Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

Mathematical Science Department, Faculty of Science, Princess Nourah Bint Abdulrahman University, Riyadh 11546, Saudi Arabia Department of Mathematics, College of Science, King Khalid University, 9004 Abha, Saudi Arabia Department of Mathematics, Science and Arts College, Rabigh Campus, King Abdulaziz University,, Jeddah 21589, Saudi Arabia Institute of Physical and Mathematical Sciences and IT, Immanuel Kant Baltic Federal University, 5A. Nevskogo st. 14, 236016 Kaliningrad, Russia


Introduction and Motivations
For geometric analysis, the work of Obata [1] becomes an essential tool of investigation. Obata [1] provided a characterization theorem for a standard sphere in terms of a differential equation, known as the Obata equation. If (M n , g) is a complete manifold with n ≥ 1, then the function φ is nonconstant and fulfills the ordinary differential equation: if and only if there is an isometry between (M n , g) and the sphere S n (c), where c denotes the sectional curvature. If c � 1, then (M n , g) and the unit sphere S n are congruent. A large number of investigations on this subject are studied. erefore, the characterization of these spaces the Euclidean space R n , the Euclidean sphere S n , and the complex projective space CP n are recognized fields in the study of differential geometry and are studied in research works such as [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In particular, the Euclidean space R n is designated through the differential equation ∇ 2 φ � cg, where c is a positive constant, which is proven by Tashiro [19]. In [20], Lichnerowicz has established that if the first nonzero eigenvalue λ 1 of the Laplace operator of the compact manifold (M n , g) with Ric ≥ n − 1 is λ 1 � n, then (M n , g) is isometric to the sphere S n . us, Obata's theorem can be utilized to address Lichnerowicz's eigenvalue equality condition derived in [20]. Deshmukh and Al-Solamy [21] proved that an n-dimensional connected Riemannian manifold (M n , g) which is compact and has a Ricci curvature satisfying 0 < Ric ≤ (n − 1)(2 − (nc/λ 1 )c) for a constant c, where λ 1 is the first eigenvalue of the Laplacian, is isometric to S n (c) if M n is allowed to be a nonzero conformal gradient vector field.
e authors also proved that if M n is an Einstein manifold, meaning that the Einstein constant is μ � (n − 1)c, then M n is isometric to S n (c) with c > 0 if a conformal gradient vector field is allowed. Taking account of ODE (1), Barros et al. [6] showed that the gradient of an almost Ricci soliton (M n , g, ∇φ, λ) that is compact is isometric to the Euclidean sphere when the Ricci tensor is Codazzi, and a constant sectional curvature is present. For more information regarding the Obata equation, see [1]. Motivated by the previous studies, we establish a number of results in the present paper which realize as characterizations of spheres. More precisely, we have the following.
then, the base B is isometric to the standard sphere S p .
From the Bochner formula, we are able to prove the following result: where λ 1 > 0 is a positive eigenvalue associated with the eigenfunction φ � ln ; f. Moreover, n � dimM, p � dimB, and q � dimF. e paper is organized as follows. In Section 2, we study some preliminaries formulas, notations, and definitions related to our study. In the same section, we prove a lemma, a key result for our main theorem. In Section 3, we demonstrate our main conclusions and provide several consequences from our main findings. We also give an example for existence C-totally real warped product submanifold in Sasakian manifolds. In Section 4, we give concluding remarks.

Preliminaries and Notations
Let (M, tg) be the odd-dimensional C ∞ -manifold equipped with an almost contact structure (ψ, ζ, η) such that for any W 1 , W 2 ∈ Γ(TM). Of course, the notations are well known: ζ is a structure vector, the (1, 1)-type tensor field is denoted by ψ, and η is the dual one-form. Moreover, the tensorial equation for a Sasakian manifold [22][23][24] with the structure (ψ, ζ, η) is given by If we choose two vector fields W 1 and W 2 at M, such that ∇ is the Riemannian connection regarding g, and assume that M 2m+1 (ε) is a Sasakian space form with a constant ψ-sectional curvature ε, then its curvature tensor R is for all W 1 , W 2 , W 3 , W 4 ∈ Γ(TM). e odd-dimensional Euclidean space R 2m+1 and odd-dimensional sphere S 2m+1 with sectional curvatures of ε � − 3 and ε � 1 are remarkable examples of Sasakian space forms in [25]. Moreover, if the structure vector field ζ belongs to the normal space of M n , then M n is said to be a C-totally real submanifold; for more 2 Journal of Mathematics details, see the work presented in [22-24, 26, 27]. It should be noted that the curvature tensor R for M n in Sasakian Suppose M n is a Riemannian submanifold of a Riemannian manifold M 2m+1 considering the induced metric g and ∇ and ∇ ⊥ are connections along TM and T ⊥ M of M n , where TM is a tangent bundle and T ⊥ M is a normal bundle of M n . erefore, for any W 1 , W 2 ∈ X(TM) and ξ ∈ X(T ⊥ M), the Gauss and Weingarten formulas are Note that B and A ξ denote the second fundamental form as well as the shape operator, respectively. ey are governed by the relation for any W 1 , W 2 , W 3 , W 4 ∈ X(M), where the curvature tensors of M 2m +1 and M n are represented by R and R. Furthermore, H, which is the mean curvature of M n , is calculated as gives the second fundamental form kernel of M n over x. If the plane section is spanned by e α and e β over x in M 2m+1 , then such a curvature is called a sectional curvature and is denoted by K αβ � K(e α ∧ e β ). e relation between the scalar curvature τ(T x M) of M 2m+1 and K(e α ∧ e β ) at some x in M 2m+1 is represented by e first equality in (11) is reciprocal to the following: e previous relation will be utilized in the subsequent proofs. Similarly, the scalar curvature τ(L x ) of an L-plan is expressed as Let e 1 , . . . , e n be an orthonormal frame of the tangent space T x M and e r � (e n+1 , . . . , e 2m+1 ) be an orthonormal frame of the normal space T ⊥ M. us, we have Let K αβ and K αβ be the sectional curvature of a submanifold M n and M 2m+1 ; then, we have following relation due to the Gauss equation (9): Furthermore, the Ricci tensor is defined as Fixing the distinct indices for vector fields from e 1 , . . . , e n on M n by e u , which is governed by W, the Ricci curvature is given as Journal of Mathematics erefore, equation (12) can be written as Ric e u . (18) us, which will be frequently used in future studies. e gradientsquared norm of the positive smooth function φ for an orthonormal basis e 1 , . . . , e n is given by Assume that B and F are Riemannian manifolds with Riemannian metrics g 1 and g 2 , respectively. Suppose f is a differentiable function on B.
en, the manifold B × F endorsed by the Riemannian metric g � g 1 + f 2 g 2 is referred to a warped product manifold and classified as notation M n � B× f F [28]. Assume that M n � B× f F is a warped product, then we have ∀W 1 ∈ Γ(B) and W 2 ∈ Γ(F ). It was proved in Section 3.3 in [28] that the following relation holds: Remark 2. Sometimes we will use the following abbreviation throughout the paper: "WPS" for Warped product submanifold, "WF" for warping function, "RM" for Riemannian manifold, and "SSF" for Sasakian space form.

Ricci Curvature for C-Totally Real Warped Products
Inspired by the work [2,3,9], we prove the following proposition which we will use in further result.

Proposition 1.
Let M n � B× f F be a C-totally real warped product submanifold into a Sasakian space form M 2m+1 (ε) having the minimal base B. en, for all unit vectors W ∈ T x M n , the following Ricci inequality holds: where p � dimB and q � dimF. (iii) e equality in (23) is satisfied for any unit tangent vectors at M n and any x ∈ M n ⟶ M n is either totally geodesic, or totally umbilical, mixed totally geodesic, and B-totally geodesic WPS such that dimF � 2.
Proof. Assume that M n is a B-minimal C-totally real warped product. An analogous technique will be used for similar cases. Utilizing the Gauss equation (9), we derive Assume e 1 , . . . , e p , e p+1 , . . . , e n is the local orthonormal frame field of M 2m+1 (ε) in which the basis e 1 , . . . , e p are tangent to B and e p+1 , . . . , e n are tangent to F. us, for the unit tangent vector W � e u ∈ e 1 , . . . , e n , we can expand (24): Journal of Mathematics which is equivalent to the following by using (8): As we assumed that the base of the warped product submanifold M n is minimal, we derive In view of (15), we obtain 2m+1 r�n+1 1≤α<β≤n From the fact that the base B is minimal and putting (28) in (27), we deduce

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On the contrary, using (11), we define From (22) and (11), we obtain From (29)-(31) and using (14), we deduce Now, we note that e u is either tangent to the base B or to the fiber F. After that, the proof of the former case is introduced. Case 1. Let e u be tangent to B. We fix the unit tangent vector from e 1 , . . . , e p to be e u and consider W � e u � e 1 . en, from (17) and (32), we obtain Substituting W 1 � W 2 � e α and W 2 � W 2 � e β for 1 ≤ α, β ≤ n in (7) and summarizing, we obtain n α,β�1 R e α , e β , e α , e β � ε + 3 4 n(n − 1).
erefore, using (34) in equation (33), we obtain e calculation of the last two terms of (35) implies 2m+1 r�n+1 1≤i<j≤p In similar way, we obtain 2m+1 r�n+1 1≤i<j≤p Using (37) in (33) leads to As for the warped product submanifold M n such that the base is minimal in M n , we compute the following simplification: At the same time, utilizing the minimality of the base manifold B p , we deduce that Utilizing (39)-(41), equation (38) will take the form e above inequality is equivalent to the following: Using (22), this gives inequality (23). For the second case, we have the following.

Case 2.
Assume that e u is tangent F. We fix a unit tangent vector field from e p+1 , . . . , e n in which W � e u � e n . Utilizing (17) to (33) and following a similar technique from (33)-(42), it implies 8

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Using (34), we obtain As the base of M n is minimal, then By using a similar technique to the first case, using (46) in (45), we obtain (49) us, (48) can be reduced, using the above relation, as follows: (51) From the minimality of the base of warped product submanifold M n , we obtain is gives the proof of inequality (23). We will use the technique adopted for case (i) to determine inequality (23) when M n is F-minimal. Now, equality (23) can be verified in a similar manner as in [2,3,29].
For a completely minimal submanifold, Proposition 1 presents the following result.