Characteristic foliation of twisted Jacobi manifolds

The Twisted Higher Harmonic Signature for Foliations

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. Replaces The Higher Harmonic Signature for Foliations I: The Untwisted Case, and contains significant improvements.

正则 Jacobi 流形的 Godbillon-Vey 类

Godbillon-Vey classes of regular Jacobi manifolds

针对具有正则叶状结构的 Jacobi 流形，本文研究其 Godbillon-Vey 示性类，并利用 Jacobi 结构给出该类的显式表达。

On Existence of Regular Jacobi Structures

We prove $h$-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on $h$ principle of contact foliations in terms of the regular Jacobi structures.

Twisted submersions in nonnegative sectional curvature

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for nonnegatively curved Riemannian submersions even without the strict positive curvature condition, and irrespective of the particular metric.

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.

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。

Characteristic classes of foliations via SAYD-twisted cocycles

We have previously shown that the truncated Weil algebra of any Lie algebra is a Hopf-cyclic type complex with nontrivial coefficients. In this paper we apply this result to transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. Our result in codimension 1 matches with the only existing explicit computation done by Connes-Moscovici. In codimension 2 case, we carry out a constructive and explicit computation, by which we present the transverse fundamental class, the Godbillon-Vey class, and the other four residual cla…

On quasi-Jacobi and Jacobi-quasi bialgebroids

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissonization.

Vanishing theorem for transverse Dirac operators on Riemannian foliations

We obtain a vanishing theorem for the half-kernel of a transverse ${\rm Spin}\sp c$ Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a sufficiently large power of a line bundle, whose curvature vanishes along the leaves and is transversely non-degenerate at any point of the ambient manifold.

Extrinsic geometric flows on foliated manifolds, III

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second order parabolic PDEs, for which the existence/uniquenes and in some cases convergence of a solution are shown. Applications to the problem of prescribing mean curvature function of a codimensio…

Foliations with geometric structures

A foliation on a manifold M can be informally thought of as a partition of M into injectively immersed submanifolds, called leaves. In this thesis we study foliations whose leaves carry some specific geometric structures. The thesis consists of two parts. In the first part we classify foliations on open manifolds whose leaves are either locally conformal symplectic or contact manifolds. These foliations can be described by some higher geometric structures - namely the Poisson and the Jacobi structures. In the second part of the thesis, we consider foliations on open contact manifolds whose lea…

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study submanifolds with constant $r$th mean curvature $S_r$. We investigate, the stability of such submanifolds in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros - Sousa and Alías - Colares, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is a $r$th Jacobi field of a submanifold with $S_{r+1}$ constant. Finally, we study relations between $r$th Jacobi fields and vector fields preserving a foliation.

H-twisted Lie algebroids

We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form $H$ with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomology two via realization as Q-structure on graded manifolds. This classifies PQ3-manifolds.

T-Duality, Jacobi Forms and Witten Gerbe Modules

In this paper, we extend the T-duality Hori maps in [arXiv:hep-th/0306062], inducing isomorphisms of twisted cohomologies on T-dual circle bundles, to graded Hori maps and show that they induce isomorphisms of two-variable series of twisted cohomologies on the T-dual circle bundles, preserving Jacobi form properties. The composition of the graded Hori map with its dual equals the Euler operator. We also construct Witten gerbe modules arising from gerbe modules and show that their graded twisted Chern characters are Jacobi forms under an anomaly vanishing condition on gerbe modules, thereby giv…

带边叶状流形的基本（Dolbeault）上同调

Basic (Dolbeault) Cohomology of Foliated Manifolds with Boundary

本文为具基本边界流形上的Riemannian和Hermitian叶状结构发展了$L^2$理论，建立了基本上同调的分解、消灭、扭曲对偶与延拓定理，并证明了复类似物及研究了$\bar{\partial}_B$-方程的整体正则性。

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $κ_μ$ (relatively to a suitable riemannian metric $μ$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where $n = \codim \mathcal{F}$. By the Poincaré Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{κ_μ}}(M/\m…

Interface Foliation Near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ ε^2Δ_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where $ε$ is a small parameter. Let $\KK\subset \MM$ be an $(N-1)$-dimensional smooth minimal submanifold that separates $\MM$ into two disjoint components. Assume that $\KK$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that $|A_{\KK}|^2+\mbox{Ric}_{\tilde g}(ν_{\KK}, ν_{\KK})$ is positive along $\KK$. Then for each integer $m\geq 2$, we esta…

The Einstein-Hilbert type action on foliated pseudo-Riemannian manifolds

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

Lifting a Weak Poisson Bracket to the Algebra of Forms

We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian fu…

$b$-接触结构的存在性与分类

Existence and classification of $b$-contact structures

本文研究$b$-流形上的$b$-接触结构，证明三维过扭叶情形的存在性$h$-原理，并提出分类方法，完成临界曲面为二球面或未打结环面的$S^3$分类。

Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface $Σ$ in the cotangent bundle $τ:T^*M\to M$ of a closed manifold $M$ endowed with a twisted symplectic structure. We show that if the characteristic foliation of $Σ$ is Anosov, then a smooth 1-form $θ$ on $M$ is exact if and only $τ^*θ$ has zero integral over every closed characteristic of $Σ$. This result is derived from a related theorem about magnetic flows which generalizes our work in \cite{DP}. Other rigidity issues are also discussed.

Foliated manifolds, algebraic K-theory, and a secondary invariant

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $η$-invariants of Dirac operators and local correction terms. Initially, the construction of the element in $\mathbb{C}/\mathbb{Z}$ involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the …

Twisted Adiabatic Limit for Complex Structures

Given a complex manifold $X$ and a smooth positive function $η$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_η$ whose principal part is $η\partial + \bar\partial$. The role of the zero-th order part is to force the integrability property $D_η^2=0$ that leads to a cohomology isomorphic to the de Rham cohomology of $X$, while the components of types $(0,\,1)$ and $(1,\,0)$ of $D_η$ induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kod…