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CONFORMALLY FLAT CONTACT THREE-MANIFOLDS

Published online by Cambridge University Press:  03 November 2016

JONG TAEK CHO*
Affiliation:
Department of Mathematics, Chonnam National University, Gwangju 61186, Korea email [email protected]
DONG-HEE YANG
Affiliation:
Department of Mathematics and Statistics, Graduate School of Chonnam National University, Gwangju 61186, Korea email [email protected]
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Abstract

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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.

Type
Research Article
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

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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