Related papers: Contact topology and CR geometry in three dimensio…
A regular contact manifold is a manifold $M$ equipped with a globally defined contact form $\eta$ such that the topological space $M/\mathcal{R}$ of orbits (trajectories) of the Reeb vector field $\mathcal{R}$ of $\eta$ carries a smooth…
In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian…
H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat…
In this paper, we consider the CPE conjecture in the frame-work of $K$-contact and $(\kappa, \mu)$-contact manifolds. First, we prove that if a complete $K$-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere…
A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Taubes has shown that ECH is isomorphic to a version of Seiberg-Witten Floer homology (and both are conjecturally isomorphic to a version of Heegaard…
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a…
In this note we show that a closed oriented contact manifold is obtained from the standard contact sphere of the same dimension by contact surgeries on isotropic and coisotropic spheres. In addition, we observe that all closed oriented…
Is the universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the universe is assumed to…
Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong…
We study generalized almost contact structures on odd-dimensional manifolds. We introduce a notion of integrability and show that the class of these structures is closed under symmetries of the Courant-Dorfman bracket, including T-duality.…
Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact…
This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological…
We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the…
A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real…
We give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. Specifically, we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo-holomorphic curves.
This work addresses a modification of the random geometric graph (RGG) model by considering a set of points uniformly and independently distributed on the surface of a $(d-1)$-sphere with radius $r$ in a $d-$dimensional Euclidean space,…
Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a…
For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are…