Related papers: Interval topology in contact geometry
Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of…
It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian…
This paper is an overview of the idea of using contact geometry to construct invariants of immersions and embeddings. In particular, it discusses how to associate a contact manifold to any manifold and a Legendrian submanifold to an…
Interactions in complex systems are widely observed across various fields, drawing increased attention from researchers. In mathematics, efforts are made to develop various theories and methods for studying the interactions between spaces.…
This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood…
A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of…
We discuss the occurrence of some notions and results from contact topology in the non-equilibrium thermodynamics. This includes the Reeb chords and the partial order on the space of Legendrian submanifolds.
In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3-manifolds such as Heegaard Floer homology and embedded contact homology. These…
In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov…
Time has entered the domain of topological phases in the field of non-Hermitian physics. Previous studies have relied on periodic modulation in time to make an intuitive connection to established spatial topological invariants, albeit with…
We introduce a pseudo-metric on the contactomorphism group of any contact manifold $(M,\xi)$ with a cooriented contact structure $\xi$. It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of…
Basic concepts and definitions in differential geometry and topology which are important in the theory of solitons and instantons are reviewed. Many examples from soliton theory are discussed briefly, in order to highlight the application…
In this paper we survey several intersection and non-intersection phenomena appearing in the realm of symplectic topology. We discuss their implications and finally outline some new relations of the subject to algebraic geometry.
By introducing various topologies on the homotopy groups of a topological space, some researchers make these well known notions in algebraic topology more useful and powerful. In this paper, first we recall and review some known topologies…
In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the…
We survey what is known about various special types of submanifolds of contact manifolds and discuss their role in the development of contact geometry.
We study the global topology of the space $\mathcal L$ of loops of contactomorphisms of a non-orderable closed contact manifold $(M^{2n+1}, \alpha)$. We filter $\mathcal L$ by a quantitative measure of the ``positivity'' of the loops and…
We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.
We construct new unbounded invariant distances on the universal cover of certain Legendrian isotopy classes. This is the first instance where unboundedness of an invariant distance is obtained without assuming the existence of a positive…
I review few conceptual steps in analytic description of topological interactions, which constitute the basis of a new interdisciplinary branch in mathematical physics, "Statistical Topology", emerged at the edge of topology and statistical…