On differential equations classifying a warped product submanifold of cosymplectic space forms

Akram Ali,Irfan Anjum Badruddin,Rifaqat Ali,Fatemah Mofarreh

doi:10.1186/s13660-021-02620-z

Abstract

In the present paper, we extend the study of (Ali et al. in J. Inequal. Appl. 2020:241, 2020) by using differential equations (García-Río et al. in J. Differ. Equ. 194(2):287–299, 2003; Pigola et al. in Math. Z. 268:777–790, 2011; Tanno in J. Math. Soc. Jpn. 30(3):509–531, 1978; Tashiro in Trans. Am. Math. Soc. 117:251–275, 1965), and we find some necessary conditions for the base of warped product submanifolds of cosymplectic space form widetilde{M}^{2m+1}(epsilon ) to be isometric to the Euclidean space mathbb{R}^{n} or a warped product of complete manifold N and Euclidean space mathbb{R}.

Highlights

Background and motivationIn [11, 12, 17,18,19], the authors gave the characterizations of Euclidean spaces by analyzing a differential equation
Soc. 117:251–275, 1965), and we find some necessary conditions for the base of warped product submanifolds of cosymplectic space form M2m+1( ) to be isometric to the Euclidean space Rn or a warped product of complete manifold N and Euclidean space R
If and only if ( n, g) is isometric to the Euclidean spaces Rn, where c is any positive constant. There is another characterization by using differential equation which was discovered by Río, Kupeli, and Unal [12]. They demonstrated that the complete Riemannian manifold ( n, g) is isometric to the warped product of a complete Riemannian manifold N and an Euclidean line R with warping function θ accomplishes the differential equation d2θ dt2 + λ1θ = 0 (1.2)

Summary

Introduction

Background and motivationIn [11, 12, 17,18,19], the authors gave the characterizations of Euclidean spaces by analyzing a differential equation. Matsuyama [14] derived a characterization such that the complete totally real submanifold n of the complex projective space CPn with bounded Ricci curvature admits a function ψ satisfying (1.3) for λ1 ≤ n, n is isometric to the hyperbolic space component that is connected if (∇ψ)x = 0 or it is isometric to the warped product of the complete Riemannian manifold and the Euclidean line if ∇ψ is non-vanishing, where the warping function θ on R ensures equation (1.2).

Results

Conclusion