The fundamental theorem for hypersurfaces in Heisenberg groups

Abstract

We study the horizontally regular curves in the Heisenberg groups Hn. We show the fundamental theorem of curves in Hn (n ≥ 2) and define the orders of horizontally regular curves. We also show that the curve γ is of order k if and only if, up to a Heisenberg rigid motion, γ lies in H k but not in H k−1 ; moreover, two curves with the same order differ from a rigid motion if and only if they have the same invariants: p-curvatures and contact normality. Thus, combining with our previous work [1] we have completed the classification of horizontally regular curves in Hn for n ≥ 1.

The Fundamental and Rigidity Theorems for Pseudohermitian Submanifolds in the Heisenberg Groups

“…In this section, we give a brief review of Cartan's method of moving frame and Calculus on Lie groups. For the details, we refer the reader to [5]. Let (X,G) be a Klein geometry.…”

“…The contact bundle is ξ = kerΘ. We refer the reader to [2,3,5] for the details about the Heisenberg groups, and to [6,11,12,15,16] for pseudohermitian geometry.…”

“…where R= A −B B A ∈SO(2n). In [5], we showed that each symmetry Φ∈ PSH(n) has the unique decomposition Φ = L p •Φ R , for some p ∈ H n and R ∈ SO(2n). Since the action of PSH(n) on H n is transitive, the associated geometry is a kind of Klein geometry.…”

Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds

“…As a flat pseudohermitian manifold, the Heisenberg group plays an important role in pseudohermitian geometry. We refer the reader to [2] and [12] for the details about the Heisenberg group, and to [13], [19], [20] and [30] for pseudohermitian geometry. Denote by H n the Heisenberg group, which is the space R 2n+1 with coordinates (x β , y β , t) as a set.…”

A characterization of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$

“…We let P SH (1) be the group of Heisenberg rigid motions, that is, the group of all pseudohermitian transformations in H 1 . For details of this group, we refer readers to [4], which is the first published paper where the fundamental theorem in the Heisenberg groups has been studied. We say that two surfaces are congruent if they differ by an action of a Heisenberg rigid motion.…”