Related papers: Homogeneous integrable Legendrian contact structur…
Unit tangent bundles $UM$ of semi-Riemannian manifolds $M$ are shown to be examples of dynamical Legendrian contact structures, which were defined in recent work [25] of Sykes-Zelenko to generalize leaf spaces of 2-nondegenerate CR…
We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R^3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea…
We investigate the geometry of Legendrian complex projective manifolds $X\subset\PP V$. By definition, this means $V$ is a complex vector space of dimension $2n+2$, endowed with a symplectic form, and the affine tangent space to $X$ at each…
Bethe/Gauge correspondence as it is usually stated is ill-defined in five dimensions and needs a "non-perturbative" completion; a related problem also appears in three dimensions. It has been suggested that this problem, probably due to…
The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the…
A $n$-dimensional Lie algebra $g=(V,\mu)$ is called $2$-compatible if it is isomorphic to a quadratic deformation of a Lie algebra $g_0=(V,\mu_0)$. By quadratic deformation we means a formal deformation $\mu_t=\mu_0+t\varphi_1+t^2\varphi_2$…
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are…
Motivated by the problem of global stability of thermodynamical equilibria in non-equilibrium thermodynamics formulated in a recent paper [12], we introduce some mechanisms for constructing semi-infinite orbits of contact Hamiltonian…
We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable…
Odd exact Courant algebroids constitute a simple class of transitive Courant algebroids. Their underlying vector bundle is of odd rank and differs from a generalized tangent bundle by the addition of a line bundle. In this article we study…
A quantization over a manifold can be seen as a way to construct a differential operator with prescribed principal symbol. The quantization map is moreover required to be a linear bijection. It is known that there is in general no natural…
Suppose $K$ is a knot in a 3-manifold $Y$, and that $Y$ admits a pair of distinct contact structures. Assume that $K$ has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin…
We show that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a $C^0$-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the…
To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical…
We express the first jet bundle of curves in Euclidean space as homogeneous spaces associated to a Galilean-type group. Certain Cartan connections on a manifold with values in the Lie algebra of the Galilean group are characterized as…
In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,\xi)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from…
M5-branes on an ADE singularity are described by certain six-dimensional "conformal matter" superconformal field theories. Their Higgs moduli spaces contain information about various dynamical processes for the M5s; however, they are not…
We investigate a family of integrable Hamiltonian systems on Lie-Poisson spaces $\mathcal{L}_+(5)$ dual to Lie algebras $\mathfrak{so}_{\lambda, \alpha}(5)$ being two-parameter deformations of $\mathfrak{so}(5)$. We integrate corresponding…
Ozsvath and Stipsicz showed that the LOSS invariant is natural under +1 contact surgery. We extend their result and prove the naturality of the LOSS invariant of a Legendrian L under any positive integer contact surgery along another…
The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'''=f(x,u,u',u'')$ which admits the maximal ten-dimensional contact symmetry Lie algebra. The…