Warped Product Submanifolds in Locally Golden Riemannian Manifolds with a Slant Factor

: In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in a locally Golden Riemannian manifold.


Introduction
B.-Y. Chen studied CR-submanifolds of a Kähler manifold, which are warped products of holomorphic and totally real submanifolds, respectively [1][2][3]. In addition, in his new book [4], he presents a multitude of properties for warped product manifolds and submanifolds, such as: warped product of Riemannian and Kähler manifolds, warped product submanifolds of Kähler manifolds (with the particular cases: warped product CRsubmanifolds, warped product semi-slant or hemi-slant submanifolds of Kähler manifolds), CR-warped products in complex space forms and so on.
Metallic Riemannian manifolds and their submanifolds were defined and studied by C. E. Hretcanu, M. Crasmareanu and A. M. Blaga in [5,6], as a generalization of Golden Riemannian manifolds studied in [7][8][9]. The authors of the present paper studied some properties of invariant, anti-invariant and slant submanifolds [10], semi-slant submanifolds [11] and, respectively, hemi-slant submanifolds [12] in metallic and Golden Riemannian manifolds and they obtained integrability conditions for the distributions involved in these types of submanifolds. Moreover, properties of metallic and Golden warped product Riemannian manifolds were presented in the two previous works of the authors [13,14]. Lately, the study of submanifolds in metallic Riemannian manifolds has been continued by many authors [15][16][17], which introduced the notion of a lightlike submanifold of a metallic semi-Riemannian manifold.
In the present paper, we study warped product pointwise semi-slant and hemi-slant submanifolds in locally Golden Riemannian manifolds. In Section 2, we recall the main properties of Golden Riemannian manifolds and of their submanifolds, and we prove some immediate consequences of the Gauss and Weingarten equations for an isometrically immersed submanifold in a Golden Riemannian manifold. In Section 3, we give some properties of pointwise slant submanifolds in Golden Riemannian manifolds. In Section 4, we study some properties of pointwise bi-slant submanifolds in Golden Riemannian manifolds. In Section 5, we discuss warped product pointwise bi-slant submanifolds in Golden Riemannian manifolds, and in Section 6, we find some properties of pointwise semi-slant and hemi-slant submanifolds in locally metallic Riemannian manifolds. We also provide examples of pointwise slant and pointwise bi-slant submanifolds, of warped product semi-slant, hemi-slant and pointwise bi-slant submanifolds in Golden Riemannian manifolds.

Preliminaries
The Golden number φ = 1+ is the positive solution of the equation It is a member of the metallic numbers family introduced by Spinadel [18], given by the positive solution σ p,q = p+ √ p 2 +4q 2 of the equation x 2 − px − q = 0, where p and q are positive integer values.
The Golden structure J is a particular case of polynomial structure on a manifold [19,20], which satisfies where I is the identity operator on Γ(TM).
If (M, g) is a Riemannian manifold endowed with a Golden structure J such that the Riemannian metric g is J-compatible, i.e., for any X, Y ∈ Γ(TM), then (M, g, J) is called a Golden Riemannian manifold [7]. In this case, g verifies for any X, Y ∈ Γ(TM). Let M be an isometrically immersed submanifold in a Golden Riemannian manifold (M, g, J). The tangent space T x M of M in a point x ∈ M can be decomposed into the direct Let i * be the differential of the immersion i : M → M. Then, the induced Riemannian metric g on M is given by g(X, Y) = g(i * X, i * Y), for any X, Y ∈ Γ(TM). In the rest of the paper, we shall denote by X the vector field i * X for any X ∈ Γ(TM).
For any X ∈ Γ(TM), let TX := (JX) T and NX := (JX) ⊥ be the tangential and normal components, respectively, of JX, and for any V ∈ Γ(T ⊥ M), let tV := (JV) T and nV := (JV) ⊥ be the tangential and normal components, respectively, of JV. Then, we have for any X ∈ Γ(TM) and V ∈ Γ(T ⊥ M).
Let ∇ and ∇ be the Levi-Civita connections on (M, g) and on its submanifold (M, g), respectively. The Gauss and Weingarten formulas are given by for any X, Y ∈ Γ(TM) and V ∈ Γ(T ⊥ M), where h is the second fundamental form and A V is the shape operator, which satisfy For any X, Y ∈ Γ(TM), the covariant derivatives of T and N are given by For any X ∈ Γ(TM) and V ∈ Γ(T ⊥ M), the covariant derivatives of t and n are given by From (2), we obtain for any X, Y, Z ∈ Γ(TM), which implies [21] g for any X, Y, Z ∈ Γ(TM) and V ∈ Γ(T ⊥ M).
The analogue concept of locally product manifold is considered in the context of Golden geometry, having the name of locally Golden manifold [14]. Thus, we say that the Golden Riemannian manifold (M, g, J) is locally Golden if J is parallel with respect to the Levi-Civita connection ∇ on M, i.e., ∇J = 0.

Remark 1.
Any almost product structure F on M induces two Golden structures on M [9]: where φ is the Golden number.
In addition, for an almost product structure F, the decompositions into the tangential and normal components of FX and FV are given by Moreover, the maps f and C are g-symmetric [22]: for any X, Y ∈ Γ(TM) and U, V ∈ Γ(T ⊥ M).

Remark 2 ([11]
). If M is a submanifold in the almost product Riemannian manifold (M, g, F) and J is the Golden structure induced by F on M, then for any X ∈ Γ(TM), we have

Pointwise Slant Submanifolds in Golden Riemannian Manifolds
We shall state the notion of pointwise slant submanifold in a Golden Riemannian manifold, following Chen's definition [23,24] of pointwise slant submanifold of an almost Hermitian manifold.

Definition 1.
A submanifold M of a Golden Riemannian manifold (M, g, J) is called pointwise slant if, at each point x ∈ M, the angle θ x (X) between JX and T x M (called the Wirtinger angle) is independent of the choice of the nonzero tangent vector X ∈ T x M \ {0}, but it depends on x ∈ M. The Wirtinger angle is a real-valued function θ (called the Wirtinger function), verifying for any x ∈ M and X ∈ T x M \ {0}.
A pointwise slant submanifold of a Golden Riemannian manifold is called slant submanifold if its Wirtinger function θ is globally constant.
In a similar manner as in [23], we obtain for some real-valued function x → θ x , for x ∈ M.
From (8) and (21), we have for any X, Y ∈ T x M \ {0} and any x ∈ M.
From (21), by a direct computation, we obtain for any X, Y ∈ T x M \ {0} and any x ∈ M.

Pointwise Bi-Slant Submanifolds in Golden Riemannian Manifolds
In this section, we introduce the notion of pointwise bi-slant submanifold in the Golden Riemannian context. (iii) The distributions D 1 and D 2 are pointwise slant, with slant functions θ 1x and θ 2x , for x ∈ M.
The pair {θ 1 , θ 2 } of slant functions is called the bi-slant function. A pointwise bi-slant submanifold M is called proper if its bi-slant functions θ 1 , θ 2 = 0; π 2 and both θ 1 and θ 2 are not constant on M.
In particular, if θ 1 = 0 and θ 2 = 0; π 2 , then M is called a pointwise semi-slant submanifold; if θ 1 = π 2 and θ 2 = 0; π 2 , then M is called a pointwise hemi-slant submanifold. If M is a pointwise bi-slant submanifold of M, then the distributions D 1 and Now, we provide an example of a pointwise bi-slant submanifold in a Golden Riemannian manifold. Example 1. Let R 6 be the Euclidean space endowed with the usual Euclidean metric ·, · . Let i : M → R 6 be the immersion given by 2 )}. We can find a local orthogonal frame on TM given by We define the Golden structure J : R 6 → R 6 by J(X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) := (φX 1 , φX 2 , φX 3 , φX 4 , φX 5 , φX 6 ), is the Golden number and φ = 1 − φ. We remark that J verifies J 2 = J + I and JX, Y = X, JY , for any X, Y ∈ R 6 . Additionally, we have We remark that JZ 1 , On the other hand, we get We denote by D 1 := span{Z 1 } the pointwise slant distribution with the slant angle θ 1 , where is a pointwise bi-slant submanifold in the Golden Riemannian manifold (R 6 , ·, · , J).

Example 3.
On the other hand, if, in Example 2, we consider f = 0 (i.e., tan v = ±φ), then cos θ 1 = 0, and we remark that M is a hemi-slant submanifold in the Golden Riemannian manifold (R 6 , ·, · , J), with the slant angle θ = arccos 1 If we denote by P i the projections from TM onto D i for i ∈ {1, 2}, then X = P 1 X + P 2 X for any X ∈ Γ(TM). In particular, if X ∈ D i , then X = P i X, for i ∈ {1, 2}.
If we denote by T i = P i • T for i ∈ {1, 2}, then, from (5), we obtain In a similar manner as in [24], we obtain Lemma 1. Let M be a pointwise bi-slant submanifold of a locally Golden Riemannian manifold (M, g, J) with pointwise slant distributions D 1 and D 2 and slant functions θ 1 and θ 2 , respectively. Then (i) for any X, Y ∈ D 1 and Z ∈ D 2 , we have (ii) for any X ∈ D 1 and Z, W ∈ D 2 , we have Proof. From (2), we have for any X, Y ∈ D 1 and Z ∈ D 2 . By using (3) and (∇ X J)Y = 0, we obtain From (25), we obtain JX = T 1 X + NX, JY = T 1 Y + NY and JZ = T 2 Z + NZ for any X, Y ∈ D 1 and Z ∈ D 2 and, from here, we obtain Thus, we obtain By using g(T 1 Y + Y, Z) = 0, we obtain By using (8) and (23), we find and from we have (sin 2 θ 1 − sin 2 θ 2 )g(∇ X Y, Z) = (1 − sin 2 θ 1 )g(∇ X Y, T 2 Z) and from here, we obtain (26).
In the same manner, we find (27).
for any X, Y ∈ D T and Z ∈ D θ , and sin 2 θg(∇ Z W, T 1 X + X) = g(∇ Z W, T 1 X) + g(∇ Z X, T 2 W) for any X ∈ D T and Z, W ∈ D θ . (ii) If the slant functions are θ 1 = θ and θ 2 = 0, we obtain for any X, Y ∈ D θ and Z ∈ D T , and for any X ∈ D θ and Z, W ∈ D T . (i) If the slant functions are θ 1 = π 2 and θ 2 = θ, we obtain for any X, Y ∈ D ⊥ and Z ∈ D θ , and for any X ∈ D ⊥ and Z, W ∈ D θ . (ii) If the slant functions are θ 1 = θ and θ 2 = π 2 , we obtain for any X, Y ∈ D θ and Z ∈ D ⊥ , and for any X ∈ D θ and Z, W ∈ D ⊥ .

Warped Product Pointwise Bi-Slant Submanifolds in Golden Riemannian Manifolds
In [13], the authors of this paper introduced the Golden warped product Riemannian manifold and provided a necessary and sufficient condition for the warped product of two locally Golden Riemannian manifolds to be locally Golden. Moreover, the subject was continued in the papers [14,25], where the authors characterized the metallic structure on the product of two metallic manifolds in terms of metallic maps and provided a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic.
Let (M 1 , g 1 ) and (M 2 , g 2 ) be two Riemannian manifolds (of dimensions n 1 > 0 and n 2 > 0, respectively) and let π 1 , π 2 be the projection maps from the product manifold M 1 × M 2 onto M 1 and M 2 , respectively. We denote by ϕ := ϕ • π 1 the lift to M 1 × M 2 of a smooth function ϕ on M 1 . Then, M 1 is called the base, and M 2 is called the fiber of M 1 × M 2 . The unique element X of Γ(T(M 1 × M 2 )) that is π 1 -related to X ∈ Γ(TM 1 ) and to the zero vector field on M 2 will be called the horizontal lift of X, and the unique element V of Γ(T(M 1 × M 2 )) that is π 2 -related to V ∈ Γ(TM 2 ) and to the zero vector field on M 1 will be called the vertical lift of V. We denote by L(M 1 ) the set of all horizontal lifts of vector fields on M 1 and by L(M 2 ) the set of all vertical lifts of vector fields on M 2 .
For f : M 1 → (0, ∞), a smooth function on M 1 , we consider the Riemannian metric g on M := M 1 × M 2 : Definition 3. The product manifold M of M 1 and M 2 together with the Riemannian metric g is called the warped product of M 1 and M 2 by the warping function f [26]. A warped product manifold M := M 1 × f M 2 is called trivial if the warping function f is constant. In this case, M is the Riemannian product M 1 × M 2 , where the manifold M 2 is equipped with the metric f 2 g 2 (which is homothetic to g 2 ) [4].
We remark that J verifies J 2 = J + I and JX, Y = X, JY , for any X, Y ∈ R 6 . Additionally, we have 6 .
For any X ∈ Γ(TM 1 ) and Z, W ∈ Γ(TM 2 ), by using ( On the other hand, after interchanging Z by X, we have and using (43), we obtain (41).

Warped Product Pointwise Semi-Slant or Hemi-Slant Submanifolds in Golden Riemannian Manifolds
In this section, we obtain some properties of the distributions in the case of pointwise semi-slant and pointwise hemi-slant submanifolds in locally Golden Riemannian manifolds.
If M θ and M T are the integral manifolds of the distributions D 1 and D 2 , respectively, then is a warped product semi-slant submanifold in the Golden Riemannian manifold (R 7 , ·, · , J).
A similar result valid for warped product hemi-slant submanifolds in a locally metallic Riemannian manifold [25], which can be proved following the same steps, holds in our setting, too.

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