Biharmonic integral $\mathcal{C}$-parallel submanifolds in 7-dimensional Sasakian space forms

Abstract

We find the characterization of maximum dimensional proper-biharmonic integral C-parallel submanifolds of a Sasakian space form and then classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere S 7 we explicitly find all 3-dimensional proper-biharmonic integral C-parallel submanifolds. We also determine the proper-biharmonic parallel Lagrangian submanifolds of CP 3 .

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

“…This paper corrects errors in [1], and moreover, completes the classification. Our main result is the following, which corrects Theorem 5.1 in [5]. (3λ 2 − α −1 )(3λ 4 − 2(ǫ + 1)λ + α −2 ) + λ 4 ((a + c) 2 + d 2 ) = 0, (a + c)(5λ 2 + a 2 + c 2 − 7α −1 + 4) + cd 2 = 0, d(5λ 2 + d 2 + 3c 2 + ac − 7α −1 + 4) = 0,…”

“…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.…”

“…On the other hand, following the wrong statement of Lemma 4.4 in [1], the nonflat case of Theorem 5.1 in [5] is investigated. Therefore, the classification presented in the theorem is not complete.…”

“…In Theorem 5.1 of [5], Fetcu and Oniciuc presented the classification result for proper biharmonic C-parallel Legendrian submanifolds in 7-dimensional Sasakian space forms. The case (2) of the theorem is proved by applying Lemma 4.4 in [1].…”

Biharmonic surfaces with parallel mean curvature in complex space forms

“…Some very important examples of such ambient spaces are the complex space forms, i.e., simply-connected Kähler manifolds with constant holomorphic sectional curvature. Recent papers as [12,13,21,24,27] treated the subject of proper-biharmonic submanifolds in complex space forms and several classification results and examples were found.…”