Abstract
In the present, we first obtain Chen–Ricci inequality for C-totally real warped product submanifolds in cosymplectic space forms. Then, we focus on characterizing spheres and Euclidean spaces, by using the Bochner formula and a second-order ordinary differential equation with geometric inequalities. We derive the characterization for the base of the warped product via the first eigenvalue of the warping function. Also, it proves that there is an isometry between the base mathbb{N}_{1} and the Euclidean sphere mathbb{S}^{m_{1}} under some different extrinsic conditions.
Highlights
Introduction and motivationsThe seminal work of Obata [30] has become a basic tool of investigation in geometric analysis
The Euclidean space Rn is designated through the differential equation ∇2ω = cg, where c is a positive constant, which was proven by Tashiro [32]
Motivated by the previous studies, we will establish the following results: Theorem 1.1 Let Υ : n = N1 ×f N2 −→ M2m+1( ) be a C-totally real isometric embedding of the warped product submanifold n into a cosymplectic space form M2m+1( ) with nonnegative Ricci curvature
Summary
Nc μ1 c) for a constant c, where μ1 is the first eigenvalue of the Laplacian, is isometric to Sn(c) if n admits a nonzero conformal gradient vector field. From the Bochner formula, we are able to prove the following result: Theorem 1.2 Let Υ : n = N1 ×f N2 −→ M2m+1( ) be a C-totally real isometric embedding for the warped product submanifold n to the cosymplectic space form M2m+1( ) with base. Let K γ and K γ be the sectional curvature of a submanifold n and M2m+1, respectively, we have following relation due to the Gauss equation (2.6): 2m+1. Lemma 2.1 Suppose Υ : n = N1 ×f N2 −→ M2m+1( ) is a C-totally real warped product immersed submanifold into a cosymplectic space form M2m+1 whose base N1 is minimal. Lemma 2.2 Assume ω : n = N1 ×f N2 −→ M2m+1 is a C-totally real minimal isometric embedding of a warped product n to the cosymplectic space form M2m+1. Under the assumption that the Ricci curvature is greater than or equal to zero, i.e., Ric(W) ≥ 0, the latter equation implies n2λ1
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
