Improved Chen–Ricci inequality for curvature-like tensors and its applications

Abstract

We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.2000 Mathematics Subject Classification. 53C40, 53C42, 53B25, 53C15, 53C25.Keywords and phrases: Curvature like tensor, Riemannian vector bundle, improved Chen-Ricci inequality, improved Chen-Ricci inequality, Lagrangian submanifold, Kaehlerian slant submanifold, C-totally real submanifold, complex space form and Sasakian space form.

Improved Chen inequality of Sasakian space forms with the Tanaka-Webster connection

“…On the other hand, for the above mentioned content, S. Deng [4] proved the improved Chen inequality for Lagrangian submanifolds of complex space forms just by using some crucial algebraic inequalities and also discussed the equality case, which is not discussed in Oprea's paper. Furthermore, M. M. Tripathi [9] proved Chen inequality and improved Chen inequality for curvature like tensors. He also applied these inequalities to Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.…”

Chen Inequalities for Submanifolds of Real Space Forms with a Ricci Quarter-Symmetric Metric Connection

“…In [7], Hong and Tripathi presented a general inequality for submanifolds of a Riemannian manifold by using (1.3). In [13], this inequality was named Chen-Ricci inequality by Tripathi. In fact, the general inequality obtained in [7] is a special case of Theorem 3.1 of [5].…”

“…Later, Mihai and Özgür in [10] proved Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric metric connection. Moreover, several works in this direction is studied [1,6,9,13].…”

The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

“…For contradict that warped product pseudo-slant submanifolds always not generalize CR-warped product submanifold which was show in [13]. However, some interesting inequalities have been obtained by many geometers (see [4,10,12,[16][17][18][19][20]) for distinct warped product submanifolds in the different types of ambient manifolds. In [5], Al-Solamy derived the inequality for mixed, totally geodesic warped product pseudo-slant submanifolds of type M = M θ × f M ⊥ , in a nearly cosymplectic manifold.…”