Chen’s first inequality for hemi-slant warped products in nearly trans-Sasakian manifolds

Abstract

Abstract In this paper, we prove that every hemi-slant warped product submanifold of the form N θ × f N ⊥ in a nearly trans-Sasakian manifold M͠ satisfies the following inequality: ∥h∥2 ≥ n 2cot2 θ(∥∇̂(ln f)∥2 – β 2), whereas the warped product by reversing these two factors, i.e., N ⊥ × f N θ satisfying the inequality: $\begin{array}{} \displaystyle \|h\|^2\geq \frac{n_1}{9}\cos^2\theta(\|\widehat\nabla(\ln f)\|^2-\beta^2), \end{array}$ where n 1 = dim N θ , n 2 = dim N ⊥, ∇̂(ln f) is the gradient of ln f and ∥h∥ is the length of the second fundamental form of the warped product immersion in M͠. The equality cases of these inequalities are investigated. Furthermore, we discuss some special cases of these inequalities. Finally, we construct two non-trivial examples.