Helical CR structures and sub-Riemannian geodesics

John P. D'Angelo, Jeremy T. Tyson

Research output: Contribution to journalArticlepeer-review


A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step-two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step-two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.

Original languageEnglish (US)
Pages (from-to)205-221
Number of pages17
JournalComplex Variables and Elliptic Equations
Issue number3-4
StatePublished - Mar 2009


  • Carnot group
  • Heisenberg group
  • Helical CR structure
  • Higher curvature
  • Homogeneous polynomial
  • Proper holomorphic mapping
  • Sub-Riemannian geodesic

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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