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Register a new userLeft-invariant CR structures on 3-dimensional Lie groups
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The systematic study of CR manifolds originated in two pioneering 1932 papers of Elie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on $mathrm{SU}_2 = S^3$, deforming the standard `spherical structure. In this paper, mostly expository, we illustrate and clarify Cartans results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that $mathrm{SL}_2(mathbb{R})$ admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of $mathrm{SL}_2(mathbb{R})$ provide CR embedding or immersions of these structures. The same methods apply to all other three-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for $mathrm{SU}_2$, the Heisenberg group, and the Euclidean group.
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A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to a left action of the 2-group $G$ on the Lie groupoid $G$, hence to an action of $G$ on the Lie 2-algebra $mathbb{X}(G)$. As a result we get the Lie 2-algebra $mathbb{X}(G)^G$ of left-invariant multiplicative vector fields. On the other hand there is a well-known construction that associates a Lie 2-algebra $mathfrak{g}$ to a Lie 2-group $G$: apply the functor $mathsf{Lie}: mathsf{Lie Groups} to mathsf{Lie Algebras}$ to the structure maps of the category $G$. We show that the Lie 2-algebra $mathfrak{g}$ is isomorphic to the Lie 2-algebra $mathbb{X}(G)^G$ of left invariant multiplicative vector fields.
We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n,R) which doesnt admit left invariant contact form, yet admits a contact form which is invariant by a maximal compact subgroup. We determine also Pfaffian forms on the Heisenberg $3$-dimensional group invariant by a subgroup and obtain the Transport Equation.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
We show that the compact quotient $Gammabackslashmathrm{G}$ of a seven-dimensional simply connected Lie group $mathrm{G}$ by a co-compact discrete subgroup $Gammasubsetmathrm{G}$ does not admit any exact $mathrm{G}_2$-structure which is induced by a left-invariant one on $mathrm{G}$.
The purpose of the present paper is to study the globally and locally $varphi $-${cal T}$-symmetric $left( varepsilon right) $-para Sasakian manifold in dimension $3$. The globally $varphi $-$ {cal T}$-symmetric $3$-dimensional $left( varepsilon right) $-para Sasakian manifold is either Einstein manifold or has a constant scalar curvature. The necessary and sufficient condition for Einstein manifold to be globally $varphi $-${cal T}$ -symmetric is given. A $3$-dimensional $% left( varepsilon right) $ -para Sasakian manifold is locally $varphi $-$ {cal T}$-symmetric if and only if the scalar curvature $r$ is constant. A $3 $-dimensional $left( varepsilon right) $-para Sasakian manifold with $% eta $-parallel Ricci tensor is locally $varphi $-${cal T}$-symmetric. In the last, an example of $3$-dimensional locally $varphi $-${cal T}$-symmetric $left( varepsilon right) $-para Sasakian manifold is given.
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