Examining Pseudo-Hermitian Transformations of the Heisenberg Group and the Sphere Through Integral Geometry

Abstract

This paper first examines Pseudo-Hermitian transformations of the Heisenberg Group, with much influence from Yen-Chang Huang's articles ``The Kinesmatic Formula in the 3D-Heisenberg Group", ``Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group", and ``An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group". This paper proves Theorem 1 of the ``The Kinesmatic Formula in the 3D-Heisenberg Group", which may be stated as follows: Let $D$ be a convex domain with boundary $\Sigma\equiv\partial D$ in $H_{1}$. Let $\gamma(s)$ be an oriented horizontal line $G_{p,\theta,t}$ that intersects $D$ by the length of chord $\sigma$ with respect to the Levi metric. Then \begin{equation} \int_{\forall G, G\cap D\neq\emptyset}\sigma dG=2\pi V(D) \end{equation} where $dG=dp\wedge d\theta\wedge dt$ is the density of the horizontal lines and $V(D)$ is the Lebesque volume of $D$ in $\mathds{R}^{3}$. This paper discusses some of the proof's finer details. The second part of this paper outlines the analogy of Theorem 1 on the complex sphere instead of in the Heisenberg Group. This is done by taking the particular frame of the sphere that corresponds to the frame used to prove Theorem 1 in the Heisenberg Group. Then, this paper describes how every horizontal line in the Heisenberg Group corresponds to a line in the upper half of the complex sphere, showing that properties of horizontal lines in the Heisenberg Group, (specifically, Theorem 1,) can apply in the sphere. Finally, this paper outlines what Theorem 1 would look like on the sphere.