Applications of differential equations to characterize the base of warped product submanifolds of cosymplectic space forms

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 N1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{N}_{1}$\end{document} and the Euclidean sphere Sm1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{S}^{m_{1}}$\end{document} under some different extrinsic conditions.


Introduction and motivations
The seminal work of Obata [30] has become a basic tool of investigation in geometric analysis. Obata [30] provided a characterization theorem for the standard sphere in terms of a differential equation, nowadays famous as Obata equation. If ( n , g) is a complete manifold, then the function ω is nonconstant and satisfying the ODE Hess(ω) + cωg = 0 (1.1) if and only if there is an isometry between ( n , g) and the sphere S n (c), where c denotes the sectional curvature. If c = 1, then ( n , g) and the unit sphere S n are congruent. Many investigations have been dedicated to this subject, and therefore, characterization of the Euclidean space R n , the Euclidean sphere S n , and the complex projective space CP n are recognized fields in differential geometry and studied in some researches, e.g., [1-10, 13-15, 20-25]. In particular, the Euclidean space R n is designated through the differential equation ∇ 2 ω = cg, where c is a positive constant, which was proven by Tashiro [32]. In [27], Lichnerowicz established that if the first nonzero eigenvalue μ 1 of Laplace operator of the compact manifold ( n , g) with Ric ≥ n -1 is μ 1 = n, then ( n , g) is isometric to the sphere S n . Hence, Obata's theorem can be utilized to address the equality condition of Lichnerowicz's eigenvalue. Deshmukh-Al-Solamy [25] proved that an n-dimensional Riemannian manifold ( n , g) satisfying 0 < Ric ≤ (n -1)(2 -nc μ 1 c) for a constant c, where μ 1 is the first eigenvalue of the Laplacian, is isometric to S n (c) if n admits a nonzero conformal gradient vector field. They also proved that if n is an Einstein manifold such that Einstein constant is μ = (n -1)c, then n is isometric to S n (c) with c > 0 if it admits a conformal gradient vector field. Taking account of ODE (1.1), Barros et al. [11] showed that the gradient almost Ricci soliton ( n , g, ∇ω, λ) that is compact is isometric to the Euclidean sphere with Codazzi-Ricci tensor and constant sectional curvature. For more terminology of Obata equation, see [30]. Motivated by the previous studies, we will establish the following results: ) be a C-totally real isometric embedding of the warped product submanifold n into a cosymplectic space form M 2m+1 ( ) with nonnegative Ricci curvature. Then, the compact and minimal base N 1 is isometric to the Euclidean sphere S m 1 ( λ 1 m 1 ) if the following equality holds: where λ 1 > 0 is the eigenvalue connected to the eigenfunction ω = ln f and Hess(ω) is a Hessian tensor for the function ω. Moreover, here the constant curvature c is equal to λ 1 m 1 . In particular, if λ 1 = m 1 satisfies the condition then the base N 1 is isometric to the standard sphere S m 1 .
From the Bochner formula, we are able to prove the following result: ) be a C-totally real isometric embedding for the warped product submanifold n to the cosymplectic space form M 2m+1 ( ) with base N 1 being minimal and compact. If the Ricci curvature of n is nonnegative, then N 1 is isometric to the sphere S m 1 (c) with constant curvature equal to c = λ 1 m 1 if the following equality holds: where λ 1 > 0 is an eigenvalue associated with the eigenfunction ω = ln f . Moreover, n = dim , m 1 = dim N 1 , and m 2 = dim N 2 .
For examples of C-totally real isometric immersions from warped product manifolds, see [31,34].

Preliminaries and notations
Let ( M, g) be an odd-dimensional C ∞ -manifold equipped with an almost contact structure (ϕ, κ, η) such that for any W 1 , W 2 ∈ (T M). Of course, the notations are well known: κ is a structure vector field, (1, 1)-type tensor field is denoted by ϕ, and η is the dual 1-form. Moreover, the tonsorial equation for a cosymplectic manifold [7] with the structure (ϕ, κ, η) is given by if we choose two vector fields W 1 , W 2 over M such that ∇ is the Riemannian connection corresponding to g. Assume that M 2m+1 ( ) is a cosymplectic space form with constant ϕ-sectional curvature , then its curvature tensor R is for all W 1 , W 2 , W 3 , W 4 ∈ (T M). Moreover, if the structure vector field κ belongs to the normal space of n , then n is said to be a C-totally real submanifold; for more details, see [7,26,28,32,33]. It should be noted that the curvature tensor R for n in cosymplectic space form M 2m+1 ( ) is defined as Suppose n is a Riemannian submanifold of a Riemannian manifold M 2m+1 considering induced metric g, ∇ and ∇ ⊥ are connections along T and T ⊥ of n , where T is a tangent bundle and T ⊥ is the normal bundle of n . Therefore, the Gauss and Weingarten formulae are written as and ξ ∈ X(T ⊥ ). Note that ζ and A ξ denote the second fundamental form and shape operator, respectively, for the embedding of n to M 2m+1 , and they are governed by the relation g(ζ (W 1 , where the curvature tensors of M 2m+1 and n are represented by R and R. Also, the mean curvature H of n is calculated as H = 1 n trace(ζ ), and n is totally umbilical if ζ (W 1 , W 2 ) = g(W 1 , W 2 )H and totally geodesic if ζ (W 1 , W 2 ) = 0, for any W 1 , W 2 ∈ X( ). Furthermore, n is minimal if H = 0. Here, gives the second fundamental form kernel of n over x. If the plane section is spanned by e and e γ over x in M 2m+1 then such a curvature is called sectional curvature and it is denoted by K γ = K(e ∧ e γ ). The relation between the scalar curvature τ (T x M) of M 2m+1 and K(e ∧ e γ ) at some x in M 2m+1 is represented by The equality in (2.8) is equivalent to the following: The latter relation will be utilized in the subsequent proofs. Similarly, the scalar curvature τ (L x ) of an L-plane is expressed as Let {e 1 , . . . , e n } be an orthonormal frame of the tangent space T x and e r = (e n+1 , . . . , e 2m+1 ) be an orthonormal frame of the normal space T ⊥ . Hence we have ζ r γ = g ζ (e , e γ ), e r and Let K γ and K γ be the sectional curvature of a submanifold n and M 2m+1 , respectively, then we have following relation due to the Gauss equation (2.6): 2τ Furthermore, the Ricci tensor is defined for an orthonormal basis {e 1 , . . . , e n } of n as Using the distinct indices for vector fields {e 1 , . . . , e n } on n from e u , which is governed by W , then the Ricci curvature is given as (2.14) Therefore, equation (2.9) can be written as Ric(e u ), (2.16) which will be frequently used in the following study. For a k-plane L k of T x n , suppose The gradient squared-norm of the positive smooth function ω of the orthonormal basis {e 1 , . . . , e n } is given by Assume that N 1 and N 2 are Riemannian manifolds with Riemannian metrics g 1 and g 2 , respectively. Suppose f is a differentiable function in N 1 . Then, the manifold N 1 × N 2 equipped with the Riemannian metric g = g 1 + f 2 g 2 is referred to as the warped product manifold and defined as n = N 1 × f N 2 (for details, see [17]). Assume that n = N 1 × f N 2 is a nontrivial warped product, then ∀ W 1 ∈ (N 1 ) and W 2 ∈ (N 2 ), we have The following relation was proved in (see (3.3) in [17]) as follows: (2.20) x ∈ n ⇐⇒ n is totally geodesic, and either an N 2 -totally geodesic WPS, or an N 2 -totally umbilical WPS in x ∈ M 2m+1 such that dim N 2 = 2.
(2) The equality in (2.21) is satisfied for any unit tangent vectors at n and any x ∈ n ⇐⇒ n is either totally geodesic or totally umbilical, mixed totally geodesic and N 1 -totally geodesic WPS such that dim N 2 = 2.
Case II. Assume that e u is tangent N 2 . Fix a unit tangent vector field from e m 1 +1 , . . . , e n in which W = e u = e n . Utilizing (2.14)-(2.29) and following a similar technique as in Case I implies Using ( From the minimality of the base of the warped product submanifold n , we get This gives the proof of inequality (2.21). We will use the technique adopted for case (1) to get inequality (2.21) when n is N 2 -minimal. Now equality (2.21) can be verified similarly as in [3,4,29].
For completely minimal submanifolds, Lemma 2.1 will lead to the following result.

Lemma 2.2
Assume ω : n = N 1 × f N 2 − → M 2m+1 is a C-totally real minimal isometric embedding of a warped product n to the cosymplectic space form M 2m+1 . Then, for any unit vector W ∈ T x n , the following Ricci inequality is satisfied: where m 1 = dim N 1 and m 2 = dim N 2 .

Conclusions
In brief, it is well known that a cosmological model of the universe consisting of a perfect fluid whose molecules are galaxies is a Robertson-Walker spacetime. For example, if S 3 indicates a three-dimensional manifold with constant curvature κ = -1, 0, 1 and I denotes an open interval in the real line R, then a warped product of the form (κ, f ) = I × f S 3 with its metric ds 2 = -dt 2 + f 2 ds 2 S is a Robertson-Walker spacetime. Therefore, the concept of a warped product submanifold is useful because of its importance in mathematical physics [5,6,12,16,18,19]. In the present work, we have combined the ordinary differential equation with warped product submanifolds. Therefore, the paper presents excellent combinations of ordinary differential equation with Riemannian geometry.