In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.

We prove extremality results concerning specific quasiconformal mappings in the hyperbolic plane, namely, hyperbolic spiral and stretch maps.

We consider the Tanaka-Webster geometry of surfaces embedded in a 3-dimensional Lie group with a CR structure inherited by a contact form. We define the notions of Gauss and mean curvature and give specific examples.

Utilizing the framework of quaternionic contact geometry, we define a sequence of Riemannian metrics $\{g_L\}$ on the quaternionic Heisenberg group $\mathfrak{H}_{\mathbb{H}}$ by rescaling the vertical directions. By analyzing the limit of this sequence, we characterize the Carnot-Carathéodory geodesics and provide the explicit description of the Carnot-Carathéodory distance and spheres in $\mathfrak{H}_{\mathbb{H}}$ . Furthermore, we derive a general formula for the horizontal mean curvature of hypersurfaces.

We define linear and radial stretch maps in the affine-additive group, and prove that they are minimizers of the mean quasiconformal distortion functional. For the proofs we use a method based on the notion of modulus of a curve family and the minimal stretching property (MSP) of the afore-mentioned maps. MSP relies on certain given curve families compatible with the respective geometric settings of the stretch maps.

We calculate the modulus of curve families inside a hyperbolic quadrilateral and a hyperbolic annulus.

We consider the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group in contrast to the fact that both of these groups are globally contactomorphic to the affine-additive group. Moreover, each quasiregular map, from the Heisenberg group to the affine-additive group must be constant.

In this paper, we endow the right half plane with warped product metrics. Of our interest are two of those metrics: they have zero and unbounded negative sectional curvature, respectively, and both of them are not complete. The group of holomorphic isometries of both metrics is isomorphic to the real additive group.

In this paper we describe the geodesics on the Kähler cone of the Heisenberg group. Furthermore we also prove that this is not a complete manifold.

We show that metric bisectors with respect to the Koranyi metric in the Heisenberg group are spinal spheres and vice versa. We also calculate explicitly their horizontal mean curvature.

Let ℌ be the first Heisenberg group equipped with the Korányi metric d . We prove that the equilateral dimension of ℌ is 4.

We study X-valued CR functions of Theodoresco class B-1 on the Heisenberg group H-1 = R x C equipped with the non-embeddable CR structure discovered by Nirenberg [On a question of Hans Lewy. Russian Math Surveys. 1974;29:251- 262] where X is an arbitrary complex Frechet space. If (L) over bar (phi) = phi partial derivative/partial derivative x is Nirenberg's perturbation of Lewy's operator (L) over bar =partial derivative/partial derivative w - i w partial derivative/partial derivative x, we show that for every open neighbourhood U subset of H-1 of the origin, no two solutions f(a) : U -> X, a is an element of {1, 2}, of Theodoresco class B-1, to the tangential Cauchy-Riemann equations (L) over bar (phi)f = 0 are functionally independent.

Let ℌ be the Heisenberg group. From the standard CR structure ℋ of ℌ we construct the complex hyperbolic structure of the Siegel domain. Additionally, using the same minimal data for ℌ, that is, its Sasakian structure, we provide the Siegel domain with yet another Kähler structure: this structure is of unbounded negative sectional curvature, and its complex structure does not commute with the standard complex structure. However, we show that those two Kähler structures are PCR Kähler equivalent, that is to say, essentially the same when restricted to ℋ.

. In this paper, we endow the right half plane with a metric with non constant negative curvature. We show that it is not a complete metric. The geodesics and the isometries of this metric are also studied here.

We show that an open subset $${\mathfrak {F}}_4''$$ of the $$\mathrm{PU}(2,1)$$ configuration space of four points in $$S^3$$ is in bijection with an open subset of $${\mathfrak {H}}^{\star }\times {\mathbb {R}}_{>0}$$ , where $${\mathfrak {H}}^\star $$ is the affine-rotational group. Since the latter is a Sasakian manifold, the cone $${\mathfrak {H}}^\star \times {\mathbb {R}}_{>0}$$ is Kähler and thus $${\mathfrak {F}}_4''$$ inherits this Kähler structure.

We show that an open subset 𝔉_4” of the PU(2,1) configuration space of four points in S^3 is in bijection with an open subset of ℌ^⋆×ℝ_>0 , where ℌ^⋆ is the affine-rotational group. Since the latter is a Sasakian manifold, the cone ℌ^⋆×ℝ_>0 is Kähler and thus 𝔉_4” inherits this Kähler structure.

In this paper, we endow the right half plane with warped product metrics. The group of holomorphic isometries of all such metrics is isomorphic to the real additive group. Of our interest are two of those metrics: they have zero and unbounded negative sectional curvature, respectively, and both of them are not complete.