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A note on doubly warped product contact CR-submanifolds in trans-Sasakian manifolds. (English) Zbl 1135.53319

Summary: Warped product \(CR\)-submanifolds in Kählerian manifolds were intensively studied only since 2001 after the impulse given by B. Y. Chen [Monatsh. Math. 133, 177-195 (2001; Zbl 0996.53044)], [Monatsh. Math. 134, 103–119 (2002; Zbl 0996.53045)]. Immediately after, another line of research, similar to that concerning Sasakian geometry as the odd dimensional version of Kählerian geometry, was developed, namely warped product contact \(CR\)-submanifolds in Sasakian manifolds cf. I. Mihai [Geom. Dedicata 109, 165–173 (2004; Zbl 1114.53052)], M. I. Munteanu [Publ. Math. Debrecen 66, 75–120 (2005; Zbl 1063.53052)]. In this note we prove that there exists no proper doubly warped product contact \(CR\)-submanifolds in trans-Sasakian manifolds.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C40 Global submanifolds
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References:

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