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

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 ̃ M2m+1( ) to be isometric to the Euclidean space Rn or a warped product of complete manifold N and Euclidean space R.


Background and motivation
In [11,12,[17][18][19], the authors gave the characterizations of Euclidean spaces by analyzing a differential equation. They have shown that a function ψ which is non-constant on a complete manifold ( n , g) agrees with the following equation: if and only if ( n , g) is isometric to the Euclidean spaces R n , 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 if and only if there exists a real-valued non-constant function ψ associated with the negative eigenvalue λ 1 < 0 that has the solution of the following differential equation: These types of complete space classifications are extremely significant and were researched by several mathematicians (see [2,6,[9][10][11]16]). For example, by using (1.1), Al-Dayel, Deshmukh, and Belova [3] showed that a connected and complete Riemannian manifold ( n , g) is isometric to R n if and only if the nontrivial concircular vector field u along the function ψ satisfies R(∇ψ, ∇ψ) = 0 or u = 0. In [7], Chen and Deshmukh proved that a complete Riemannian manifold admits a concurrent vector field if and only if it is isometric to an Euclidean space by using (1.1). Similarly, in [8] it has been shown that ( n , g) is isometric to the Euclidean space if and only if ( n , g) permits the nontrivial gradient conformal vector field which is named Jacobi-type vector field. On the other hand, Matsuyama [14] derived a characterization such that the complete totally real submanifold n of the complex projective space CP n with bounded Ricci curvature admits a function ψ satisfying (1.3) for λ 1 ≤ n, then 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). Also, similar results have been obtained for generalized Sasakian space forms by Jamali and Shahid [13].
In the present paper, by using the Chen-Ricci inequality which was derived in [4], and influenced by the studies in [1-3, 6, 13], we derive similar characterization for C-totally real warped product submanifolds of cosymplectic space forms as rigidity theorems.

Notations and formulas
The almost contact metric manifold ( M, g) with Riemannian metric g satisfies the conditions for the almost contact structure (φ, η, ξ ) and ∀ W 1 , W 2 ∈ (T M). The manifold M 2m+1 is defined to be a cosymplectic manifold if the following relation holds: where ∇ denotes the Riemannian connection with concern of the metric g. A cosymplectic space form is the cosymplectic manifold considering the constant φ-sectional curvature , it is also defined as M 2m+1 ( ); see [Eq. (6.2) in [5]], where the Riemannian curvature tensor of M 2m+1 ( ) is defined in detail.
In case the structure field ξ is perpendicular to the submanifold n in M 2m+1 ( ), n is a C-totally real submanifold of M 2m+1 ( ); also this case φ maps any tangent space of n into its correspondent to the normal space (see [3,5]). Now, we remember the Bochner formula for a differential function on a Riemannian manifold n , that is, ψ : n → R, we have such that the Ricci tensor of n is denoted by Ric. Now, let ψ be a differential function defined on . Thus, the gradient ∇f is given as follows: The Laplacian ψ of ψ is also given by Theorem 2.1 (Green's theorem [20]) Let be a compact oriented Riemannian manifold without boundary, and let ψ be a differential function, then the following formula holds: where dV denotes the volume of .

The main results
To prove our main result, the next lemma which is proved in [4], is stated.
for every unit vector X ∈ T x n , where p = dim B and q = dim F. The qualities in the above inequality have been discussed in detail in [4].
The next compositions will be used to the end of this paper: 'CSF' as cosymplectic space form, 'WF' as warping function, and 'WPS' as warped product submanifold. More precisely, we give the next theorem.
Assuming that the Ricci curvature is bounded below with any positive constant K > 0, that is, Ric(X) ≥ K , we have One of the most famous results connecting the curvature and topology of the complete Riemannian manifold n is the famous theorem of Myers [15], which states that if the Ricci curvature with regard to unit vectors on n is bounded with a positive constant K > 0, then n is compact. This implies that n is compact, then taking integration (3.3) and using Green's lemma, we find that This can be written as On the other hand, we have Hess(ψ) -tI 2 = Hess(ψ) 2 + t 2 |I| 2 -2tg I, Hess(ψ) , which leads to the following: g(Hess(ψ), I * ) = tr(I * Hess(ψ)) = trHess(ψ) =ψ and (2.3) Hess(ψ) -tI 2 = 2t ψ + t 2 p + Hess(ψ) 2 .
Substituting t = λ 1 p and integrating the preceding equation with respect to the volume element dV , we obtain As we assumed that Ric(∇ψ, ∇ψ) ≥ K for K > 0, then we proceed to the next expression n Hess(ψ) - Inserting Eq. (3.4) into the above equation, we derive n Hess(ψ) - If (3.2) is satisfied, then (3.7) implies that Hence, we get for any V ∈ (B) with constant c = λ 1 p . Therefore, by implementing Tashiro theorems [17,19], we analyze that B is isometric to an Euclidean space R p .
The next result comes from the motivation by the study of Río, Kupeli, and Unal [12]. We prove the following.
If ψ is an eigenfunction associated with the eigenvalue λ 1 such that ψ = λ 1 ψ with implemented integration, then we get the following from the preceding equation: On the other hand, we obtain Applying Green's theorem along with ψ = λ 1 ψ, we arrive at The above equation implies that n Hess(ψ) + λ 1 p ψI (3.14) Our assumption (3.8) is satisfied, that is, From (3.14) and (3.15), we get the following: The above equation gives us Tracing the preceding equation, we derive ψ + λ 1 ψ = 0. (3.16) According to [12], the base B is isometric to the connected components of hyperbolic space if (∇ψ) x = 0. But (∇ψ) x = 0 leads to a contradiction as n is a nontrivial warped product. Hence B is isometric to the warped product of the type R × θ N , where N is the complete Riemannian manifold, also R is an Euclidean line. Moreover, the warping function θ ensures the following differential equation: This proof is completed.
For the minimality case, that is, H 2 = 0, we give the following corollary. Then B is isometric to an Euclidean space R p .
We give the following corollary by using Theorem 3.2.