An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms

New characterizations of the Whitney spheres and the contact Whitney spheres

“…Without introducing the notion of contact Whitney spheres as canonical examples, Pitiş [16] proved that results for Lagrangian submanifolds of the complex space forms in [6,17] hold analogously for those Legendrian submanifolds of the Sasakian space forms which satisfy (1.9). Moreover, an analogue of the result for Whitney spheres in complex space forms by Li-Vrancken [15] was established for contact Whitney spheres in Sasakian space forms by Hu-Yin [14]. It follows that a corresponding version of Theorem 1.1 for contact Whitney spheres in Sasakian space forms is already known.…”

“…Remark 1.1. The classification of Lagrangian submanifolds with parallel second fundamental form in N n (4c) has been fulfilled for each c, see [11,14] for details.…”

“…where a is a positive real number and c = 4 a − 3. Then, as were introduced in [14], the contact Whitney spheres in Ñ 2n+1 (c) for c > −3 are a family of Legendrian immersions Ψθ : S n → S 2n+1 for θ > 0, that are explicitly given by…”

“…Then, B n ×R has a Sasakian structure { φ, ξ, η, ḡ} with constant φ-sectional curvature −4, defined as follows: and c = − 1 a − 3, where a is a positive number. As were introduced in [14], the contact Whitney spheres in Ñ 2n+1 (c) for c < −3 are a one-parameter family of Legendrian immersions Φθ : S n → B n × R for θ > 0, that are explicitly given by…”

“…According to Proposition 2 of Blair-Carriazo [1] and Theorem 4.2 of Hu-Yin [14], for each contact Whitney sphere M n in any Sasakian space form Ñ 2n+1 (c), the second fundamental form h and the mean curvature vector field H satisfy the relation , H)ϕX for any tangent vector fields X, Y ∈ T M n . Without introducing the notion of contact Whitney spheres as canonical examples, Pitiş [16] proved that results for Lagrangian submanifolds of the complex space forms in [6,17] hold analogously for those Legendrian submanifolds of the Sasakian space forms which satisfy (1.9).…”