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 ${\mathbb{M}}^{2m+1}\left(\epsilon \right)$ . 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 $\mathbb{B}$ of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere ${\mathbb{S}}^p$ .