On conformally flat minimal Legendrian submanifolds in the unit sphere

This paper is concerned with the study on an open problem of classifying conformally flat minimal Legendrian submanifolds in the $(2n+1)$ -dimensional unit sphere S²ⁿ⁺¹ admitting a Sasakian structure (φ,\,ξ,\,η,\,g) for n≥ 3 , motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we completely classify such Legendrian submanifolds by assuming that the tensor K:=-φ h is semi-parallel, which is introduced as a natural extension of C -parallel second fundamental form h . Secondly, such submanifolds have also been determined under the condition that the Ricci tensor is semi-parallel, generalizing the Einstein condition. Finally, as direct consequences, new characterizations of the Calabi torus are presented.