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CONFORMALLY FLAT CONTACT THREE-MANIFOLDS

  • JONG TAEK CHO (a1) and DONG-HEE YANG (a2)

Abstract

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$ , where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$ . We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$ . We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$ . Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2014R1A1A2053665).

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CONFORMALLY FLAT CONTACT THREE-MANIFOLDS

  • JONG TAEK CHO (a1) and DONG-HEE YANG (a2)

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