On the slit tangent manifold of a Finsler manifold M are given the vertical and the Liouville foliations. In this paper we define some new types of vertical forms with respect to the Liouville foliation on TM^0. We define a cohomology group of TM^0 using these new forms. We prove a de Rham type theorem.
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}$ is a singular Riemannian foliation on the Riemannian manifold $(M,\mathtt{h} )$. As a direct consequence we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind $W$ is an infinitesimal homothety of $\mathtt{h}$ …
We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the canonical variation of the metric.
On k-nullity foliations in Finsler geometry and completeness
这里，考虑Finsler流形（M，F）与相应的曲率张量，视为2-形式的非零切向量的丛。在M的切空间中引入了由曲率决定的子空间，称之为曲率算子的k-零叶理。证明了当叶理维数为常数时，分布是对合的，且每个极大积分流形是全测地线的。给出了k-零叶理的特征，以及关于旗曲率的恒定性和它们的积分流形的完备性的一些结果，给出了（M，F）的完备性.引入的k-零空间是黎曼几何中零空间的自然扩张，由S. S. Chern和N. H.柯伊伯和扩大到芬斯勒设置H。阿克巴尔-扎德，并包含它作为一个特殊的情况。
A relevant property of equifocal submanifolds is that their parallel sets are still immersed submanifolds, which makes them a natural generalization of the so-called isoparametric submanifolds. In this paper, we prove that the regular fibers of an analytic map $π:M^{m+k}\to B^{k}$ are equifocal whenever $M^{m+k}$ is endowed with a complete Finsler metric and there is a restriction of $π$ which is a Finsler submersion for a certain Finsler metric on the image. In addition, we prove that when the fibers provide a singular foliation on $M^{m+k}$, then this foliation is Finsler.
We compute the de Rham cohomology of the weak stable foliation of the geodesic flow of a connected orientable closed hyperbolic surface with various coefficients. For most of the coefficients, we also give certain "Hodge decompositions" of the corresponding de Rham complexes, which are not obtained by the usual Hodge theory of foliations. These results are based on unitary representation theory of $\mathrm{PSL}(2,\mathbb{R})$. As an application we obtain an answer to a problem considered by Haefliger and Li around 1980.
A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal F$, and that of the basic differential forms on $M$. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases: when $\mathcal F$ is a regular foliation, when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of $M$, and, as a special case of the l…
We study extrinsic geometry of a codimension-one foliation ${\cal F}$ of a closed Finsler space $(M,F)$, in particular, of a Randers space $(M,α+β)$. Using a unit vector field $ν$ orthogonal (in the Finsler sense) to the leaves of ${\cal F}$ we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(α,β)$. For that $g$ we derive several geometric invariants of ${\cal F}$ (e.g. the Riemann curvature and the shape operator) in terms of $F$, then under natural assumptions on $β$ which simplify derivations, we express them in terms of corresponding invariants arising …
The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the Earth can be in principle reconstructed from single scattering measurements at the surface.
It is shown that the characteristic classes of foliations that were defined by Losik and that take values in the de~Rham cohomology of the space of infinite order frames over the leaf space may be mapped to the characteristic classes with values in the Čech-de~Rham cohomology of the leaf space studied in details by Crainic and Moerdijk. This map is in general non-injective. All constructions are done using Losik's approach to Gelfand formal geometry. A similar result is obtained for the exotic characteristic classes as well as for the group actions of the diffeomorphisms. As illustrating examp…
A similarity structure on a connected manifold M is a Riemannian metric on its universal cover such that the fundamental group of M acts by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham decomposition with at most two factors, one of which is Euclidean. Very recently, after Belgun and Moroianu conjectured that the number of factors was at most one, Matveev and Nikolayevsky found an example with two factors. When the non-flat factor has dimension 2, we give a complete classification of the examples with two factors. In greater dimensions, we make t…
This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham's theory of currents, and Sullivan's theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. N…
We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context.
This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties of PMCTs, which already show that they are the analogues of compact symplectic manifolds, thus placing them in a prominent position among all Poisson manifolds. For instance, their Poisson cohomology behaves very much like the de Rham cohomology of compact symplectic manifolds (Hodge decomposition, non-degener…
