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...

For the first time in dimension 9, the Goursat distributions are not locally smoothly classified by their small growth vector at a point. As shown in [M1], in dimension 9 of the underlying manifold 93 different local behaviours are possible and four irregular pairs of them have coinciding small growth vectors. In the present paper we distinguish geometrically objects in three of those pairs. Smooth functions in three variables - contact hamiltonians in the terminology of Arnold, [A] - help to do...

Quantization together with quantum dynamics can be simultaneously formulated as the problem of finding an appropriate flat connection on a Hilbert bundle over a contact manifold. Contact geometry treats time, generalized positions and momenta as points on an underlying phase-spacetime and reduces classical mechanics to contact topology. Contact quantization describes quantum dynamics in terms of parallel transport for a flat connection; the ultimate goal being to also handle quantum systems in terms...