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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 (M ¯,ϕ,ξ,η,g) be an almost contact metric manifold with constant ϕ-sectional curvature c, and curvature tensor given by

R ¯(X,Y)Z=f 1 {g(Y,Z)X-g(X,Z)Y}+f 2 {g(X,ϕZ)ϕY-g(Y,ϕZ)ϕX+2g(X,ϕY)ϕZ}+f 3 {η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ},

where f 1 , f 2 and f 3 are differentiable functions on M ¯. The authors consider four types of generalized Sasakian space forms determined by the f i , which are functions of the constant ϕ-sectional curvature: Sasakian, Kenmotsu, cosymplectic, and almost C(α). 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 ξ-horizontal CR-submanifolds.

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