Abstract
In this paper, we give an estimate of the first eigenvalue of the Laplace operator on minimally immersed Legendrian submanifold N n in Sasakian space forms N ˜ 2 n + 1 ( ϵ ) . We prove that a minimal Legendrian submanifolds in a Sasakian space form is isometric to a standard sphere S n if the Ricci curvature satisfies an extrinsic condition which includes a gradient of a function, the constant holomorphic sectional curvature of the ambient space and a dimension of N n . We also obtain a Simons-type inequality for the same ambient space forms N ˜ 2 n + 1 ( ϵ ) .
Highlights
In 1959, Yano and Nagano [1] proved that if a complete Einstein space of dimension strictly greater than 2 admits a 1-parameter group of non-homothetic conformal transformations, it is isometric to a sphere
Utilizing Obata Equation (1), Barros et al [14] have shown that a compact gradient almost Ricci soliton ( N n, g, ∇ψ, λ) with the Codazzi Ricci tensor and constant sectional curvature is isometric to the Euclidean sphere, and ψ is a height function in this case
As a generalization in the case of an odd-dimensional sphere, a minimally immersed Legendrian submanifold into a Sasakian space form of constant holomorphic sectional curvature e should be considered in order to obtain Simon’s-like inequality theorem
Summary
Introduction and MotivationsIn 1959, Yano and Nagano [1] proved that if a complete Einstein space of dimension strictly greater than 2 admits a 1-parameter group of non-homothetic conformal transformations, it is isometric to a sphere. Recall that a complete manifold ( N n , g) admits a non-constant function ψ satisfying the Obata differential equation Utilizing Obata Equation (1), Barros et al [14] have shown that a compact gradient almost Ricci soliton ( N n , g, ∇ψ, λ) with the Codazzi Ricci tensor and constant sectional curvature is isometric to the Euclidean sphere, and ψ is a height function in this case.
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