Applications of differential equations to characterize the base of warped product submanifolds of cosymplectic space forms

Abstract

In the present, we first obtain Chen–Ricci inequality for C-totally real warped product submanifolds in cosymplectic space forms. Then, we focus on characterizing spheres and Euclidean spaces, by using the Bochner formula and a second-order ordinary differential equation with geometric inequalities. We derive the characterization for the base of the warped product via the first eigenvalue of the warping function. Also, it proves that there is an isometry between the base $\mathbb{N}_{1}$ N 1 and the Euclidean sphere $\mathbb{S}^{m_{1}}$ S m 1 under some different extrinsic conditions.

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

“…, e n in which W � e u � e n . Utilizing (17) to (33) and following a similar technique from (33)-(42), it implies…”

“…en, the embedding Φ: (− (π/2), (π/2))× cos;t S ⟶ T 1 M such that Φ(t, x) � sintN + costx, where N is the unit vector perpendicular with the linear subspace including T 1 M, which is a C-totally real isometric embedding. For more classification, see [31][32][33].…”

Characterization of Skew CR-Warped Product Submanifolds in Complex Space Forms via Differential Equations

“…e categorization of differential equations on Riemannian manifold has become a fascinating topic of research and has been investigated by numerous researchers, for instance, [6][7][8][9].…”

“…Latterly, Ali et al [7] characterized warped product submanifolds in Sasakian space form by the approach of the differential equation. e purpose of this paper is to study the impact of differential equation on skew CR-warped product submanifolds in the framework of the complex space form.…”

Study of Differential Equations on Warped Product Semi-Invariant Submanifolds of the Generalized Sasakian Space Forms

“…The categorization of differential equations on Riemannian manifolds turns into an attractive research subject that has been explored by various researchers, for example, [7][8][9][10][11].…”