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CR-submanifolds of generalized Sasakian space forms. (English) Zbl 1141.53043
The paper is a study of CR-submanifolds of a generalized Sasakian space form. Let $(\overline{M},\varphi ,\xi,\eta,g) $ be an almost contact metric manifold with constant $\varphi$-sectional curvature $c$, and curvature tensor given by $$\align \overline{R}(X,Y)Z= &f_1\{g( Y,Z)X- g(X,Z)Y\}\\ &+f_2\{g(X,\varphi Z)\varphi Y- g(Y,\varphi Z)\varphi X+ 2g(X,\varphi Y)\varphi Z\}\\ &+f_3\{\eta(X)\eta(Z)Y- \eta(Y)\eta(Z)X+ g(X,Z)\eta(Y)\xi- g(Y,Z)\eta(X)\xi\}, \endalign$$ where $f_1$, $f_2$ and $f_3$ are differentiable functions on $\overline{M}$. The authors consider four types of generalized Sasakian space forms determined by the $f_i$, which are functions of the constant $\varphi$-sectional curvature: Sasakian, Kenmotsu, cosymplectic, and almost $C(\alpha)$. The authors obtain formulas for the sectional curvature of CR-submanifolds in each case, as well as formulas for the Ricci tensor and scalar curvature for minimal $\xi$-horizontal CR-submanifolds.

53C40Global submanifolds (differential geometry)
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)