Abstract
In the present work, we consider two types of bi-warped product submanifolds, M=MT×f1M⊥×f2Mϕ and M=Mϕ×f1MT×f2M⊥, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.
Highlights
Let M = M1 × M2 × M3 × · · · × Mk be the Cartesian product of Riemannian manifolds M1, M2, · · ·, Mk and let πi : M −→ Mk denote the canonical projection maps for i = 1, · · ·, k
If we choose two fibers of a multiple warped product M1 × f1 M2 × · · · × f k Mk, such that M = M1 × f1 M2 × f2 M3, M is defined as a bi-warped product submanifold, which satisfies the following: Published: 13 April 2021
We provide a relationship between the squared norm of the second fundamental form and the warping function for the bi-warped product
Summary
(Theorem 4.1 of [19]) Let M = M T × f Mφ be a warped product semi-slant submanifold of a nearly trans-Sasakian M, inequality (40) implies the following inequality:. If we put α = 0, β = 0 into Theorem 1, the following is obtained for the bi-warped product submanifold of a nearly cosymplectic manifold: Theorem 6. Cot φ + 1 csc φ k∇(ln f 2 )k , By inserting β = 0 and α = 1, β = 0 into Theorem 1, the following results for the bi-warped product submanifolds of a nearly α-Sasakian manifold and a nearly Sasakian manifold, respectively, can be obtained: Theorem 9. For the second type of bi-warped product submanifold, Mφ × f1 M T × f2 M⊥ , we prove the following result: Theorem 11.
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