2018/11/07 by Chen, Bang-Yen, Uddin, Siraj, Alghanemi, Azeb +2
#53C15 #53C25 #53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1811.02767
In this paper, we prove that there are no proper CRS bi-warped product submanifolds other than contact CR-biwarped products in Sasakian manifolds. On the other hand, we prove that if M is a CRS bi-warped product of the form M=NT ×f1N^n1_⊥×f2 N^n2θ in a cosymplectic manifold \widetilde M, then its second fundamental form h satisfies the inequality: ‖h‖2≥ 2n1‖∇(ln f1)‖2+2n2(1+2\cot2θ)‖∇(ln f2)‖2, where NT, N^n1_⊥ and N^n2θ are invariant, anti-invariant and proper pointwise slant submanifolds of \widetilde M, respectively, and ∇(ln f1) and ∇(ln f2) denote the gradients of ln f1 and ln f2, respectively. Several applications of this inequality are given. At the end, we provide a non-trivial example of bi-warped products satisfying the equality case.