Revista de la
Unión Matemática Argentina
A remark on trans-Sasakian 3-manifolds
Yaning Wang and Wenjie Wang
Volume 60, no. 1 (2019), pp. 257–264

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Let $M$ be a trans-Sasakian $3$-manifold of type $(\alpha,\beta)$. In this paper, we give a negative answer to the question proposed by S. Deshmukh [Mediterr. J. Math. 13 (2016), no. 5, 2951–2958], namely we prove that the differential equation $\nabla\beta=\xi(\beta)\xi$ on $M$ does not necessarily imply that $M$ is homothetic to either a Sasakian or cosymplectic manifold even when $M$ is compact. Many examples are constructed to illustrate this result.