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New classes of almost contact metric structures. (English) Zbl 0611.53032
Let M be an almost contact metric manifold with structure tensors ($\phi$,$\xi$,$\eta$,g). As is well known an almost contact structure ($\phi$,$\xi$,$\eta)$ is said to be normal if the almost complex structure J on $M\times {\bbfR}$ defined by $$J[X,a\frac{d}{dt}]=[\phi X-a\xi,\eta (X)\frac{d}{dt}]$$ where a is a $C\sp{\infty}$ function on $M\times {\bbfR}$, is integrable. Let h be the product metric on $M\times {\bbfR}$ and $h\circ =e\sp{2t}h$, t being the coordinate on ${\bbfR}$. The author first shows that ($\phi$,$\xi$,$\eta$,g) is cosymplectic if and only if (J,h) is Kaehlerian and that ($\phi$,$\xi$,$\eta$,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) {\it A. Gray} and {\it 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$\circ))$ belong to the classes $\omega\sb 4$ and $\omega\sb 2\oplus \omega\sb 4$ 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 $\phi$ 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 {\it K. Kenmotsu} in Tohoku Math. J., II. Ser. 24, 93-103 (1972; Zbl 0245.53040) are also trans-Sasakian.
Reviewer: D.Blair

MSC:
 53C15 Differential geometric structures on manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)