Noncommutative geometry of foliations

Noncommutative Hamiltonian dynamics on foliated manifolds

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.

Noncommutative geometry and lower dimensional volumes in Riemannian geometry

In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes don't involve noncommutative geometry or spin structures at all.

CR geometry and conformal foliations

We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.

Noncommutative residue, conformal invariants and lower dimensional volumes in Riemannian geometry

The results of this paper are outdated. Finer versions of them will appear elsewhere.

On the geometry of warped foliations

We discuss the geometry of warped foliations. After examining the Levi-Civita connection, we describe the formulae for sectional, Ricci and scalar curvatures. In the final part of this note, we present some examples.

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.

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.

Towards Noncommutative Topological Quantum Field Theory: New invariants for 3-manifolds

We define some new invariants for 3-manifolds using the space of taut codim-1 foliations along with various techniques from noncommutative geometry. These invariants originate from our attempt to generalise Topological Quantum Field Theories in the Noncommutative geometry / topology realm.

Transversely Hessian foliations and information geometry

A family of probability distributions parametrized by an open domain $Λ$ in $R^n$ defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannian metric on $Λ$. If we replace the "positive definite" assumption by the existence of a suitable torsion-free connection, a foliation with a transversely Hessian structure appears naturally. In the paper we develop the study of transversely Hessian foliations in view of applicati…

The geometry of the space of leaf closures of a transversely almost Kähler foliation

We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space.

Towards Noncommutative Topological Quantum Field Theory: Tangential Hodge-Witten cohomology

We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian

Geometry of Kaehler metrics and holomorphic foliation by discs

The purpose of this paper is to establish a partial regularity theory on certain homogeneous complex Monge-Ampere equations. As consequences of this new theory, we prove the uniqueness of extremal Kaehler metrics and give an necessary condition for existence of extremal Kaehler metrics.

Analysis on Riemannian foliations of bounded geometry

A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.

紧复环面上的湍流全纯叶状结构与横截全纯Cartan几何

Turbulent holomorphic foliations on compact complex tori and transversely holomorphic Cartan geometry

本研究定义了紧复环面上推广Ghys余维一情形的高余维非奇异全纯湍流叶状结构，证明其所有横截全纯Cartan几何均平坦，并给出了该几何的唯一性结果。

Geometry of Kähler Metrics and Foliations by Holomorphic Discs

The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, we will prove the uniqueness of extremal Kähler metrics and give an necessary condition for existence of extremal Kähler metrics. Further applications will be discussed in our forthcoming papers.

微扰Schwarzschild时空中常质量面函数叶化的全局存在性与几何性质

Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime

本文证明了真空微扰Schwarzschild时空中常质量面函数叶化的全局存在性，并通过与球对称叶化比较研究其几何性质，为应用零Penrose不等式奠定核心基础。

Classical and quantum dynamics in transverse geometry of Riemannian foliations

First, we survey some results on classical and quantum dynamical systems associated with transverse Dirac operators on Riemannian foliations. Then we illustrate these results by two examples of Riemannian foliations: a foliation given by the fibers of a fibration and a linear foliation on the two-dimensional torus.

On k-nullity foliations in Finsler geometry and completeness

Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant then the distribution is involutive and each maximal integral manifold is totally geodesic. Characterization of the k-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral…

The Geometry of K-Orbits of a Subclass of MD5-Groups and Foliations Formed by their Generic K-Orbits

The present paper is a continuation of Le Anh Vu's ones [13], [14], [15]. Specifically, the paper is concerned with the subclass of connected and simply connected MD5-groups such that their MD5-algebras $\mathcal{G}$ have the derived ideal ${\mathcal{G}}^{1} : = [ \mathcal{G},\mathcal{G} ]\equiv$ ${\bf{R}}^{3}$. We shall describe the geometry of K-orbits of these MD5-groups. The foliations formed by K-orbits of maximal dimension of these MD5-groups and their measurability are also presented in the paper.

Riemannian foliations of bounded geometry

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will given in a forthcoming paper.

Nearly Kaehler geometry and Riemannian foliations

We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a …

洛伦兹叶状结构的横向几何及其在洛伦兹轨形中的应用

Transverse Geometry of Lorentzian foliations with applications to Lorentzian orbifolds

该研究在洛伦兹叶状结构背景下证明横向直径定理，引入叶空间新因果结构，将结果应用于洛伦兹轨形并推导其类时直径定理。

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.

Local Invariants and Geometry of the sub-Laplacian on H-type Foliations

$H$-type foliations $(\mathbb{M},\mathcal{H},g_{\mathcal{H}})$ are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping $\mathbb{M}$ with the Bott connection we consider the scalar horizontal curvature $κ_{\mathcal{H}}$ as well as a new local invariant $τ_{\mathcal{V}}$ induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in su…

Omniscient foliations and the geometry of cosmological spacetimes

We identify certain general geometric conditions on a foliation of a spacetime (M,g) by timelike curves that will impede the existence of null geodesic lines, especially if (M,g) possesses a compact Cauchy hypersurface. The absence of such lines, in turn, yields well-known restrictions on the geometry of cosmological spacetimes, in the context of Bartnik's splitting conjecture. Since the (non)existence of null lines is actually a conformally invariant property, such conditions only need to apply for some suitable conformal rescaling of g.

On the Noncommutative Geometry of the Endomorphism Algebra of a Vector Bundle

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.

Noncommutative Symplectic Geometry of the Endomorphism Algebra of a Vector Bundle

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a vector bundle, and give the full description of the family of Poisson structures for this algebra.

等参数流形与有界几何

Isoparametric foliations and bounded geometry

本文证明了在固定维度n≠5且有限基本群的情况下，仅有有限多个等参数流形具有有界几何，同时构造了多个互不等参数同胚的等参数流形家族。

对称泊松几何、完全测地叶分解与雅可比-乔丹代数

Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras

本文研究了对称泊松结构与雅可比-乔丹代数的关系，揭示了其在几何和动力学中的应用。

Riemannian geometry of noncommutative super surfaces

In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the connections are metric-compatible and have zero torsion when the super metric is symmetric, giving rise to the corresponding super Riemann curvature. The latter also satisfies the noncommutative super analogue of the first and second Bianchi identities. We also give some examples and study them in details.

Isometry groups and geodesic foliations of Lorentz manifolds. Part II: Geometry of analytic Lorentz manifolds with large isometry groups

This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; that is, they are (at least locally) warped products of constant curvature Lorentz manifolds by Riemannian manifolds.

Cartan Spaces and Natural Foliations on the Cotangent Bundle

In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M,K) are studied. It is shown that the geometry of these foliations is closely related to the geometry of the Cartan space (M,K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.

A h-principle for symplectic foliations

We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a $h$-principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spirit of Haefliger's criterion for homotoping a distribution to a foliation. We give an example to show that this criterion is not too unsharp.

数值平坦叶状结构与全纯泊松几何

Numerically flat foliations and holomorphic Poisson geometry

本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。

Jets and connections in commutative and noncommutative geometry

It is emphasized that equivalent definitions of connections on modules over commutative rings are not so in noncommutative geometry.

The index of operators on foliated bundles

We compute the equivariant cohomology Chern character of the index of elliptic operators along the leaves of the foliation of a flat bundle. The proof is based on the study of certain algebras of pseudodifferential operators and uses techniques for analizing noncommutative algebras similar to those developed in Algebraic Topology, but in the framework of cyclic cohomology and noncommutative geometry.

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an …

Basic automorphism group of complete Cartan foliations covered by fibration

We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension of this group depending on the transverse geometry are found. Further, we investigate Cartan foliations covered by fibrations. When the global holonomy group of that foliation is discrete, we obtain the explicit new formula for determining its basic automorphism Lie group. Example…

Noncommutative integrability and action-angle variables in contact geometry

We introduce a notion of the noncommutative integrability within a framework of contact geometry.

横截仿射叶状结构的堆叠几何与对数拓扑

Stacky geometry and logarithmic topology of transversely affine foliations

研究横截仿射叶状结构关联的堆叠与对数拓扑结构，通过holonomy群与商堆叠给出乘性发展坐标的几何基础，并利用Kato-Nakayama空间刻画边界线性动力学与对数提升。

SU(n)-Connections and Noncommutative Differential Geometry

We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.

Submanifolds and Quotient Manifolds in Noncommutative Geometry

We define and study noncommutative generalizations of submanifolds and quotient manifolds, for the derivation-based differential calculus introduced by M.~Dubois-Violette and P.~Michor. We give examples to illustrate these definitions.

On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.

Twistor Geometry of Null Foliations in Complex Euclidean Space

We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface $\mathcal{Q}^n$ of dimension $n \geq 3$, and its twistor space $\mathbb{PT}$, defined to be the space of all linear subspaces of maximal dimension of $\mathcal{Q}^n$. Viewing complex Euclidean space $\mathbb{CE}^n$ as a dense open subset of $\mathcal{Q}^n$, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on $\mathbb{CE}^n$ can be constructed in terms of complex submanifolds of $\mathbb{PT}$. The construction is ill…

Noncommutative Geometry and Gravity

We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.

Some aspects of noncommutative differential geometry

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework.

Gauge theories in noncommutative geometry

In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features.

Necklace Lie algebras and noncommutative symplectic geometry

Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg.

On noncommutative geometry of the Standard Model: fermion multiplet as internal forms

We unveil the geometric nature of the multiplet of fundamental fermions in the Standard Model of fundamental particles as a noncommutative analogue of de Rham forms on the internal finite quantum space.