Contravariant gravity on Poisson manifolds and Einstein gravity

Yukio Kaneko, Hisayoshi Muraki, Satoshi Watamura

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


A relation between gravity on Poisson manifolds proposed in Asakawa et al (2015 Fortschr. Phys. 63 683-704) and Einstein gravity is investigated. The compatibility of the Poisson and Riemann structures defines a unique connection, the contravariant Levi-Civita connection, and leads to the idea of the contravariant gravity. The Einstein-Hilbert-type action yields an equation of motion which is written in terms of the analog of the Einstein tensor, and it includes couplings between the metric and the Poisson tensor. The study of the Weyl transformation reveals properties of those interactions. It is argued that this theory can have an equivalent description as a system of Einstein gravity coupled to matter. As an example, it is shown that the contravariant gravity on a two-dimensional Poisson manifold can be described by a real scalar field coupled to the metric in a specific manner.

Original languageEnglish
Article number115002
JournalClassical and Quantum Gravity
Issue number11
Publication statusPublished - 2017 May 18


  • Poisson geometry
  • gravity theory
  • noncommutative geometry

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)


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