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Geometric structures arising from the deformation of groups of Heisenberg type

Claudio Afeltra

Abstract

Motivated by the desire of finding a geometric interpretation to the Yamabe equation on groups of Heisenberg type, we define a geometric structure on manifolds modelled locally on these groups, which we call contact structure of Heisenberg type. In the case of the Heisenberg group is equivalent to contact Riemannian manifolds. We define a natural connection on these structures, we compute the formula for the conformal change of scalar curvature, and introduce the Yamabe problem for these manifods.

Geometric structures arising from the deformation of groups of Heisenberg type

Abstract

Motivated by the desire of finding a geometric interpretation to the Yamabe equation on groups of Heisenberg type, we define a geometric structure on manifolds modelled locally on these groups, which we call contact structure of Heisenberg type. In the case of the Heisenberg group is equivalent to contact Riemannian manifolds. We define a natural connection on these structures, we compute the formula for the conformal change of scalar curvature, and introduce the Yamabe problem for these manifods.
Paper Structure (9 sections, 14 theorems, 141 equations)

This paper contains 9 sections, 14 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Groups of Heisenberg type
  3. Contact structures of Heisenberg type
  4. Conformal change
  5. The Iwasawa sphere
  6. A natural connection
  7. Partial connection along $\mathfrak{H}$
  8. Partial connection along $\mathfrak{T}$
  9. Scalar curvature

Key Result

Theorem 2.1

The following are equivalent.

Theorems & Definitions (28)

  • Theorem 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 18 more