EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 2481-2491 ISSN 1307-5543 – ejpam.com Published by New York Business Global Chen’s Inequality for CR-Warped Products in Locally Metallic Riemannian Manifolds Lamia Saeed Alqahtani1,∗, Ebtehal M. AL-Husainy1,2 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia 2 Mathematics Department, Faculty of Science, Taibah University, 344 AL-Madinah AL- Munawara, Saudi Arabia Abstract. In this paper, we study CR-warped product submanifolds in locally metallic Rieman- nian manifolds. We provide several non-trivial examples of such submanifolds. We establish a sharp inequality known as Chen’s inequality for the squared norm of the second fundamental form. We also discuss the equality case of Chen’s inequality. 2020 Mathematics Subject Classifications: 53B25, 53C15, 53C40, 53C42, 53D10 Key Words and Phrases: Warped product, CR warped product, Metallic Riemannian structure, Golden Structure, CR submanifold 1. Introduction Warped products are considered a generalization of Cartesian products. The study of warped product manifolds was developed by Bishop and O’Neill in [1], who obtained fundamental properties of warped product manifolds and constructed a class of complete manifolds with negative curvature. Subsequently, B. Y. Chen studied CR-submanifolds of Kähler manifolds, that are warped products of complex and totally real submanifolds, and published his findings in a series of papers [6–8]. Also, He presented a multitude of properties for warped product manifolds and submanifolds in [9] and discussed the applications of these properties to differential geometry and geometric analysis. Hretcanu and Crasmareanu introduced the notion of a Golden structure on Riemannian manifolds in [15]. They showed that a Golden structure is a generalization of an almost product structure. The properties of submanifolds in Golden Riemannian manifolds were then studied in [10, 16] using the correspondences between a Golden structure and an almost product structure. The metallic structure, defined in [17], is a further generalization of the Golden structure. Different types of submanifolds in metallic Riemannian manifolds ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5492 Email addresses: Lalqahtani@kau.edu.sa (L. Alqahtani), ealhusainy@stu.kau.edu.sa (E. AL-Husainy) https://www.ejpam.com 2481 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2482 were studied in [3, 13], which involved obtaining different integrability conditions for the distributions related to these submanifolds. The metallic warped product Riemannian manifold was studied in [2, 14]. After that, Hretcanu and Blaga worked on the existence problem of proper warped product bi-slant submanifolds in locally metallic Riemannian manifolds [14]. They provided a brief overview of metallic Riemannian manifolds and their submanifolds, and then discussed slant and bi-slant submanifolds (including semi-slant and hemi-slant submanifolds) in locally metallic Riemannian manifolds. They also studied the properties of warped product bi-slant submanifolds in metallic Riemannian manifolds and investigated the existence of various types of warped products, including warped product CR submanifolds in locally metallic Riemannian manifolds [14] where they proved that there is no proper CR warped product of the form MT ×f M⊥, where MT and M⊥ are invariant and anti-invariant submanifolds, respectively, in a locally metallic Riemannian manifold. A study related to this field using other mathematical methods that may be relevant from another point of view in [18]. In this paper, we continue the research on warped product CR-submanifolds of the form M⊥ ×f MT in locally metallic Riemannian manifolds. We provide some examples of warped product CR-submanifolds in a metallic Riemannian manifold. Also, we obtain some useful lemmas that will be used to prove our main theorem. We derive a relation for the squared norm of the second fundamental form in terms of the components of the gradient of the warping function, and consider the equality case. 2. Preliminaries Let M̃ be a smooth manifold of dimension m. The metallic structure J is a (1,1) tensor field defined by the equation J2 = pJ + qI, (1) where p, q∈ N and I is the identity operator on the space of all vector fields on M̃ , denoted by Γ(TM̃), [17]. A metallic Riemannian manifold is a Riemannian manifold (M̃, g̃) where the Rieman- nian metric g̃ is J-compatible. This means that g̃(JX, Y ) = g̃(X, JY ) (2) holds for all vector fields X and Y in Γ(TM̃)[17]. Specifically, a Golden structure is a special type of a metallic structure. It is defined by the equation J2 = J + I, where p = q = 1 [11]. On a metallic Riemannian manifold M̃ , the Riemannian metric g̃ satisfies the equation g̃(JX, JY ) = pg̃(JX, Y ) + qg̃(X,Y ), (3) for all X,Y ∈ Γ(TM̃). This equation can be derived from equations (1) and (2). Now, let M be a submanifold embedded in a metallic Riemannian manifold (M̃, g̃, J). Let TX and NX be the tangential and normal components of JX, respectively, for any L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2483 X ∈ Γ(TM). Similarly, let tV and nV be the tangential and normal components of JV , respectively, for any V ∈ Γ(T⊥M). That is JX = TX +NX, (4) JV = tV + nV. (5) This implies that for any X,Y in Γ(TM) and U, V in Γ(T⊥M), we have [3] g̃(TX, Y ) = g̃(X,TY ), (6) g̃(nU, V ) = g̃(U, nV ), (7) g̃(NX,V ) = g̃(X, tV ). (8) Clearly, the maps T and n are g̃ -symmetric. Consequently, the following equations hold for any X ∈ Γ(TM) and V ∈ Γ(T⊥M), [13] T 2X = pTX + qX − tNX, pNX = NTX + nNX, (9) n2V = pnV + qV −NtV, ptV = TtV + tnV. (10) Suppose that ∇̃ and ∇ be the Levi-Civita connections on Riemannian manifolds M̃) and M , respectively. Then, for any X.Y ∈ Γ(M), V ∈ Γ(T⊥M), the Gauss and Wein- garten formulas are given by ∇̃XY = ∇XY + h(X,Y ), (11) ∇̃XV = −AV X +∇⊥ XV, (12) where h and AV are the second fundamental form and the shape operator on M , respec- tively [5]. They are related by g̃(h(X,Y ), V ) = g̃(AV X,Y ). (13) A locally metallic Riemannian manifold (M̃, g̃, J) is a manifold that has a metallic Riemannian structure such that J is parallel with respect to the Levi-Civita connection ∇̃ on M̃ , that is ∇̃J = 0, [12]. Hence, from equation (1), one can see that the following equation holds for any X,Y, Z ∈ Γ(TM), [4] g̃((∇̃XJ)Y,Z) = g̃(Y, (∇̃XJ)Z). (14) Now, let (M1, g1) and (M2, g2) be two Riemannian manifolds, then the warped product M1 ×f M2 is a Riemannian manifold with Riemannian metric g = g1 + f2g2, where f is a positive smooth function on M1, called the warping function [1]. Note that if M1 and M2 have dimension n1 and n2, respectively, then the dimension of the warped product M1 ×f M2 is n = n1 + n2. On the warped product M = M1×f M2, if X,Y ∈ Γ(TM1), and Z,W ∈ Γ(TM2), then [9], ∇XY ∈ Γ(TM1), (15) L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2484 ∇XZ = ∇ZX = X(ln f)Z, (16) ∇ZW = ∇́ZW − g(Z,W )∇⃗ ln f. (17) where ∇ is the Levi-Civita connection on M and ∇⃗ ln f is the gradient of ln f which is defined for any X ∈ Γ(TM) as g(∇⃗f,X) = X(f). (18) This implies that ∥ ∇⃗f ∥2= n∑ i=1 (ei(f)) 2 (19) where {e1, · · ·, en} is a local orthonormal frame field on M . Moreover, on any warped product M = M1 ×f M2, M1 and M2 are totally geodesic and totally umbilical submani- folds of M [1]. A submanifold M of a metallic Riemannian manifold M̃ is called invariant under the metallic structure J if J(TxM) ⊂ TxM , for any x ∈ M . It follows that the orthogonal com- plement of the tangent space of M is also invariant under J , J(TxM ⊥) ⊂ TxM ⊥, for any x ∈ M . This is because g(X, JU) = g(JX,U) = 0, for any U ∈ Γ(TM⊥) and X ∈ Γ(TM). An anti-invariant submanifold M of M̃ is a submanifold such that J(TxM) ⊂ TxM ⊥, for any x ∈ M [4]. The existence of warped product CR submanifolds in a locally metallic Riemannian manifold M̃ has been studied in[14], and proved the following: Theorem 1. ([14]) Let M = MT ×f M⊥ be a warped product CR-submanifold in a lo- cally metallic Riemannian manifold (M̃, g̃, J), where MT and M⊥ are invariant and anti- invariant submanifolds of M̃ , respectively. Then, M = MT ×f M⊥ is a non-proper warped product submanifold in M̃ , that is, the warping function f is constant on MT . 3. Basic Lemmas and Examples In this section, we consider warped product CR-submanifolds in the form M = M⊥×f MT such that M⊥ is an anti-invariant submanifold and MT is an invariant submanifold of a locally metallic Riemannian manifold M̃ . Lemma 1. Let M = M⊥×f MT be a warped product CR-submanifold in a locally metallic Riemannian manifold M̃ , then for any X,Y ∈ Γ(TMT ) and Z,W ∈ Γ(TM⊥), we have g(h (X,Y ) , JZ) = −Z (ln f) g(X, JY ), (20) g(h (X,Z) , JW ) = 0, (21) g(h (JX, Y ) , JZ) = −pZ (ln f) g(JX, Y ) + qZ (ln f) g(X,Y ). (22) L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2485 Proof. Since M̃ is a locally metallic Riemannian manifold, then by using (3), for any Z ∈ Γ(TM⊥) and X,Y ∈ Γ(TMT ), we have g(h (X,Y ) , JZ) = 1 p g(∇̃XJY, JZ)− q p g(∇̃XY,Z) (23) Also, by Gauss formula (11) and (16), we have g(h (X,Y ) , JZ) = 1 p g(h (X, JY ) , JZ) + q p Z (ln f) g(X,Y ). (24) By interchanging Y by JY in (24), we have g(h (X,JY ) , JZ) = g(h (X,JY ) , JZ) + q p g(h (X,Y ) , JZ)− q p Z (ln f) g(X, JY ). (25) Then, we get g(h (X,Y ) , JZ) = −Z (ln f) g(X, JY ), (26) which proves (20). Now, by using the same strategies, g(h (X,Z) , JW ) = 1 p g(∇̃ZJX, JW )− q p g(∇̃ZX,W ). (27) On the other hand, we derive g(h (X,Z) , JW ) = 1 p g(h (Z, JX) , JW ). (28) By replacing X with JX, and applying (3), we get g(h (JX,Z) , JW ) = g(h (Z, JX) , JW ) + q p g(h (Z,X) , JW ). (29) This implies that g(h (X,Z) , JW ) = 0. By rewriting the equation (20) with JX instead of X, and by using (3), we can see that (22) holds. This completes the proof of the lemma. Corollary 1. Let M = M⊥ ×f MT be a warped product CR-submanifold M in a locally metallic Riemannian manifold M̃ , then g(h (JX, JY ) , JZ) = qg(h (X,Y ) , JZ) + pg(h (JX, Y ) , JZ). (30) for any X,Y ∈ Γ(TMT ) and Z ∈ Γ(TM⊥), Proof. After interchanging Y by JY in (22), and applying (3), we obtain g(h (JX, JY ) , JZ) = −(p2 + q)Z (ln f) g(JX, Y )− pqZ (ln f) g(X,Y ). (31) Using equation (31)and (20) along with (22), we get (30). Here, we provide some examples of a CR-warped product manifold M = M⊥ ×f MT in a locally metallic Riemannian manifold (M̃, g̃, J). L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2486 Example 1. Let M be a submanifold of M̃ defined by the immersion i as follows: i(f, α, θ) = (f cos θ sinα, f cos θ cosα, f sin θ sinα, f sin θ cosα, f, σ̃q −1 2 f, σq −1 2 f), where f > 0, α, θ ∈ (0, π2 ), σ is a metallic number, defined by σ = p+ √ p2+4q 2 , and σ̃ = p−σ, for some positive integers p and q. It is straightforward to compute that the tangent bundle ofM is spanned by the vectors {Z1, Z2, Z3}, where Z1 = cos θ sinα ∂ ∂x1 + cos θ cosα ∂ ∂x2 + sin θ sinα ∂ ∂x3 + sin θ cosα ∂ ∂x4 + ∂ ∂x5 + σ̃ √ q ∂ ∂x6 + σ √ q ∂ ∂x7 , Z2 = f cos θ cosα ∂ ∂x1 − f cos θ sinα ∂ ∂x2 + f sin θ cosα ∂ ∂x3 − f sin θ sinα ∂ ∂x4 , Z3 = −f sin θ sinα ∂ ∂x1 − f sin θ cosα ∂ ∂x2 + f cos θ sinα ∂ ∂∂x3 + f cos θ cosα ∂ ∂x4 . By using the metallic structure J of M̃ which is J(x1, x2, x3, x4, x5, x6, x7) = (σx1, σx2, σx3, σx4, σ̄x5, σx6, σ̄x7), Then, we find that JZ1 = σ cos θ sinα ∂ ∂x1 + σ cos θ cosα ∂ ∂x2 + σ sin θ sinα ∂ ∂x3 + σ sin θ cosα ∂ ∂x4 + σ̃ ∂ ∂x5 + σσ̃ √ q ∂ ∂x6 + σ̃σ √ q ∂ ∂x7 , JZ2 = fσ cos θ cosα ∂ ∂x1 − fσ cos θ sinα ∂ ∂x2 + fσ sin θ cosα ∂ ∂x3 − fσ sin θ sinα ∂ ∂x4 , JZ3 = −fσ sin θ sinα ∂ ∂x1 − fσ sin θ cosα ∂ ∂x2 + fσ cos θ sinα ∂ ∂x3 + fσ cos θ cosα ∂ ∂x4 , Now, we define two vector spaces DT and D⊥, where DT = Span{Z2, Z3} is the invariant distribution and D⊥ = Span{Z1} is the anti-invariant distribution, which are preserved by the action of J . Hence, the Riemannian metric of the warped product CR- submanifold M is given by the following: g = (2 + σ̃2q−1 + σ2q−1)d2f + f2(d2θ + d2α). Then, M = M⊥ × fMT is a warped product CR-submanifold in the metallic Riemannian manifold M̃ . L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2487 Example 2. Let M be a submanifold of M̃ defined by the immersion i as follows: i(u, v, θ, ϕ) =(u cos θ, u sin θ, v cos θ, v sin θ, u cosϕ, u sinϕ, v cosϕ, v sinϕ, 1 √ q σu, 1 √ q σ̃v, 1 √ q σ̃u, 1 √ q σv), where u, v > 0, θ, ϕ ∈ (0, π2 ), σ is a metallic number, defined by σ = p+ √ p2+4q 2 , and σ̃ = p− σ, for some positive integers p and q. By some computation, it is easy to find that the tangent bundle of M is spanned by {Z1, Z2, Z3, Z4}, where Z1 = cos θ ∂ ∂x1 + sin θ ∂ ∂x2 + cosϕ ∂ ∂x5 + sinϕ ∂ ∂x6 + σ √ q ∂ ∂x9 + σ̃ √ q ∂ ∂x11 , Z2 = cos θ ∂ ∂x3 + sin θ ∂ ∂x4 + cosϕ ∂ ∂x7 + sinϕ ∂ ∂x8 + σ̃ √ q ∂ ∂x10 + σ √ q ∂ ∂x12 , Z3 = −u sin θ ∂ ∂x1 + u cos θ ∂ ∂x2 − v sin θ ∂ ∂x3 + v cos θ ∂ ∂x4 , Z4 = −u sinϕ ∂ ∂x5 + u cosϕ ∂ ∂x6 − v sinϕ ∂ ∂∂x7 + v cosϕ ∂ ∂x8 , By using the metallic structure J of M̃ which defines as J(x1, x2, x3, x4, x5, x6, x7, x8,x9, x10, x11, x12) = (σx1, σx2, σx3, σx4, σ̄x5, σ̄x6, σ̄x7, σ̄x8, σ̄x9, σx10, σx11, σ̄x12). Following this, we obtain that JZ1 = σ cos θ ∂ ∂x1 + σ sin θ ∂ ∂x2 + σ̃ cosϕ ∂ ∂x5 + σ̃ sinϕ ∂ ∂x6 + σ̃σ √ q ∂ ∂x9 + σ̃σ √ q ∂ ∂x11 , JZ2 = σ cos θ ∂ ∂x3 + σ sin θ ∂ ∂x4 + σ̃ cosϕ ∂ ∂x7 + σ̃ sinϕ ∂ ∂x8 + σ̃σ √ q ∂ ∂x10 + σ̃σ √ q ∂ ∂x12 , JZ3 = −uσ sin θ ∂ ∂x1 + uσ cos θ ∂ ∂x2 − vσ sin θ ∂ ∂∂x3 + vσ cos θ ∂ ∂x4 , JZ4 = −uσ̃ sinϕ ∂ ∂x5 + uσ̃ cosϕ ∂ ∂x6 − vσ̃ sinϕ ∂ ∂∂x7 + vσ̃ cosϕ ∂ ∂x8 , Now, we define two vector spaces DT and D⊥ such that the invariant distribution DT is spanned by {Z3, Z4} and the anti-invariant distribution D⊥ is spanned by {Z1, Z2}. These distributions are preserved by the action of J . The Riemannian metric of the warped product CR-submanifold M is given by the following L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2488 g = (2 + 1 q σ̃2 + 1 q σ2)(d2u+ d2v) + (u2 + v2)(d2θ + d2ϕ). In other words, M = M⊥ ×f MT is a warped product CR-submanifold in the metallic Riemannian manifold M̃ . 4. Main Theorem In this section, we prove the main result, which is based on Lemma 1. Now, we can define a canonical frame field for an n-dimensional warped product CR-submanifold M = M⊥ × fMT of an m-dimensional locally metallic Riemannian manifold M̃ . Let dim(MT ) = n1 and dim(M⊥) = n2, and so n = n1 + n2. Also, let D and D⊥ be the tangent bundles of MT and M⊥, respectively. The canonical frame field of D and D⊥ are given by the orthonormal vectors {e1, e2, · · · , et, et+1 = Je1 σ , et+2 = Je2 σ , · · · , e2t = en1 = Jet σ }, (32) {en1+1, en1+2, · · · , en=n1+n2}, (33) respectively, where σ is a metallic number. On the other hand, the orthonormal frame fields of the normal subbundles of JD⊥ and µ are respectively {e∗1 = Jen1+1, e ∗ 2 = Jen1+2, · · · , e∗n2 = Jen=n1+n2} (34) {e∗n2+1 = en+n2+1, e ∗ n2+2 = en+n2+2, ..., e ∗ m−n = em} (35) Theorem 2. Let M be a warped product CR-submanifold of a locally metallic Riemannian manifold (M̃, g̃, J), where MT and M⊥ are invariant and anti-invariant submanifolds of M̃ , respectively. In accordance with this, we have (i) The squared norm of the second fundamental form of M satisfies the following in- equality ∥h∥2 ≥ ( p2 2 + q2 σ2 )n1∥∇⃗⊥ ln f∥2, (36) where dim(MT ) = n1, dim(M⊥) = n2, and ∇⃗⊥ ln f is gradient of ln f in the normal direction to M . (ii) If equality sign, in the above inequality, holds identically, then M⊥ and MT are totally geodesic and totally umbilical submanifolds of M̃ , respectively . Proof. As can be inferred from the definition of h, we have ∥h∥2 = n∑ i,j=1 g(h(ei, ej), h(ei, ej)) = m−n∑ r=1 n∑ i,j=1 g(h(ei, ej), e ∗ r) 2. (37) L. Alqahtani, E. AL-Husainy / Eur. J. Pure Appl. Math, 17 (4) (2024), 2481-2491 2489 In view of the above equation and the definitions of the frame fields of D, D⊥, JD⊥ and µ, we can derive the following: ∥h∥2 = n2∑ r=1 n1∑ i,j=1 g(h(ei, ej), e ∗ r) 2 + m−n∑ r=n2+1 n1∑ i,j=1 g(h(ei, ej), e ∗ r) 2 + 2 n2∑ r=1 n1∑ i=1 n∑ j=n1+1 g(h(ei, ej), e ∗ r) 2 + 2 m−n∑ r=n2+1 n1∑ i=1 n∑ j=n1+1 g(h(ei, ej), e ∗ r) 2+ n2∑ r=1 n∑ i,j=n1+1 g(h(ei, ej), e ∗ r) 2 + m−n∑ r=n2+1 n∑ i,j=n1+1 g(h(ei, ej), e ∗ r) 2. (38) Leaving the µ-components in (38), and using (21), we obtain that ∥h∥2 ≥ n2∑ r=1 n1∑ i,j=1 g(h(ei, ej), e ∗ r) 2. With the help of equations (32) then (34), we can find ∥h∥2 ≥ n∑ r=n1+1 t∑ i,j=1 g(h(ei, ej), Jer) 2 + 2 n∑ r=n1+1 t∑ i=1 n1∑ j=t+1 g(h(ei, ej), Jer) 2+ n∑ r=n1+1 n1∑ i,j=t+1 g(h(ei, ej), Jer) 2. (39) Using (20) with the help the canonical frame field of D for all r = 1, · · · , n1, we can simplify the last inequality as follows ∥h∥2 ≥ n∑ r=n1+1 n1∑ j=t+1 t∑ i=1 (σ(er ln f)g(ei, ej)) 2 + 2 n∑ r=n1+1 t∑ i,j=1 ((er ln f)g(ei, J2ej σ ))2+ n∑ r=n1+1 t∑ i,j=1 ((er ln f)g( Jei σ , J2ej σ ))2. From (1), we obtain the following result ∥h∥2 ≥ ( p2 2 + q2 σ2 )2t∥∇⃗⊥ ln f∥2 = ( p2 2 + q2 σ2 )n1∥∇⃗⊥ ln f∥2. Focusing on the fifth and sixth terms of (38), we get h(D⊥, D⊥) = 0. (40) Similarly, we obtain the following from leaving the second term of (34) h(D,D) ⊆ JD⊥. (41) REFERENCES 2490 While the other terms in (34) vanish, the fourth term gives the following: h(D,D⊥) ⊆ JD⊥. (42) Then we can find that M⊥ is totally geodesic in M̃ due to its totally geodesic in M and (40). Similarly, equations (41) and (42) imply that MT is totally umbilical in M̃ due to MT being totally umbilical in M , which ends the proof. Clearly, Theorem 2 is true for the Golden Riemannian manifolds i.e. p = q = 1, and locally product Riemannian manifolds p = 0, q = 1. 5. Conclusion The exploration of warped product submanifolds in locally metallic Riemannian man- ifolds opens several intriguing avenues for future research. In particular, focusing on warped products formed by the product of a proper slant submanifold with an invariant submanifold, called warped product semi-slant, or with an anti-invariant submanifold, called warped product hemi-slant, opens significant aspects for upcoming studies. Further investigation into the geometric properties and characteristics of these submanifolds can deepen our understanding of their structure and the implications of the local metallic- ity of the ambient manifold. Additionally, establishing a comprehensive classification of warped product submanifolds, alongside concrete examples, will enhance the literature and provide benchmarks for further studies. 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