An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms
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An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms Zejun Hu1
· Jiabin Yin1
Received: 12 September 2018 © Mathematica Josephina, Inc. 2019
Abstract Let N 2n+1 (c) be an (2n +1)-dimensional Sasakian space form with Sasakian structure (ϕ, ξ, η, g) and constant ϕ-sectional curvature c. An n-dimensional submanifold M n of N 2n+1 (c) is called integral if the contact form η restricted to M n vanishes. In this paper, we established a general inequality for n-dimensional integral submanifolds in Sasakian space forms involving the norm of the covariant differentiation of both the second fundamental form h and the mean curvature vector field H . Our result is optimal in that we can classify all integral submanifolds realizing the equality case of the inequality. As direct consequence, we give a characterization of the contact Whitney spheres in the Sasakian space forms. Keywords Sasakian space form · Integral submanifold · Contact Whitney sphere · C-parallel second fundamental form · Lagrangian submanifold Mathematics Subject Classification Primary 53C24; Secondary 53C25 · 53C42
1 Introduction Let N 2n+1 (c) be a Sasakian space form with constant ϕ-sectional curvature c, i.e., it is a complete, simply connected, (2n + 1)-dimensional contact manifold equipped with a Sasakian contact metric structure {ϕ, ξ, η, g}. As is well known from Tanno [31] (cf. also [3] for details), N 2n+1 (c) = R2n+1 for c = −3; N 2n+1 (c) = S2n+1 [c] for c > −3; N 2n+1 (c) = Bn × R for c < −3. The authors were supported by NSF of China, Grant Number 11771404.
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Zejun Hu [email protected] Jiabin Yin [email protected]
1
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China
123
Z. Hu, J. Yin
In this paper, we study n-dimensional integral (or equivalently, C-totally real) submanifolds of N 2n+1 (c), which by definition are n-dimensional submanifolds of N 2n+1 (c) having the property that when restricted to them the contact form η of N 2n+1 (c) vanishes. From the view point of submanifolds geometry, the study of integral submanifolds in a contact manifold is one of the most interesting topics. In particular, the study of integral submanifolds in Sasakian space forms has attracted much attention of geometers, and many results have been established, see, e.g., [1,3,4,12,13,19,32] among others. The initial motivation of this paper is related to the characterization of the classical Whitney spheres. Recall that the well-known Whitney spheres are usually defined as a family of Lagrangian immersions from the unit sphere Sn → Rn+1 , centered at the origin of Rn+1 , into the complex Euclidean space Cn ∼ = R2n , given by r ,B : Sn → 2n R with r ,B (u 0 , u 1 , . . . , u n ) =
r (u , . . . , u n , u 0 u 1 , . . . , u 0 u n ) + 1+u 20 1
B,
(1.1)
where r is a positive number and B is a vector of Cn . The number r and the vector B are called the radius and the center of the Whitney sphere, respectively. The Whitney spheres have many interesting g
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