Torsion and deformation of contact metric structures on 3-manifolds

Samuel I. Goldberg, Gabor Toth

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

S.-S. Chern raised the question of determining those compact 3-manifolds M admitting a contact metric structure whose characteristic vector field generates a one-parameter group of isometries. S. Tachibana showed that the first betti number of these spaces must be even, and H. Sato proved that the second homotopy group of M is zero unless M is homotopy equivalent to S1×S2. A. Weinstein pointed out that M is a Seifert fibre space over an orientable surface. In this paper, it is shown as a consequence of a more general theorem that if, in addition, the scalar curvature is suitably bounded below by a negative constant, then the metric may be deformed to a metric of positive constant sectional curvature. Thus, if the manifold is simply connected it is diffeomorphic with the 3-sphere.

Original languageEnglish (US)
Pages (from-to)365-372
Number of pages8
JournalTohoku Mathematical Journal
Volume39
Issue number3
DOIs
StatePublished - Sep 1987

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Contact Riemannian manifolds
  • Curvature
  • Torsion

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