Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

Abstract

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold M n of Sasakian space forms M 2 m + 1 ε . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product M n via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere S p .

Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

“…e geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1).…”

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

“…More recently, several results have been derived on topological and differentiable structures of submanifolds when imposing certain conditions on the second fundamental form, Ricci curvatures, and sectional curvatures in a series of articles [4,10,11,[18][19][20][21][22][23] by different geometers. For the warped product structure, we refer to [20,[24][25][26][27][28][29][30].…”

Analysis of Obata’s Differential Equations on Pointwise Semislant Warped Product Submanifolds of Complex Space Forms via Ricci Curvature

“…Furthermore, Al-Dayael and Khan [19] proved that, under certain conditions, the base of contact CR-warped product submanifolds N T × f N ⊥ is isometric to a sphere. Recently, Mofarreh et al [20] used Obata's differential equation on warped product submanifolds of Sasakian space form and established some characterization.…”