arXiv research

Proves conjecture about foliations on curved spaces.

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.

Geometric framework for inverse problems using foliations and dual connections.

A formula calculates the Euler class of foliations using dual graphs.

The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree $d$ and we describe some explicit examples. These results allow us to show that…

Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.

A global twistor correspondence is established for neutral self-dual conformal structures with alpha-surface foliation when the structure is close to the standard structure on S^2 times S^2. We need to introduce some singularity for the alpha-surface foliation such that the leaves intersect on a fixed two sphere. In th…

Totally geodesic dual leaves on curved manifolds are also curved.

We introduce the foliated anti-self dual equation for higher dimensional smooth manifolds with codimension-4 Riemannian foliations. Several fundamental results are established, towards the defining of a Donaldson type invariant for such foliations.

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…

The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.

Paper studies statistical manifolds with logarithmic divergences.

Haefliger cohomology characterizes taut foliated manifolds by Haefliger's theorem. We show that Haefliger cohomology characterizes strongly tense foliated manifolds, namely, foliated manifolds which admit a Riemannian metric such that the mean curvature form of the leaves is closed and basic. We show that Haefliger coh…

We study transversely Lorentzian foliations on the closed 3-manifolds. We classify them under a completeness hypothesis and we deduce the dual classification of codimension 1 geodesically complete timelike totally geodesic foliations. Besides we provide an example of a Lorentzian foliation on a compact 3-manifold which…

Investigates dual foliations of polygon spaces based on area and perimeter.

The study establishes a criterion for the holomorphy of curvature in smooth webs and applies it to dual webs of homogeneous foliations.

Paper classifies fibers of fat Riemannian submersions with non-negative curvature.

Characterizes elliptic operators on singular foliations.

The Whitehead link exterior lacks most Euler class taut foliations.

If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discu…

Benedetti and Guadagnini have conjectured that the marked lenght spectrum of the constant mean curvature foliation $M_τ$ in a 2+1 dimensional flat spacetime $V$ with compact hyperbolic Cauchy surfaces converges, in the direction of the singularity, to that of the marked measure spectrum of the R-tree dual to the measur…

Dual pairs constructed for volume preserving diffeomorphisms using symplectic geometry.

Study simplicial volume via foliated simplices and duality.

Let $(M,ω)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion forms vanish identically. uppose that $L_0$ is a compact leaf which Weinstein connection …

A holomorphic foliation on $\mathbb P^2_{\mathbb C}$, or a real analytic foliation on $\mathbb{P}^{2}_{\mathbb{R}},$ is said to be convex if its leaves other than straight lines have no inflection points. The classification of the convex foliations of degree $2$ on $\mathbb P^2_{\mathbb C}$ has been established in $201…

We consider control-linear left-invariant time-optimal problems on step 2 Carnot groups with strictly convex set of control parameters (in particular, sub-Finsler problems). We describe all linear-in-momenta Casimirs on the dual of the Lie algebra. In the case of rank 3 Lie groups we describe the symplectic foliation o…

We determine explicitly the foliated cohomology $H_{\cal F}^\ast (M)$ of the affine Reeb flow ${\cal F}$ on the Hopf manifold ${\Bbb S}^n\times {\Bbb S}^1$. The vector space $H_{\cal F}^1(M)$ contains exactly the obstructions to solve the cohomological equation $X\cdot f=g$ where $f$ and $g$ are $C^\infty$-functions a…

The paper studies homeotopy groups of leaf spaces for specific foliations.

Affine manifolds linked to integrable equations and geometric structures.

Using twistor methods, we explicitly construct all local forms of four--dimensional real analytic neutral signature anti--self--dual conformal structures $(M,[g])$ with a null conformal Killing vector. We show that $M$ is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural pr…

The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.

The Gronwall conjecture states that a planar 3-web of foliations which admits more than one distinct linearizations is locally equivalent to an algebraic web. We propose an analogue of the Gronwall conjecture for the 3-web of foliations by Legendrian curves in a contact three manifold. The Legendrian Gronwall conjectur…

We investigate various structures associated with the hyperbolic Markov and homological spectra of a pseudoAnosov map $φ$ on a surface. Each unstable eigenvalue of the action of $φ$ on first cohomolgy yields an eigen-cocycle that is transverse and holonomy invariant to the stable foliation $\mathcal{F}^s$ of $φ$. Each …

For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that corresp…

The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.

We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining an interesting class of symplectic groupoids coming from their symplectic foliati…

A trace formula for foliated flows on closed manifolds.

We construct rigid supersymmetric gauge theories on Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of a six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals. …

The paper studies how hyperbolic surfaces degenerate along harmonic map rays.

The present paper unifies some aspects concerning the vertical Liouville distributions on the tangent (cotangent) bundle of a Finsler (Cartan) space in the context of generalized geometry. More exactly, we consider the big-tangent manifold $\mathcal{T}M$ associated to a Finsler space $(M,F)$ and of its $\mathcal{L}$-du…

Let $Γ$ be a surface group of higher genus. Let $ρ\_0: Γ\to {PGL}(V)$ be a discrete faithful representation with image contained in the natural embedding of ${SL}(2, {\mathbb R})$ in ${PGL}(3, {\mathbb R})$ as a group preserving a point and a disjoint projective line in the projective plane. We prove that such a repres…

We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual ($α$-planes) or anti-self-dual ($β$-planes) and so we consider $α$-surfaces and $β$-surfaces. The metric of the examples we study, which include the spaces of oriente…

Study of Ricci flow equations in topological quantum gravity.

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit red…

We study the tangential Poisson cohomology (TP-cohomology) of regular Poisson manifolds, first defined by Lichnerowicz using contravariant tensor fields. We show that for a regular Poisson manifold M, the TP-cohomology coincides with the leafwise de Rham (or Cech) cohomology of the symplectic foliation of M. Its comput…

We give a proof, using harmonic maps from disks to real trees, of Skora's theorem (Morgan-Otal (1993), Skora (1990), originally conjectured by Shalen): if G is the fundamental group of a surface of genus at least 2, then any small minimal G-action on a real tree is dual to the lift of a measured foliation. Analytic too…

The paper disproves a conjecture about 3D manifolds using even lattice points.

The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces…