Abstract
In this paper, it is shown that for a 3-dimensional compact simply connected trans-Sasakian manifold of type \({(\alpha,\beta)}\), the smooth functions \({\alpha,\beta}\) satisfy the Poisson equations \({\Delta \alpha = \beta}\), \({\Delta \alpha = \alpha ^{2}\beta}\) and \({\Delta \beta = \alpha ^{2}\beta}\), respectively, if and only if it is homothetic to a Sasakian manifold. We also find a necessary and sufficient condition for a connected 3-dimensional trans-Sasakian manifold of type \({(\alpha,\beta)}\) in terms of a differential equation satisfied by the smooth function \({\alpha}\) to be homothetic to a Sasakian manifold.
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Deshmukh, S. Trans-Sasakian Manifolds Homothetic to Sasakian Manifolds. Mediterr. J. Math. 13, 2951–2958 (2016). https://doi.org/10.1007/s00009-015-0666-4
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DOI: https://doi.org/10.1007/s00009-015-0666-4