Application of Bochner formula to generalized Sasakian space forms

The base of warped product submanifolds of Sasakian space forms characterized by differential equations

“…On the other hand, Matsuyama [24] derived a characterization stating that if the complete totally real submanifold n for the complex projective space CP n with bounded Ricci curvature admits a function ψ satisfying (3), for λ 1 ≤ n, then n is isometric to the hyperbolic space component that is connected if (∇ψ) x = 0 or if it is isometric to the warped product of a complete Riemannian manifold and the Euclidean line if ∇ψ is nonvanishing, where the warping function θ on R satisfies equation (2). Furthermore, similar results have been obtained for generalized Sasakian space forms by Jamali and Shahid [22]. In this study, inspired by [1-3, 5, 7, 9-12, 22, 30, 35], we derive a similar characterization for C-totally real warped product submanifolds of Sasakian space forms as rigidity theorems.…”

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

“…According to Obata, if (M n , g) is a complete Riemannian manifold, then the nonconstant function f on M n satisfies the differential equation ∇ 2 f + cfg � 0 or Hessian(f) + cfg � 0 if and only if M n is isometric to n-dimensional sphere of radius c. A significant number of studies have been conducted on this topic. As a result, the Euclidean space, Euclidean sphere, and complex projective space are recognized domains in the analysis of differential geometry of manifolds, for instance, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. As a special case, the differential equation ∇ 2 f � cg signifies the Euclidean space, where c is a constant; infact, this was proved by Tashiro [17].…”

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].…”