On Two Different Classes of Warped Product Submanifolds of Kenmotsu Manifolds
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Authors: S. K. HUI, MD. H. SHAHID, T. PAL AND J. ROY
DOI: 10.46793/KgJMat2306.965H
Abstract:
Warped product skew CR-submanifold of the form M = M1 ×fM⊥ of a Kenmotsu manifold (throughout the paper), where M1 = MT ×M???? and MT, M⊥, M???? represents invariant, anti-invariant and proper slant submanifold of , studied in [?] and another class of warped product skew CR-submanifold of the form M = M2 ×fMT of , where M2 = M⊥×M???? is studied in [?]. Also the warped product submanifold of the form M = M3 ×fM???? of , where M3 = MT × M⊥ and MT, M⊥, M???? represents invariant, anti-invariant and proper point wise slant submanifold of , were studied in [?]. As a generalization of the above mentioned three classes, we consider a class of warped product submanifold of the form M = M4 ×fM????3 of , where M4 = M????1 × M????2 in which M????1 and M????2 are proper slant submanifolds of and M????3 represents a proper pointwise slant submanifold of . A characterization is given on the existence of such warped product submanifolds which generalizes the characterization of warped product submanifolds of the form M = M1 ×fM⊥, studied in [?], the characterization of warped product submanifolds of the form M = M2 ×fMT, studied in [?], the characterization of warped product submanifolds of the form M = M3 ×fM????, studied in [?] and also the characterization of warped product pointwise bi-slant submanifolds of , studied in [?]. Since warped product bi-slant submanifolds of does not exist (Theorem 4.2 of [?]), the Riemannian product M4 = M????1 × M????2 cannot be a warped product. So, for studying the bi-warped product submanifolds of of the form M????1 ×f1M????2 ×f2M????3, we have taken M????1, M????2, M????3 as pointwise slant submanifolds of of distinct slant functions ????1, ????2, ????3 respectively. The existence of such type of bi-warped product submanifolds of is ensured by an example. Finally, a Chen-type inequality on the squared norm of the second fundamental form of such bi-warped product submanifolds of is obtained which also generalizes the inequalities obtained in [?], [?] and [?], respectively.
Keywords:
Kenmotsu manifold, pointwise slant submanifolds, warped product, submanifolds, bi-warped product submanifolds.
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