An Application of the Moving Frame Method to Integral Geometry in the Heisenberg Group

Abstract

We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.

References in corpus (5)

• The fundamental theorems for curves and surfaces in 3d Heisenberg group
• The kinematic formula in the 3D-Heisenberg group
• The fundamental theorem of curves and classifications in the Heisenberg groups
• A Codazzi-like equation and the singular set for $C^1$ smooth surfaces in the Heisenberg group
• The topology of a subspace of the Legendrian curves in a closed contact 3-manifold

Cited by in corpus (5)

• Tubular neighborhoods in the sub-Riemannian Heisenberg groups
• The kinematic formula in the 3D-Heisenberg group
• The fundamental theorem of curves and classifications in the Heisenberg groups
• On sub-Riemannian geodesic curvature in dimension three
• The existence of horizontal envelopes in the 3D-Heisenberg group