The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we use the abstract theory of filtered and graded Lie algebras. We also describe all differential and integral invariants of new Lie algebras found in the paper and discuss the infinite-dimensional case.
We study Lie algebras of generators of infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds. The almost-cosymplectic-contact structure admits on the sheaf of pairs of 1-forms and functions the structure of a Lie algebra. We describe Lie subalgebras in this Lie algebra given by pairs generating infinitesimal symmetries of basic tensor fields given by the almost-cosymplectic-contact structure.
In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riemannain manifold.
We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Sasaki structure. We also present a list of 25 classes of seven-dimensional nilpotent Lie groups admitting a $K$-contact structure, but not the pseudo-Sasaki structure. All the contact structures considered are central extensions of six-dime…
In this paper, left-invariant almost contact metric structures on three-dimensional non-unimodular Lie groups are investigated. It is proved that for every Riemannian Lie group, there is one of these structures. In addition, left-invariant normal almost contact metric structures on three dimensional non-unimodular Lie groups are classified.
Gromov的一个结果保证了任意连通非紧奇维李群上存在切触结构。但一般来说，这样的结构在李群的左平移下不是不变的。寻找哪些李群具有左不变切触结构（接触李群）的问题仍然是一个悬而未决的问题。而具有左不变辛结构的李群被许多作者广泛研究（其中A. Lichnerowicz; E.B. Vinberg; I.I. Pjateckii-Sapiro; S. G. Gindikin; A. Medina; Ph. Revoy; M.作者声明：J. Nakajima等），接触李群仍然是未被探索的。我们执行一个“接触化”的方法来构建，在每一个奇数维，许多接触李群与一个离散的中心，并讨论了这样的建设的一些应用和后果。我们在低维的分类结果。在任意大于或等于7的维数上，存在无穷多个局部非同构可解接触李群。我们还对具有某些特定黎曼或半黎曼结构的切触李群进行了分类和刻画，并得到了一些障碍结果。
In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with invariant metric and admissible almost complex structure, we show that the bracket annihilate…
In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to $SU(2,1)$.
In this paper, we show that the Lie superalgebra $\mathfrak{spo}(2l+2|n)$ is into the intersection of Lie superalgebra of contact vector fields $\mathcal{K}(2l+1|n)$ and the Lie superalgebra of projective vector fields $\mathfrak{pgl}(2l+2|n)$. We use mainly the embedding used by P. Mathonet and F. Radoux in "\textit{ Projectively equivariant quantizations over superspace $\mathbb{R}^{p|q}$. Lett. Math. Phys, 98: 311-331, 2011}". Explicitly, we use the embedding of a Lie superalgebra constituted of matrices belonging to $\gl(2l+2|n)$ into $\mathrm{Vect}(\R^{2l+1|n})$. We generalize thus in sup…
Ricci soliton contact metric manifolds with certain nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood, they provided a list of candidates for the nongradient case.These candidates can be realized as Lie groups, but one only knows the structures of the underlying Lie algebras, which are hard to be analyzed apart from the three-dimensional case. In this paper, we study these Lie groups with dimension greater than three, and prove that the connected, simply-connected, and complete ones can be realized as homogeneous real hypersurfaces …
Let $Ω$ and $Ω'$ be open subsets of a flat $(2,3,5)$-distribution. We show that a $C^1$-smooth contact mapping $f : Ω\to Ω'$ is a $C^\infty$-smooth contact mapping. Ultimately, this is a consequence of the rigidity of the associated stratified Lie group (the Tanaka prolongation of the Lie algebra is of finite-type). The conclusion is reached through a careful study of some differential identities satisfied by components of the Pansu-derivative of a $C^1$-smooth contact mapping.
We introduce the notion of abelian almost contact structures on an odd dimensional real Lie algebra $\mathfrak g$. This a sufficient condition for the structure to be normal. We investigate correspondences with even dimensional real Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak g$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce and study abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$. These are given by triple…
A first classification of para-Kähler structures on four-dimensional Lie algebras was obtained by D. Calvaruzo in 2015. In this paper, we propose another classification based on the classification of symplectic Lie algebras. For each four-dimensional symplectic Lie algebra, compatible para-complex structures and the corresponding pseudo-Riemannian metrics are found in explicit form. This leads to the classification of Sasaki para-contact structures on five-dimensional contact Lie algebras with a nontrivial center.
We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. We discuss, in addition, their relation with strictly contact vector fields in dimension three. Analogous results for diffeomorphisms are also given.
There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a seven-manifold. We present a natural geometric construction of a Lie contact structure on a seven-dimensional bundle over a five-manifold endowed with a $(2,3,5)$-distribution. For a class of distributions the induced Lie contact structure is constructed explicitly and we determine its symmetries. We further study the relation between the canonical normal Cartan con…
We consider locally homogeneous $CR$ manifolds and show that, under a condition only depending on their underlying contact structure, their $CR$ automorphisms form a finite dimensional Lie group.
We provide a classification of $ts$-invariant sub-Lorentzian structures on $3$ dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.
We study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.
A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M together with a contact distribution C and a bundle S of twisted cubics that is compatible with the conformal symplectic form on C. In Engel's classical work, the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry algebra of the most symmetric contact twisted cubic structure; we thus refer to this one as the contact Engel structure. In the present paper we equip the contact Engel structure with a smooth section s: M-> S that `marks' a point in each twisted cubic. We study the local geometry of the re…
In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators, it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.
In this paper we study contact structure on 2-step nilpotent, Heisenberg type Lie groups. We decompose this Lie groups to center and orthogonal complement, then investigate properties of both orthogonal Lie subgroups. Finally, we provide a connection between matchings in groups and field extensions and 2-step nilpotent Lie groups.
