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

Authors: Fatemah Mofarreh; Akram Ali; Nadia Alluhaibi; Olga Belova
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifoldMn of Sasakian space forms M2m+1ε. As Chen–Ricci inequality applications, we found the characterization of the base of the warped product Mn via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere Sp.


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## Body

## 1. 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 Mn,g is a complete manifold with n≥1, then the function φ is nonconstant and fulfills the ordinary differential equation:(1)Hessφ+cφg=0,or∇2φ+cφg=0,if and only if there is an isometry between Mn,g and the sphere Snc, where c denotes the sectional curvature. If c=1, then Mn,g and the unit sphere Sn are congruent. A large number of investigations on this subject are studied. Therefore, the characterization of these spaces the Euclidean space ℝn, the Euclidean sphere Sn, and the complex projective space ℂPn are recognized fields in the study of differential geometry and are studied in research works such as [2–18]. In particular, the Euclidean space ℝn is designated through the differential equation ∇2φ=cg, where c is a positive constant, which is proven by Tashiro [19]. In [20], Lichnerowicz has established that if the first nonzero eigenvalue λ1 of the Laplace operator of the compact manifold Mn,g with Ric≥n−1 is λ1=n, then Mn,g is isometric to the sphere Sn. Thus, 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 Mn,g which is compact and has a Ricci curvature satisfying 0<Ric≤n−12−nc/λ1c for a constant c, where λ1 is the first eigenvalue of the Laplacian, is isometric to Snc if Mn is allowed to be a nonzero conformal gradient vector field. The authors also proved that if Mn is an Einstein manifold, meaning that the Einstein constant is μ=n−1c, then Mn is isometric to Snc 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 Mn,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.Theorem 1.
Assume thatΨ:Mn=B×fF⟶M2m+1ε is a C-totally real isometric embedding from a warped product submanifold Mn into a Sasakian space form M2m+1ε with a nonnegative Ricci curvature. Then, the compact and minimal base B is isometric to the Euclidean sphere Spλ1/p if the following equality holds:(2)Hessφ2=λ14pqε+31−n−pq−n2ℍ2,where λ1>0 is the eigenvalue connected to an eigenfunction φ=ln;f for the Laplacian operator and Hessφ is a Hessian tensor for the function φ. Moreover, here, the constant curvature c is equal to λ1/p. In a particular case, if λ1=p satisfies the condition(3)Hessφ2=14qε+31−n−λ1q−n2ℍ2,then, the base B is isometric to the standard sphere Sp.From the Bochner formula, we are able to prove the following result:Theorem 2.
LetΨ:Mn=B×fF⟶M2m+1ε be a C-totally real isometric embedding from a warped product submanifold Mn into a Sasakian space form M2m+1ε having the nonnegative Ricci curvature. Then, compact and minimal base B is isometric to the sphere Spc with a constant sectional curvature equal to c=λ1/p if the following equality holds:(4)ℍ2=ε+31−pq−nn2,where λ1>0 is a positive eigenvalue associated with the eigenfunction φ=ln;f. Moreover, n=dimM, p=dimB, and q=dimF.The paper is organized as follows. In Section2, 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.


## 2. Preliminaries and Notations

LetM˜,g be the odd-dimensional C∞-manifold equipped with an almost contact structure ψ,ζ,η such that(5)ψ2=−I+η⊗ζ,ηζ=1,ψζ=0,ηψ=0,gψW1,ψW2=gW1,W2−ηW1ηW2,ηW1=gW1,ζ,for any W1,W2∈ΓTM˜. 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 [22–24] with the structure ψ,ζ,η is given by(6)∇˜W1ψW2=gW1,W2ζ−ηW2W1,∇˜W1ζ=−ψW1.If we choose two vector fieldsW1andW2 at M˜, such that ∇˜ is the Riemannian connection regarding g, and assume that M˜2m+1ε is a Sasakian space form with a constant ψ-sectional curvature ε, then its curvature tensor R˜ is(7)R˜W1,W2,W3,W4=ε+34gW2,W3gW1,W4−gW1,W3gW2,W4+ε−14ηW1ηW3gW2,W4+ηW4ηW2gW1,W3−ηW2ηW3gW1,W4−ηW1gW2,W3ηW4+gψW2,W3gψW1,W4−gψW1,W3gψW2,W4+2gW1,ψW2gψW3,W4,for all W1,W2,W3,W4∈ΓTM˜. The odd-dimensional Euclidean space ℝ2m+1 and odd-dimensional sphere S2m+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 Mn, then Mn is said to be a C-totally real submanifold; for more details, see the work presented in [22–24, 26, 27]. It should be noted that the curvature tensor R˜ for Mn in Sasakian space form M˜2m+1ε is defined as(8)R˜W1,W2,W3,W4=ε+34gW2,W3gW1,W4−gW1,W3gW2,W4.SupposeMn is a Riemannian submanifold of a Riemannian manifold M2m+1 considering the induced metric g and ∇ and ∇⊥ are connections along TM and T⊥M of Mn, where TM is a tangent bundle and T⊥M is a normal bundle of Mn. Therefore, for any W1,W2∈XTM and ξ∈XT⊥M, the Gauss and Weingarten formulas are written as ∇˜W1W2=∇W1W2+BW1,W2and∇˜W1ξ=−AξW1+∇W1⊥ξ, respectively. Note that B and Aξ denote the second fundamental form as well as the shape operator, respectively. They are governed by the relation gBW1,W2,N=gAξW1,W2. The Gauss equation is(9)RW1,W2,W3,W4=R˜W1,W2,W3,W4+gBW1,W4,BW2,W3−gBW1,W3,BW2,W4,for any W1,W2,W3,W4∈XM˜, where the curvature tensors of M2m+1 and Mn are represented by R˜ and R. Furthermore, H, which is the mean curvature of Mn, is calculated as H=1/ntraceB. Mn is totally umbilical if BW1,W2=gW1,W2H and totally geodesic if BW1,W2=0, for any W1,W2∈XM. Furthermore, Mn is minimal if H=0. Here,(10)Nx=X∈TxM|BW1,W2=0,for allW2∈TxMgives the second fundamental form kernel of Mn over x. If the plane section is spanned by eα and eβ over x in M˜2m+1, then such a curvature is called a sectional curvature and is denoted by K˜αβ=K˜eα∧eβ. The relation between the scalar curvature τ˜TxM˜ of M2m+1 and K˜eα∧eβ at some x in M2m+1 is represented by(11)τ˜TxM˜=∑1≤α<β≤2m+1K˜αβ.The first equality in (11) is reciprocal to the following:(12)2τ˜TxMn=∑1≤α<β≤nK˜αβ,1≤α,β≤n.The previous relation will be utilized in the subsequent proofs. Similarly, the scalar curvatureτ˜Lx of an L-plan is expressed as(13)τ˜Lx=∑1≤α<β≤mK˜αβ.Lete1,…,en be an orthonormal frame of the tangent space TxM and er=en+1,…,e2m+1 be an orthonormal frame of the normal space T⊥M. Thus, we have(14)Bαβr=gBeα,eβ,er,B2=∑α,β=1ngBeα,eβ,Beα,eβ=∑α,β=1nBαβr2.LetKαβ and K˜αβ be the sectional curvature of a submanifold Mn and M2m+1; then, we have following relation due to the Gauss equation (9):(15)2τTxMn=Kαβ=2τ˜TxMn+∑r=n+12m+1BααrBββr−Bαβr2=K˜αβ+∑r=n+12m+1BααrBββr−Bαβr2.Furthermore, the Ricci tensor is defined as(16)S˜W1,W2=∑i=1ng˜R˜eα,W1W2eα,W1,W2∈ΓTxMn.Fixing the distinct indices for vector fields frome1,…,en on Mn by eu, which is governed by W, the Ricci curvature is given as(17)RicW=∑α=1α≠unKeα∧eu.Therefore, equation (12) can be written as(18)τ˜TxMn=∑1≤α<β≤nKeα∧eβ=12∑A=1nRiceu.Thus,(19)2τ˜TxMn=∑1≤α<β≤nKeα∧eβ=12∑u=1nRiceu,which will be frequently used in future studies. The gradient-squared norm of the positive smooth function φ for an orthonormal basis e1,…,en is given by(20)∇φ2=∑i=1neiφ2.Assume thatB and F are Riemannian manifolds with Riemannian metrics g1 and g2, respectively. Suppose f is a differentiable function on B. Then, the manifold B×F endorsed by the Riemannian metric g=g1+f2g2 is referred to a warped product manifold and classified as notation Mn=B×fF [28]. Assume that Mn=B×fF is a warped product, then we have(21)∇W2W1=∇W1W2=W1lnfW2,∀W1∈ΓB and W2∈ΓF. It was proved in Section 3.3 in [28] that the following relation holds:(22)∑α=1p∑β=1qKeα∧eβ=qΔff=qΔlnf−∇lnf2.Remark 1.
Mn=B×fF is Riemannian product manifold if f is a constant.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.


## 3. 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.
LetMn=B×fF be a C-totally real warped product submanifold into a Sasakian space form M2m+1ε having the minimal base B. Then, for all unit vectors W∈TxMn, the following Ricci inequality holds:(23)ℛicW+qΔlnf≤n24H2+q∇lnf2+ε+34pq+n−1,where p=dimB and q=dimF.(i)
In caseHx=0, for x∈Mn, there exists a unit vector W satisfying the equality in (23) if and only if Mn is mixed totally geodesic and W lies in Nx at x.(ii)
IfMn is B-minimal, thus(a)
The equality in (23) remains for any unit tangent vectors at B and any x∈Mn⟶Mn is totally geodesic and B-totally geodesic WPS in M2m+1ε.(b)
The equality in (23) remains for any unit tangent vectors at F and any x∈Mn⟶Mn is totally geodesic, either a F-totally geodesic WPS or a F-totally umbilical WPS in M2m+1ε such that dimF=2.(iii)
The equality in (23) is satisfied for any unit tangent vectors at Mn and any x∈Mn⟶Mn is either totally geodesic, or totally umbilical, mixed totally geodesic, and B-totally geodesic WPS such that dimF=2.Proof.
Assume thatMn 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(24)n2ℍ2=2τTxMn+B2−2τ˜TxMn.
Assumee1,…,ep,ep+1,…,en is the local orthonormal frame field of M2m+1ε in which the basis e1,…,ep are tangent to B and ep+1,…,en are tangent to F. Thus, for the unit tangent vector W=eu∈e1,…,en, we can expand (24):(25)n2ℍ2=2τTxMn+12∑r=n+12m+1B11r+⋯+Bnnr−Buur2+Buur2−∑r=n+12m+1∑1≤α≠β≤nBααrBββr−2τ˜TxMn,which is equivalent to the following by using (8):(26)n2ℍ2=2τTxMn+∑r=n+12m+1B11r+⋯+Bnnr2+2Buur−B11r+⋯+Bnnr2+2∑r=n+12m+1∑1≤α<β≤nBαβr2−2∑r=n+12m+1∑1≤α<β≤nBαrBβr−ε+34nn−1.
As we assumed that the base of the warped product submanifoldMn is minimal, we derive(27)n2ℍ2+ε+34nn−1=∑r=n+12m+1Bp+1n1+1r+⋯+Bnnr2+2Buur−Bp+1p+1r+⋯+Bnnr2+2τTxMn+∑r=n+12m+1∑1≤α<β≤nBαβr2−∑r=n+12m+1∑1≤α<β≤nBααrBββr+∑r=n+12m+1∑a=1a≠uBaur2+∑r=n+1m∑1≤α<β≤nα,β≠uBαβr2−∑r=n+12m+1∑1≤α<β≤nα,β≠uBααrBββr.
In view of (15), we obtain(28)∑r=n+12m+1∑1≤α<β≤nα,β≠uBαβr2−∑r=n+12m+1∑1≤α<β≤nα,β≠uBααrBββr=∑1≤α<β≤nα,β≠AK˜αβ−∑1≤α<β≤nα,β≠AKαβ.
From the fact that the baseB is minimal and putting (28) in (27), we deduce(29)12n2ℍ2+ε+34nn−1=2τTxMn+12∑r=n+12m+12Buur−Bp+1p+1r+⋯+Bnnr2+∑r=n+12m+1∑1≤α<β≤nBαβr2−∑r=n+1m∑1≤α<β≤nα,β≠uBααrBββr+∑r=n+12m+1∑a=1,a≠uBaur2+∑1≤α<β≤nα,β≠uK˜αβ−∑1≤α<β≤nα,β≠uKαβ.
On the contrary, using (11), we define(30)τTxMn=∑1≤α<β≤nKeα∧eβ=∑i=1p∑j=p+1nKei∧ej+∑1≤i<k≤pKei∧ek+∑p+1≤l<o≤nKel∧eo.
From (22) and (11), we obtain(31)τTxMn=qΔff+τTxBp+τTxFq.
From (29)–(31) and using (14), we deduce(32)12n2ℍ2+ε+34nn−1=qΔff−2τ˜TxMn+∑1≤α<β≤nα,β≠uK˜αβ+τ˜TxBp+τ˜TxFq+∑r=n+12m+1∑1≤α<β≤nBαβr2−∑1≤α<β≤nα,β≠uBααrBββr+∑r=n+12m+1∑a=1a≠uBaur2+∑r=n+1m∑1≤i≠j≤pBiirBjjr−Bijr2+∑r=n+12m+1∑p+1≤s≠t≤nBssrBttr−Bstr2+12∑r=n+12m+12Buur−Bp+1p+1r+⋯+Bnnr2.
Now, we note thateu is either tangent to the base B or to the fiber F. After that, the proof of the former case is introduced.Case 1.
Leteu be tangent to B. We fix the unit tangent vector from e1,…,ep to be eu and consider W=eu=e1. Then, from (17) and (32), we obtain(33)12n2ℍ2≥ℛicW+qΔff−2τ˜TxMn+τ˜TxBp+τ˜TxF−ε+34nn−1+∑2≤α<β≤nK˜αβ+12∑r=n+12m+12B11r−Bp+1p+1r+⋯+Bnnr2+∑r=n+12m+1∑1≤α<β≤nBαβr2−∑r=n+12m+1∑1≤i<j≤pBijr2+∑p+1≤s<t≤nBstr2+∑r=n+12m+1∑1≤i<j≤pBiirBjjr+∑r=n+12m+1∑p+1≤s≠t≤nBssrBttr−∑2≤α<β≤nBααrhββr.
SubstitutingW1=W2=eα and W2=W2=eβ for 1≤α,β≤n in (7) and summarizing, we obtain(34)∑α,β=1nR˜eα,eβ,eα,eβ=ε+34nn−1.
Therefore, using (34) in equation (33), we obtain(35)ℛicMW≤n22ℍ2−qΔff+ε+34pq+n−1−12∑r=n+1m2B11r−Bp+1p+1r+⋯+Bnnr2+∑r=n+12m+1∑1≤i<j≤pBijr2+∑p+1≤s<t≤nBstr2−∑r=n+12m+1∑1≤i<j≤pBiirBjjr+∑r=n+1m∑p+1≤s≠t≤nBssrBttr+∑r=n+12m+1∑2≤α<β≤nBααrBββr−∑r=n+12m+1∑1≤α<β≤nBαβr2.
The calculation of the last two terms of (35) implies(36)∑r=n+12m+1∑1≤i<j≤pBijr2+∑p+1≤s<t≤nBstr2−∑r=n+12m+1∑1≤α<β≤nBαβr2=∑r=n+12m+1∑α=1p∑β=p+1nBαβr2.
In similar way, we obtain(37)∑r=n+12m+1∑1≤i<j≤pBiirBjjr+∑r=n+1m∑p+1≤s≠t≤nBssrBttr−∑2≤α<β≤nBαrBβr=∑r=n+12m+1∑j=2pB11rBjjr−∑α=2p∑β=p+1nBααrBββr.
Using (37) in (33) leads to(38)ℛicW≤12n2ℍ2−qΔff+ε+34pq+n−1−∑r=n+12m+1∑α=1p∑β=p+1nBαβ2+∑b=2n1B11rBbb2−∑α=2p∑β=p+1nBααrBββr−12∑r=n+1m2B11r−Bp+1p+1r+⋯+Bnnr2.
As for the warped product submanifoldMn such that the base is minimal in Mn, we compute the following simplification:(39)∑r=n+12m+1∑α=2p∑β=p+1nBααrBββr=−∑r=n+12m+1∑β=p+1nB11rBββr.
Similarly, we have(40)∑r=n+12m+1∑b=2pB11rBbbr=−∑r=n+12m+1B11r2.
At the same time, utilizing the minimality of the base manifoldBp, we deduce that(41)12∑r=n+12m+12B11r−Bp+1p+1r+⋯+Bnnr2+∑r=n+12m+1∑β=p+1nB11rBββr=2∑r=n+12m+1B11r2+12n2ℍ2.
Utilizing (39)–(41), equation (38) will take the form(42)ℛicX≤ε+34pq+n−1−qΔff+∑r=n+12m+1∑β=p+1nB11rBββr−B112.
The above inequality is equivalent to the following:(43)ℛicW≤12n2ℍ2+ε+34pq+n−1−qΔff+∑r=n+12m+1∑β=2nB11rBββr.
Using (22), this gives inequality (23). For the second case, we have the following.Case 2.
Assume thateu is tangent F. We fix a unit tangent vector field from ep+1,…,en in which W=eu=en. Utilizing (17) to (33) and following a similar technique from (33)–(42), it implies(44)12n2ℍ2≥ℛicW+qΔff−2τ˜TxMn+τ˜TxBp+τ˜TxFq+∑1≤α<β≤n−1K˜αβ+12∑r=n+12m+12Bnnr−Bp+1p+1r+⋯+Bnnr2+∑r=n+12m+1∑β=1n−1BnnrBββr+∑r=n+1m∑α=1p∑β=p+1nBαβr2−∑r=n+12m+1∑α=1p∑β=p+1n−1BααrBββr.
Using (34), we obtain(45)ℛicW≤12n2ℍ2−qΔff+ε+34pq+n−1+−12∑r=n+12m+12Bnnr−Bp+1p+1r+⋯+Bnnr2−∑r=n+12m+1∑β=1n−1BnnrBββr−∑r=n+12m+1∑α=1p∑β=p+1nBαβr2+∑r=n+12m+1∑α=1p∑β=p+1n−1BααrBββr.
As the base ofMn is minimal, then(46)∑r=n+12m+1∑α=1p∑β=p+1n−1BααrBββr=0.
By using a similar technique to the first case, using (46) in (45), we obtain(47)ℛicW+qΔff≤12n2ℍ2+ε+34pq+n−1−12∑r=n+12m+12Bnnr−Bp+1q+1r+⋯+Bnnr2−∑r=n+12m+1∑β=1n−1BnnrBββr−∑r=n+1m∑α=1p∑β=p+1nBαβr2.
After some calculations, we obtain(48)∑r=n+12m+112Bp+1p+1r+⋯+Bnnr−2Bnnr2+∑β=n+1n−1BnnrBββ=∑r=n+12m+112Bp+1p+1r+⋯+Bnnr2+2∑r=n+1mBnnr2−∑r=n+12m+1∑β=p+1nBnnrBββr+∑r=n+1m∑β=n+1n−1BnnrBββ−∑r=n+12m+1∑β=p+1nBnnrBββr.
Performing other calculations for the last two terms gives(49)∑r=n+12m+1∑β=n+1n−1BnnrBββ−∑r=n+1m∑β=p+1nBnnrBββr=−∑r=n+12m+1Bnnr2.
Thus, (48) can be reduced, using the above relation, as follows:(50)∑r=n+12m+112Bp+1p+1r+⋯+Bnnr−2Bnnr2+∑β=p+1n−1BnnrBββ=∑r=n+12m+112Bp+1p+1r+⋯+Bnnr2+Bnnr2−∑β=p+1n−1BnnrBββ.
Therefore, using (50) in inequality (47), we deduce that(51)ℛicW≤12n2ℍ2+ε+34pq+n−1−qΔff−14∑r=n+12m+1Bp+1p+1r+⋯+Bnnr2−∑r=n+12m+1Bnnr2−∑β=p+1n−1BnnrBββ+14Bp+1p+1r+⋯+Bnnr2.
From the minimality of the base of warped product submanifoldMn, we obtain(52)ℛicX≤14n2ℍ2+ε+34pq+n−1−qΔff−∑r=n+12m+1Bnnr−∑β=p+1n−1BnnrBββ.
This gives the proof of inequality (23). We will use the technique adopted for case (i) to determine inequality (23) when Mn is F-minimal. Now, equality (23) can be verified in a similar manner as in [2, 3, 29].For a completely minimal submanifold, Proposition1 presents the following result.Lemma 1.
Assumeφ:Mn=B×fF⟶M˜2m+1ε is a C-totally real minimal isometric embedding from a warped product Mn into a Sasakian space form M˜2m+1ε. Therefore, for any unit vector W∈TxMn, the following Ricci inequality is satisfied:(53)ℛicW+qΔlnf≤q∇lnf2+ε+34pq+n−1,where p=dimB and q=dimF.


## 4. Application to Differential Equations

### 4.1. Proof of Theorem1

Consider the following equation withφ=lnf:(54)Hessφ+tφI2=Hessφ2+t2φ2I2+2tφgHessφ,I.However, we know thatI2=traceII∗=p as well as gHessφ,I∗=trHessφI∗=trHessφ. Then, the proceeding equation takes the form(55)Hessφ−tφI2=Hessφ2+pt2φ2−2tφΔφ.Ifλ1 is an eigenvalue of the eigenfunction, then Δφ=λ1φ. Thus, we obtain(56)Hessφ+tφI2=Hessφ2+pt2−2tλ1φ2.On the contrary, we obtain(57)∫B×qΔφ22dV=∫B×qφΔφdV−∫B×q∇φ2dV.Again, usingΔφ=λ1φ, we have(58)∫B×qφ2dV=1λ1∫B×q∇φ2dV.It follows from (56) and (58) that(59)∫B×qHessφ+tφI2dV=∫B×qHessφ2dV+pt2λ1−2t∫B×q∇φ2dV.In particular, takingt=λ1/p on (59) and integrating, we obtain(60)∫B×qHessφ+λ1pφI2dV=∫B×qHessφ2dV−λ1p∫B×q∇φ2dV.Again, integrating (23) and including the Green lemma, we have(61)∫B×qℛicMWdV≤n24∫B×qℍ2dV+q∫B×q∇φ2dV+∫B×qε+34pq+n−1dV.From (60) and (61), we derive(62)1q∫B×qℛicMWdV≤n24q∫B×qℍ2dV−pλ1∫B×qHessφ+λ1nφI2dV+pλ1∫B×qHessφ2dV+∫B×qε+34p+1+p−1qdV.Under the assumption that the Ricci curvature is greater than or equal to zero, i.e.,ℛicW≥0, the above equation implies(63)∫B×qHessφ+λ1pφI2dV≤n2λ14pq∫B×qℍ2dV+∫B×qHessφ2dV+λ1p∫B×qε+34p+1+p−1qdV,which is equivalent to the following:(64)∫B×qHessφ+λ1pφI2dV≤∫B×qHessφ2dV+λ14p∫B×qn2qℍ2+ε+3p+1+p−1qdV.If the following equality holds by assumption (2),(65)∫B×qHessφ2dV=λ14pq∫B×qε+31−n−pq−n2ℍ2dV,then equations (64) and (65) imply that(66)∫B×qHessφ+λ1pφI2dV≤0.However, it is clear that(67)∫B×qHessφ+λ1pφI2dV≥0.Combining equations (66) and (67), we obtain(68)Hessφ+λ1nφI2=0⟹Hessφ=−λ1pφI.Since the WFφ=lnf of a nontrivial WPS Mn is a nonconstant, then equation (68), reduces to Obata’s differential equation where c=λ1/p by λ1>0. Thus, B is isometric to Spλ1/p. This is a complete proof of the first part. On the contrary, if we have λ1=p, then, from (68), we have(69)HessφW1,W2=−φgW1,W2,for any W1,W2∈ΓB. The proof of this theorem is completed.


### 4.2. Proof of Theorem2

Ifφ is the positive differential function on the Riemannian manifold B, then Bochner formula is defined as(70)12Δ∇φ2=Hessφ2+ℛic∇φ,∇φ+g∇φ,∇Δφ.Taking the integration along the volume element and using the Stokes Theorem, we obtain(71)∫B×qHessφ2+ℛic∇φ,∇φ+g∇φ,∇ΔφdV=0.Assume thatλ1 is an eigenvalue of the eigenfunction; then, Δφ=λ1φ, and we have(72)∫B×qHessφ2=−∫B×qℛic∇φ,∇φdV−λ1∫B×q∇φ2dV.Inserting the above equation into (60), we find that(73)∫B×qHessφ+λ1pφI2dV=−∫B×qℛic∇φ,∇φdV−λ1p+1p∫B×q∇φ2dV.Utilizing (61) in the above equation, we arrive at(74)1q∫B×qRicXdV+pλ1p+2∫B×qHessφ+λ1pφI2dV≤n24q∫B×qℍ2dV+pq+n−1q∫B×qε+34dV−pλ1p+1∫B×qℛic∇φ,∇φdV.This can be simplified as(75)∫B×qHessφ+λ1pφI2dV+λ1p+1pq∫B×qℛicXdV+∫B×qℛic∇φ,∇φdV≤λ1n2p+14pq∫B×qℍ2dV+λ1pq+n−1p+1pq∫B×qε+34dV.Following our assumption that the Ricci curvature is greater than or equal to zero, i.e.,Ric≥0, we derive that(76)∫B×qHessφ+λ1pφI2dV≤λ1p+14pq∫B×qn2ℍ2+ε+3pq+n−1dV.If the hypothesis of the theorem regarding the extrinsic condition (4) holds, from the above equation, we obtain(77)HessφW1,W2=−λ1pφgW1,W2,for any W1,W2∈ΓB. This is again Obata’s ODE [1], which implies that the base B is isometric to the Euclidean sphere Spλ1/p. The proof is completed.Using the fact that the warped product submanifoldMn is minimal, we give the following corollary derived from Theorem 1.Corollary 1.
LetM2m+1ε be a Sasakian space form, and let Ψ:Mn=B×fF⟶M2m+1ε be a C-totally real minimal isometric embedding of the warped product submanifold Mn into M2m+1ε with a nonnegative Ricci curvature. Then, there is an isometry between the compact base B and the sphere Sp if it satisfies the following:(78)Hessφ2=14qε+31−n−λ1q.Proof.
AssumingΦ is minimal and λ1=p, then, from (64), we obtain(79)∫B×qHessφ+φI2dV≤∫B×qHessφ2dV+14∫B×qε+3λ1+1+λ1−1qdV.
If assumption (78) holds, we get the following from (79):(80)HessφW1,W2=−φgW1,W2,for a nonconstant function φ=lnf. Thus, [1] completes the proof of the corollary.Remark 3.
We know that every simply connected Sasakian space formM2m+1ε is isometric to the odd-dimensional sphere S2m+1 and odd-dimensional Euclidean space ℝ2m+1 with ϕ-constant sectional curvature ε=1 and ε=−3, respectively. For more details and examples, see the work presented in [24, 26, 27].Another consequence of Theorem1 is as follows.Corollary 2.
LetS2m+1 be a Sasakian space form and Mn be the warped product submanifold having the nonnegative Ricci curvature. If Ψ:Mn=B×fF⟶S2m+1 is a C-totally real minimal isometric embedding of Mn to S2m+1. Then, there is an isometric between the compact base B and the sphere Sp if it satisfies the following:(81)Hessφ2=2q1−n−λ1q.Proof.
Now, substitutingε=1 in (79), we derive that(82)∫B×qHessφ+φI2dV≤∫B×qHessφ2dV+2∫B×qλ1+1+λ1−1qdV.
Applying equation (81) into the above equation, we obtain(83)HessφW1,W2=−φgW1,W2.
The result follows from [1]. This completes the proof of the corollary.One result deduced from Theorem2 is as follows.Corollary 3.
Assume that thatS2m+1 is a Sasakian space form and Mn is a warped product submanifold having the nonnegative Ricci curvature and the compact base B is minimal. If Ψ:Mn=B×fF⟶S2m+1 is a C-totally real isometric embedding from Mn into S2m+1, then B is an isometric to the sphere Spc such that c=λ1/p if the following equality holds:(84)ℍ2=41−pq−nn2.Proof.
Insertingε=1 into (76), we obtain(85)∫B×qHessφ+λ1pφI2dV≤λ1p+24pq∫B×qn2|ℍ|2+4pq+n−1dV.
Following condition (84) and combining it with (85), we derive that(86)HessφW1,W2=−λ1pφgW1,W2.
Then, the Obata Theorem [1] leads to the desired result.If we chooseε=−3 that is the Sasakian space form is nothing but the Euclidean space ℝ2m+1 by using Theorem 1, we obtainCorollary 4.
SupposeΨ:Mn=B×fF⟶ℝ2m+1−3 is a C-totally real isometric embedding of the warped product submanifold Mn into ℝ2m+1−3 of a nonnegative Ricci curvature. Then, a compact minimal base B is isometric to the sphere Sp if the following condition holds:(87)Hessφ2+n24qℍ2=0.Proof.
Puttingε=−3 into (64), we obtain(88)∫B×qHessφ+λ1pφI2dV≤∫B×qHessφ2dV+n24q∫B×qℍ2dV.
Equations (87) and (88) imply that(89)HessφW1,W2=−φgW1,W2.
Again, from Obata’s Theorem [1], we reached the desired result.Similarly, from Theorem2, we deriveCorollary 5.
LetMn be a warped product submanifold and Ψ:Mn=B×fF⟶ℝ2m+1−3 be a C-totally real minimal isometric embedding from Mn to the Sasakian space form ℝ2m+1−3. Then, the compact base B is isometric to the sphere Sp.Proof.
According to the hypothesis of the theorem, we know thatMn is minimal and ε=−3. Therefore, from (76), we determined the ordinary differential equation (89). Thus, we have completed the proof.Remark 4.
For examples of C-totally real isometric immersions from warped product manifolds, see the work presented in [27, 30].For existence such a warped product, we provided an example is the followingExample 1.
(see [30]). At point x, there exists no less than two types C-totally real submanifolds going within x of the non-Sasakianκ,μ-manifold. Assume the foliations are denoted by eigendistributions of h; thus, their leaves would be totally geodesic C-totally real submanifolds for the given non-Sasakian κ,μ-manifold. Suppose a C-totally real submanifold in a contact metric manifold. Let us consider gAζX,Y=−g∇˜Xζ,Y=gΦX+ΦhX,Y, so Aζ=ΦhT, such that ΦhTX is tangential part of ΦhX to any X∈TM. Assume that S is the minimal C-totally real surface of T1M. Construct the warped product manifold −π/2,π/2×cos;tS. Then, the embedding Φ:−π/2,π/2×cos;tS⟶T1M such that Φt,x=sintN+costx, where N is the unit vector perpendicular with the linear subspace including T1M, which is a C-totally real isometric embedding. For more classification, see [31–33].Remark 5.
We provided example1 which shows the existence of C-totally real isometric immersion from warped product into an almost contact metric manifolds. However, we are not claiming that this example will satisfy our assumption.


## 4.1. Proof of Theorem1

Consider the following equation withφ=lnf:(54)Hessφ+tφI2=Hessφ2+t2φ2I2+2tφgHessφ,I.However, we know thatI2=traceII∗=p as well as gHessφ,I∗=trHessφI∗=trHessφ. Then, the proceeding equation takes the form(55)Hessφ−tφI2=Hessφ2+pt2φ2−2tφΔφ.Ifλ1 is an eigenvalue of the eigenfunction, then Δφ=λ1φ. Thus, we obtain(56)Hessφ+tφI2=Hessφ2+pt2−2tλ1φ2.On the contrary, we obtain(57)∫B×qΔφ22dV=∫B×qφΔφdV−∫B×q∇φ2dV.Again, usingΔφ=λ1φ, we have(58)∫B×qφ2dV=1λ1∫B×q∇φ2dV.It follows from (56) and (58) that(59)∫B×qHessφ+tφI2dV=∫B×qHessφ2dV+pt2λ1−2t∫B×q∇φ2dV.In particular, takingt=λ1/p on (59) and integrating, we obtain(60)∫B×qHessφ+λ1pφI2dV=∫B×qHessφ2dV−λ1p∫B×q∇φ2dV.Again, integrating (23) and including the Green lemma, we have(61)∫B×qℛicMWdV≤n24∫B×qℍ2dV+q∫B×q∇φ2dV+∫B×qε+34pq+n−1dV.From (60) and (61), we derive(62)1q∫B×qℛicMWdV≤n24q∫B×qℍ2dV−pλ1∫B×qHessφ+λ1nφI2dV+pλ1∫B×qHessφ2dV+∫B×qε+34p+1+p−1qdV.Under the assumption that the Ricci curvature is greater than or equal to zero, i.e.,ℛicW≥0, the above equation implies(63)∫B×qHessφ+λ1pφI2dV≤n2λ14pq∫B×qℍ2dV+∫B×qHessφ2dV+λ1p∫B×qε+34p+1+p−1qdV,which is equivalent to the following:(64)∫B×qHessφ+λ1pφI2dV≤∫B×qHessφ2dV+λ14p∫B×qn2qℍ2+ε+3p+1+p−1qdV.If the following equality holds by assumption (2),(65)∫B×qHessφ2dV=λ14pq∫B×qε+31−n−pq−n2ℍ2dV,then equations (64) and (65) imply that(66)∫B×qHessφ+λ1pφI2dV≤0.However, it is clear that(67)∫B×qHessφ+λ1pφI2dV≥0.Combining equations (66) and (67), we obtain(68)Hessφ+λ1nφI2=0⟹Hessφ=−λ1pφI.Since the WFφ=lnf of a nontrivial WPS Mn is a nonconstant, then equation (68), reduces to Obata’s differential equation where c=λ1/p by λ1>0. Thus, B is isometric to Spλ1/p. This is a complete proof of the first part. On the contrary, if we have λ1=p, then, from (68), we have(69)HessφW1,W2=−φgW1,W2,for any W1,W2∈ΓB. The proof of this theorem is completed.


## 4.2. Proof of Theorem2

Ifφ is the positive differential function on the Riemannian manifold B, then Bochner formula is defined as(70)12Δ∇φ2=Hessφ2+ℛic∇φ,∇φ+g∇φ,∇Δφ.Taking the integration along the volume element and using the Stokes Theorem, we obtain(71)∫B×qHessφ2+ℛic∇φ,∇φ+g∇φ,∇ΔφdV=0.Assume thatλ1 is an eigenvalue of the eigenfunction; then, Δφ=λ1φ, and we have(72)∫B×qHessφ2=−∫B×qℛic∇φ,∇φdV−λ1∫B×q∇φ2dV.Inserting the above equation into (60), we find that(73)∫B×qHessφ+λ1pφI2dV=−∫B×qℛic∇φ,∇φdV−λ1p+1p∫B×q∇φ2dV.Utilizing (61) in the above equation, we arrive at(74)1q∫B×qRicXdV+pλ1p+2∫B×qHessφ+λ1pφI2dV≤n24q∫B×qℍ2dV+pq+n−1q∫B×qε+34dV−pλ1p+1∫B×qℛic∇φ,∇φdV.This can be simplified as(75)∫B×qHessφ+λ1pφI2dV+λ1p+1pq∫B×qℛicXdV+∫B×qℛic∇φ,∇φdV≤λ1n2p+14pq∫B×qℍ2dV+λ1pq+n−1p+1pq∫B×qε+34dV.Following our assumption that the Ricci curvature is greater than or equal to zero, i.e.,Ric≥0, we derive that(76)∫B×qHessφ+λ1pφI2dV≤λ1p+14pq∫B×qn2ℍ2+ε+3pq+n−1dV.If the hypothesis of the theorem regarding the extrinsic condition (4) holds, from the above equation, we obtain(77)HessφW1,W2=−λ1pφgW1,W2,for any W1,W2∈ΓB. This is again Obata’s ODE [1], which implies that the base B is isometric to the Euclidean sphere Spλ1/p. The proof is completed.Using the fact that the warped product submanifoldMn is minimal, we give the following corollary derived from Theorem 1.Corollary 1.
LetM2m+1ε be a Sasakian space form, and let Ψ:Mn=B×fF⟶M2m+1ε be a C-totally real minimal isometric embedding of the warped product submanifold Mn into M2m+1ε with a nonnegative Ricci curvature. Then, there is an isometry between the compact base B and the sphere Sp if it satisfies the following:(78)Hessφ2=14qε+31−n−λ1q.Proof.
AssumingΦ is minimal and λ1=p, then, from (64), we obtain(79)∫B×qHessφ+φI2dV≤∫B×qHessφ2dV+14∫B×qε+3λ1+1+λ1−1qdV.
If assumption (78) holds, we get the following from (79):(80)HessφW1,W2=−φgW1,W2,for a nonconstant function φ=lnf. Thus, [1] completes the proof of the corollary.Remark 3.
We know that every simply connected Sasakian space formM2m+1ε is isometric to the odd-dimensional sphere S2m+1 and odd-dimensional Euclidean space ℝ2m+1 with ϕ-constant sectional curvature ε=1 and ε=−3, respectively. For more details and examples, see the work presented in [24, 26, 27].Another consequence of Theorem1 is as follows.Corollary 2.
LetS2m+1 be a Sasakian space form and Mn be the warped product submanifold having the nonnegative Ricci curvature. If Ψ:Mn=B×fF⟶S2m+1 is a C-totally real minimal isometric embedding of Mn to S2m+1. Then, there is an isometric between the compact base B and the sphere Sp if it satisfies the following:(81)Hessφ2=2q1−n−λ1q.Proof.
Now, substitutingε=1 in (79), we derive that(82)∫B×qHessφ+φI2dV≤∫B×qHessφ2dV+2∫B×qλ1+1+λ1−1qdV.
Applying equation (81) into the above equation, we obtain(83)HessφW1,W2=−φgW1,W2.
The result follows from [1]. This completes the proof of the corollary.One result deduced from Theorem2 is as follows.Corollary 3.
Assume that thatS2m+1 is a Sasakian space form and Mn is a warped product submanifold having the nonnegative Ricci curvature and the compact base B is minimal. If Ψ:Mn=B×fF⟶S2m+1 is a C-totally real isometric embedding from Mn into S2m+1, then B is an isometric to the sphere Spc such that c=λ1/p if the following equality holds:(84)ℍ2=41−pq−nn2.Proof.
Insertingε=1 into (76), we obtain(85)∫B×qHessφ+λ1pφI2dV≤λ1p+24pq∫B×qn2|ℍ|2+4pq+n−1dV.
Following condition (84) and combining it with (85), we derive that(86)HessφW1,W2=−λ1pφgW1,W2.
Then, the Obata Theorem [1] leads to the desired result.If we chooseε=−3 that is the Sasakian space form is nothing but the Euclidean space ℝ2m+1 by using Theorem 1, we obtainCorollary 4.
SupposeΨ:Mn=B×fF⟶ℝ2m+1−3 is a C-totally real isometric embedding of the warped product submanifold Mn into ℝ2m+1−3 of a nonnegative Ricci curvature. Then, a compact minimal base B is isometric to the sphere Sp if the following condition holds:(87)Hessφ2+n24qℍ2=0.Proof.
Puttingε=−3 into (64), we obtain(88)∫B×qHessφ+λ1pφI2dV≤∫B×qHessφ2dV+n24q∫B×qℍ2dV.
Equations (87) and (88) imply that(89)HessφW1,W2=−φgW1,W2.
Again, from Obata’s Theorem [1], we reached the desired result.Similarly, from Theorem2, we deriveCorollary 5.
LetMn be a warped product submanifold and Ψ:Mn=B×fF⟶ℝ2m+1−3 be a C-totally real minimal isometric embedding from Mn to the Sasakian space form ℝ2m+1−3. Then, the compact base B is isometric to the sphere Sp.Proof.
According to the hypothesis of the theorem, we know thatMn is minimal and ε=−3. Therefore, from (76), we determined the ordinary differential equation (89). Thus, we have completed the proof.Remark 4.
For examples of C-totally real isometric immersions from warped product manifolds, see the work presented in [27, 30].For existence such a warped product, we provided an example is the followingExample 1.
(see [30]). At point x, there exists no less than two types C-totally real submanifolds going within x of the non-Sasakianκ,μ-manifold. Assume the foliations are denoted by eigendistributions of h; thus, their leaves would be totally geodesic C-totally real submanifolds for the given non-Sasakian κ,μ-manifold. Suppose a C-totally real submanifold in a contact metric manifold. Let us consider gAζX,Y=−g∇˜Xζ,Y=gΦX+ΦhX,Y, so Aζ=ΦhT, such that ΦhTX is tangential part of ΦhX to any X∈TM. Assume that S is the minimal C-totally real surface of T1M. Construct the warped product manifold −π/2,π/2×cos;tS. Then, the embedding Φ:−π/2,π/2×cos;tS⟶T1M such that Φt,x=sintN+costx, where N is the unit vector perpendicular with the linear subspace including T1M, which is a C-totally real isometric embedding. For more classification, see [31–33].Remark 5.
We provided example1 which shows the existence of C-totally real isometric immersion from warped product into an almost contact metric manifolds. However, we are not claiming that this example will satisfy our assumption.


## 5. Concluding Remarks

The paper dealed with ordinary differential equation on C-totally real warped product submanifolds from the optimization on the warping function of a C-totally real warped product submanifold of Sasakian space forms. First, we obtained a Ricci curvature inequality in the setting a C-totally real warped product submanifold which is a generalization of Theorem 2.1 in [24]. Then, we studied some characterizations theorems for a C-totally real warped product submanifold of a Sasakian space forms. Therefore, the paper has excellent combinations of ordinary differential equation with Riemnnian geometry. We hope that the paper will get influence in mathematical science because we first applied differential equation in product manifolds.



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