*(English)*Zbl 0611.53032

Let M be an almost contact metric manifold with structure tensors ($\varphi $,$\xi $,$\eta $,g). As is well known an almost contact structure ($\varphi $,$\xi $,$\eta )$ is said to be normal if the almost complex structure J on $M\times \mathbb{R}$ defined by

where a is a ${C}^{\infty}$ function on $M\times \mathbb{R}$, is integrable. Let h be the product metric on $M\times \mathbb{R}$ and $h\circ ={e}^{2t}h$, t being the coordinate on $\mathbb{R}$. The author first shows that ($\varphi $,$\xi $,$\eta $,g) is cosymplectic if and only if (J,h) is Kaehlerian and that ($\varphi $,$\xi $,$\eta $,g) is Sasakian if and only if (J,h$\circ )$ is Kaehlerian.

In Ann. Mat. Pura Appl., IV. Ser. 123, 35-58 (1980; Zbl 0444.53032) *A. Gray* and *C. M. Hervella* identified sixteen classes of almost Hermitian structures and the reader is referred to this paper for the definitions of these classes. The author defines the notions of a trans- Sasakian structure and an almost trans-Sasakian structure by requiring that (J,h) (or, equivalently, (J,h$\circ \left)\right)$ belong to the classes ${\omega}_{4}$ and ${\omega}_{2}\oplus {\omega}_{4}$ of Gray and Hervella, respectively. The author shows that an almost contact metric structure is trans-Sasakian if and only if the covariant derivative of $\varphi $ is of a particular form. Under the assumption of normality other characterizations are given. The author also discusses the relationship between trans-Sasakian and quasi-Sasakian structures [cf. the reviewer’s paper in J. Differ. Geom. 1, 331-345 (1967; Zbl 0163.439)] and gives an example of a trans-Sasakian structure which is not quasi-Sasakian and hence, in particular, neither Sasakian nor cosymplectic. The reviewer wishes to point out that the almost contact metric manifolds studied by *K. Kenmotsu* in Tohoku Math. J., II. Ser. 24, 93-103 (1972; Zbl 0245.53040) are also trans-Sasakian.

##### MSC:

53C15 | Differential geometric structures on manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |