The relative cohomology ${\mathrm{H}}_{\mathrm{diff}}^1(𝕂\left(1\right|3),\mathrm{𝔬𝔰𝔭}(2,3);{𝒟}_{\lambda ,\mu }\left(S^{1|3}\right))$ of the contact Lie superalgebra $𝕂\left(1\right|3)$ with coefficients in the space of differential operators ${𝒟}_{\lambda ,\mu }\left(S^{1|3}\right)$ acting on tensor densities on $S^{1|3}$, is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s\left(X_f\right)=D_1{D}_2{D}_3\left(f\right){\alpha }_3^{1/2}$, $X_f\in 𝕂\left(1\right|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathrm{𝔬𝔰𝔭}(2,3)$ of $𝕂\left(1\right|3)$. In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group...

We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.

This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification computes the analytic torsion explicitly in terms of Seifert data.

In the present paper we have obtained the necessary condition for the existence of almost pseudo symmetric and almost pseudo Ricci symmetric Sasakian manifold admitting a type of quarter symmetric metric connection.

To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact...

We construct the CR invariant canonical contact form $can\left(J\right)$ on scalar positive spherical CR manifold $(M,J)$, which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold $\Omega \left(\Gamma \right)/\Gamma$, where $\Gamma$ is a convex cocompact subgroup of ${Aut}_{CR}{\text{S}}^{2n+1}=PU(n+1,1)$ and $\Omega \left(\Gamma \right)$ is the discontinuity domain of $\Gamma$. This contact form can be used to prove that $\Omega \left(\Gamma \right)/\Gamma$ is scalar positive (respectively, scalar negative, or scalar vanishing) if and...

In this paper we have studied conformal curvature tensor, conharmonic curvature tensor, projective curvature tensor in Lorentzian $\alpha$-Sasakian manifolds admitting conformal Ricci soliton. We have found that a Weyl conformally semi symmetric Lorentzian $\alpha$-Sasakian manifold admitting conformal Ricci soliton is $\eta$-Einstein manifold. We have also studied conharmonically Ricci symmetric Lorentzian $\alpha$-Sasakian manifold admitting conformal Ricci soliton. Similarly we have proved that a Lorentzian $\alpha$-Sasakian...

The paper gives an account of the recent development in 3-dimensional contact geometry. The central result of the paper states that there exists a unique tight contact structure on $S^3$. Together with the earlier classification of overtwisted contact structures on 3-manifolds this result completes the classification of contact structures on $S^3$.

We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables), viewed as hypersurfaces $ℰ$ in the Lagrangian Grassmann bundle $M^{\left(1\right)}$ over a $(2n+1)$-dimensional contact manifold $(M,𝒞)$. We develop the theory of characteristics of $ℰ$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $ℰ$. After specializing such results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of...