Abstract
The present paper aims to construct an inequality for bi-warped product submanifolds in a special class of almost metric manifolds, namely nearly Kenmotsu manifolds. As geometric applications, some exceptional cases that generalized several other inequalities are discussed. We also deliberate some applications in the context of mathematical physics and derive a new relation between the Dirichlet energy and the second fundamental form. Finally, we present a constructive remark at the end of this paper which shows the motive of the study.
Highlights
Background and MotivationsA new series of bi-warped product submanifolds is constructed and some examples about it first appeared in [1]
It is founded that the bi-warped product submanifolds of types Ωφ × f1 Ω T × f2 Ω⊥, Ω⊥ × f1 Ω T × f2 Ωφ and Ω T × f1 Ωφ × f2 Ω⊥ do not exist in Kaehler manifolds
An example was used to show the existence of a non-trivial bi-warped product submanifold of the type Ω T × f1 Ω⊥ × f2 Ωφ, and the necessary and sufficient conditions are given for this submanifold to be locally trivial
Summary
A new series of bi-warped product submanifolds is constructed and some examples about it first appeared in [1]. In [4], authors developed the sharp inequality in terms of the second fundamental form with its squared norm, for a bi-warped product submanifold Ω = Ω T × f1 Ω⊥ × f2 Ωφ in a Kenmotsu manifold with giving a non-trivial example They presented the following inequality and a few applications. More interesting is that the bi-warped product submanifolds of the form Ω = Ω⊥ × f1 Ω T × f2 Ωφ in Kenmotsu manifolds are discussed in [10] and the following inequality is presented kBk2 ≥ 2t k∇(ln f 1 )k2 − 1 + 2k cos φ k∇(ln f 2 )k2 − 1. Proposition 1 and Proposition 2 insure that there do not exists any non-trivial proper bi-warped product submanifold Ω = Ω T × f1 Ω⊥ × f2 Ωφ of a nearly Kenmotsu manifold Ωwith D ⊕ D φ -mixed totally geodesic and D ⊕ D ⊥ -mixed totally geodesic restrictions.
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