arXiv research

Optimal Reeb graphs identified for polygon decomposition.

Characterizes smooth functions on manifolds with simple Reeb spaces.

Paper connects dynamics of mechanical systems to Reeb dynamics.

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is cl…

Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of $Reeb(f,X)$ be all homologically critical points where any homology of the corresponding c…

Uniform foliations with Reeb components on 3-manifolds.

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it pr…

New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.

These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers…

Geodesics spiral around Reeb orbits in 3D contact manifolds.

New problems defined for maps onto graphs related to Reeb spaces.

We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

The paper explores the structure of Reeb spaces for smooth functions on manifolds.

The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.

The study estimates Reeb chords using sheaf theory and persistence.

Study of Poincaré-Reeb graphs for algebraic domains.

New method realizes planar graphs as Reeb graphs of algebraic functions.

The study finds a special type of smooth function on connected sums of manifolds.

Study Morse functions on projective plane using Reeb graphs.

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mil…

Reeb flow made transverse to foliations without invariant measures.

We prove a version the local Reeb-Thurston stability theorem for symplectic foliations.

The paper examines a modified Godbillon-Vey class for Reeb foliations and finds it non-trivial for some foliations.

We review the standard Hopf construction of Reeb components with leafwise complex structure and almost determine the group of leafwise holomorphic smooth automorphisms for Reeb components of certain type in the case of complex leaf dimension one. In particular, it contains an infinite dimensional vector space.

Study of special subgroups of automorphism groups of Kronrod-Reeb graphs for Morse functions on 2-torus.

The study of Reeb dynamics on contact manifolds without periodic orbits.

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold $M$. Along the way, we show that the Reeb number $\mathcal{R}(M)$, i.e. the maximum cycle rank among all Reeb graphs of…

We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a c…

The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.

New example disproves complex contact theory for fat distributions with Reeb directions.

Smooth manifolds have functions with exactly two critical values.

We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $Γ$ admitting the so called good orientation there exis…

The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.

Constructing a conformal product structure on $S^3$ using the Reeb foliation.

The paper solves graph realization problems for Reeb graphs of Morse functions.

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

Invites study of contact structures and Reeb flows dynamics.

In this paper we study real hypersurfaces in the complex quadric space $Q^m$ whose structure Jacobi operator commutes with their structure tensor field. We show that the Reeb curvature $α$ of such hypersurfaces is constant and if $α$ is non-zero then the hypersurface is a tube around a totally geodesic submanifold $\ma…

Complex foliation cohomology is shown to be simple.

We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $φ$. For the normal case, we prove that a $φ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $φ$-invariant submanifold $N$ everyw…

Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

Study shows how to realize Ricci curvature as Reeb vector field for contact 3-manifolds.

The paper tightens bounds on distances between Reeb graphs.

Proves existence of elliptic Reeb orbit on real projective 3-space using ECH.

New proof for surface groups using Reeb graphs and Morse functions.

The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.