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CONFORMALLY FLAT CONTACT THREE-MANIFOLDS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 103 / Issue 2 / October 2017
- Published online by Cambridge University Press:
- 03 November 2016, pp. 177-189
- Print publication:
- October 2017
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- Article
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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.
