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 $$M\times {\mathbb{R}}$$ defined by $J[X,a\frac{d}{dt}]=[\phi X-a\xi,\eta (X)\frac{d}{dt}]$ 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 ($$\phi$$,$$\xi$$,$$\eta$$,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) 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$$\circ))$$ 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 $$\phi$$ 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 K. Kenmotsu in Tohoku Math. J., II. Ser. 24, 93-103 (1972; Zbl 0245.53040) are also trans-Sasakian.
Reviewer: D.Blair

MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)