliation of the projective plane by Riemann surfaces, even locally. We determine how a complex submanifold may arise as an exceptional leaf of a non-trivial singular Riemannian foliation of maximal dimension. Gray's tube formula is applied to obtain a volu…

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 …

Riemannian foliations on contractible manifolds

We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions we prove that the foliation is also simple.

On closed geodesics in the leaf spaces of singular Riemannian foliations

In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

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。

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.

Riemannian foliations with parallel or harmonic basic forms

In this paper, we consider a Riemannian foliation whose normal bundle carries a parallel or harmonic basic form. We estimate the norm of the O'Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.

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.

Rigidity of Riemannian foliations with complex leaves on Kaehler manifolds

We study Riemannian foliations with complex leaves on Kaehler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.

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。

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.

Geometric resolution of singular Riemannian foliations

We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.

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。

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

The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of an almost quaternion Kaehler product manifold.

Pseudo-Riemannian geodesic foliations by circles

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an $S^1$-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by $S^1\times \R$. It follows that every Lorentzian surface contains a non-closed geodesic.

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 algebras and new singular Riemannian foliations in spheres

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.

Transverse LS-Category for Riemannian Foliations

We study the transverse Lusternik-Schnirelmann category of a Riemannian foliation on a compact manifold. We obtain a necessary and sufficient condition when the transverse LS category is finite. We also introduce a variation on the concept of transverse LS category, the essential transverse category, and show that this is finite for every Riemannian foliation and coincides with the transverse category if the latter is finite. Moreover we prove that the essential transverse category is a lower bound for the number of critical leaf closures of a basic differentiable function on M.

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.

Structure des feuilletages kähleriens en courbure semi-négative

This paper is concerned with (transversally) Kähler foliations. We proved here that the holonomy pseudo-group's lie algebra is semi-simple unde negativity assumptions of the Ricci tensor.

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.

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.

Traces of heat operators on Riemannian foliations

We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace of this operator has a particular short time asymptotic expansion. The coefficients in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also, we explicitly calculate the first two nontrivial coeff…