A class of biminimal Legendrian submanifolds in Sasakian space forms

Abstract

A biminimal submanifold is a critical point of the bienergy functional for any normal variation. We give a complete description of all nonminimal biminimal Legendrian submanifolds in Sasakian space forms which are Legendrian H‐umbilical.

Classification of biharmonic $\mathcal{C}$-parallel Legendrian submanifolds in 7-dimensional Sasakian space forms

“…Proof of Theorem 1.1: The case (1) of Theorem 1.1 has been proved in (1) Theorem 5.1 of [5]. By Theorem 5.5 in [10] for n = 3 and Proposition 3.2, we obtain the case (2) of Theorem 1.1.…”

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

“…We know that every simply connected Sasakian space form M 2m+1 (ε) is isometric to the odd-dimensional sphere S 2m+1 and odd-dimensional Euclidean space R 2m+1 with ϕ-constant sectional curvature ε � 1 and ε � − 3, respectively. For more details and examples, see the work presented in [24,26,27].…”

“…For examples of C-totally real isometric immersions from warped product manifolds, see the work presented in [27,30].…”

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

“…( 6)], the Riemannian curvature tensor of M 2m+1 ( ) is defined in detail, which is also usually defined from R. If the structure field ξ is perpendicular to the submanifold n in M 2m+1 ( ), then n is a C-totally real submanifold of M 2m+1 ( ). Furthermore, in this case, φ maps any tangent space of n into its corresponding normal space (see [2,8,23,25,32,36]). Now, we recall the Bochner formula [8] for a differentiable function on a Riemannian manifold n , that is, ψ : n → R. Then, we have that…”