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

Abstract

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.

1. 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 CR-submanifolds, 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.

2. Preliminaries

The Golden number $\varphi =\frac{1+\sqrt{5}}{2}$ is the positive solution of the equation

$$x^2-x-1=0.$$

(1)

It is a member of the metallic numbers family introduced by Spinadel [18], given by the positive solution ${\sigma }_{p,q}=\frac{p+\sqrt{p^2+4q}}{2}$ 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

$$J^2=J+I,$$

(2)

where I is the identity operator on $\Gamma \left(T\overline{M}\right)$.

If ($\overline{M},\overline{g}$) is a Riemannian manifold endowed with a Golden structure J such that the Riemannian metric $\overline{g}$ is J-compatible, i.e.,

$$\overline{g}(JX,Y)=\overline{g}(X,JY),$$

(3)

for any X, $Y\in \Gamma \left(T\overline{M}\right)$, then $(\overline{M},\overline{g},J)$ is called a Golden Riemannian manifold [7].

In this case, $\overline{g}$ verifies

$$\overline{g}(JX,JY)=\overline{g}(J^{2}X,Y)=\overline{g}(JX,Y)+\overline{g}(X,Y),$$

(4)

for any X, $Y\in \Gamma \left(T\overline{M}\right)$.

Let M be an isometrically immersed submanifold in a Golden Riemannian manifold ($\overline{M},\overline{g},J)$. The tangent space $T_x\overline{M}$ of $\overline{M}$ in a point $x\in M$ can be decomposed into the direct sum $T_x\overline{M}=T_{x}M\oplus T_x^{\perp }M,$ for any $x\in M$, where $T_x^{\perp }M$ is the normal space of M in x. Let $i_{*}$ be the differential of the immersion $i:M\to \overline{M}$. Then, the induced Riemannian metric g on M is given by $g(X,Y)=\overline{g}(i_{*}X,i_{*}Y)$, for any X, $Y\in \Gamma \left(TM\right)$. In the rest of the paper, we shall denote by X the vector field $i_{*}X$ for any $X\in \Gamma \left(TM\right)$.

For any $X\in \Gamma \left(TM\right)$, let $TX:={\left(JX\right)}^T$ and $NX:={\left(JX\right)}^{\perp }$ be the tangential and normal components, respectively, of $JX$, and for any $V\in \Gamma \left(T^{\perp }M\right)$, let $tV:={\left(JV\right)}^T$ and $nV:={\left(JV\right)}^{\perp }$ be the tangential and normal components, respectively, of $JV$. Then, we have

$$JX=TX+NX,JV=tV+nV,$$

(5)

for any $X\in \Gamma \left(TM\right)$ and $V\in \Gamma \left(T^{\perp }M\right)$.

The maps T and n are $\overline{g}$-symmetric [10]:

$$\overline{g}(TX,Y)=\overline{g}(X,TY),\overline{g}(nU,V)=\overline{g}(U,nV),$$

(6)

$$\overline{g}(NX,V)=\overline{g}(X,tV),$$

(7)

for any X, $Y\in \Gamma \left(TM\right)$ and U, $V\in \Gamma \left(T^{\perp }M\right)$. Moreover, from [12] for $p=q=1$ in the metallic structure, we obtain

$$T^{2}X=TX+X-tNX,NX=NTX+nNX,$$

(8)

$$n^{2}V=nV+V-NtV,tV=TtV+tnV,$$

(9)

for any $X\in \Gamma \left(TM\right)$ and $V\in \Gamma \left(T^{\perp }M\right)$.

Let $\overline{\nabla }$ and ∇ be the Levi-Civita connections on $(\overline{M},\overline{g})$ and on its submanifold $(M,g)$, respectively. The Gauss and Weingarten formulas are given by

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),{\overline{\nabla }}_{X}V=-A_{V}X+{\nabla }_X^{\perp }V,$$

(10)

for any X, $Y\in \Gamma \left(TM\right)$ and $V\in \Gamma \left(T^{\perp }M\right)$, where h is the second fundamental form and $A_V$ is the shape operator, which satisfy

$$\overline{g}(h(X,Y),V)=\overline{g}(A_{V}X,Y).$$

(11)

For any X, $Y\in \Gamma \left(TM\right)$, the covariant derivatives of T and N are given by

$$\left({\nabla }_{X}T\right)Y={\nabla }_{X}TY-T\left({\nabla }_{X}Y\right),\left({\overline{\nabla }}_{X}N\right)Y={\nabla }_X^{\perp }NY-N\left({\nabla }_{X}Y\right).$$

(12)

For any $X\in \Gamma \left(TM\right)$ and $V\in \Gamma \left(T^{\perp }M\right)$, the covariant derivatives of t and n are given by

$$\left({\nabla }_{X}t\right)V={\nabla }_{X}tV-t\left({\nabla }_X^{\perp }V\right),\left({\overline{\nabla }}_{X}n\right)V={\nabla }_X^{\perp }nV-n\left({\nabla }_X^{\perp }V\right).$$

(13)

From (2), we obtain

$$\overline{g}(\left({\overline{\nabla }}_{X}J\right)Y,Z)=\overline{g}(Y,\left({\overline{\nabla }}_{X}J\right)Z),$$

(14)

for any X, Y, $Z\in \Gamma \left(T\overline{M}\right)$, which implies [21]

$$\overline{g}(\left({\nabla }_{X}T\right)Y,Z)=\overline{g}(Y,\left({\nabla }_{X}T\right)Z),\overline{g}(\left({\overline{\nabla }}_{X}N\right)Y,V)=\overline{g}(\left({\nabla }_{X}t\right)V,Y),$$

(15)

for any X, Y, $Z\in \Gamma \left(TM\right)$ and $V\in \Gamma \left(T^{\perp }M\right)$.

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 $(\overline{M},\overline{g},J)$ is locally Golden if J is parallel with respect to the Levi-Civita connection $\overline{\nabla }$ on $\overline{M}$, i.e., $\overline{\nabla }J=0$.

Remark 1.

Any almost product structure F on $\overline{M}$ induces two Golden structures on $\overline{M}$ [9]:

$$J=\pm \frac{2\varphi -1}{2}F+\frac{1}{2}I,$$

(16)

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

$$FX=fX+\omega X,FV=BV+CV,$$

(17)

for any $X\in \Gamma \left(TM\right)$ and $V\in \Gamma \left(T^{\perp }M\right)$, where $fX:={\left(FX\right)}^T$, $\omega X:={\left(FX\right)}^{\perp }$, $BV:={\left(FV\right)}^T$ and $CV:={\left(FV\right)}^{\perp }$.

Moreover, the maps f and C are $\overline{g}$-symmetric [22]:

$$\overline{g}(fX,Y)=\overline{g}(X,fY),\overline{g}(CU,V)=\overline{g}(U,CV),$$

(18)

for any $X,Y\in \Gamma \left(TM\right)$ and $U,V\in \Gamma \left(T^{\perp }M\right)$.

Remark 2

([11]). If M is a submanifold in the almost product Riemannian manifold $(\overline{M},\overline{g},F)$ and J is the Golden structure induced by F on $\overline{M}$, then for any $X\in \Gamma \left(TM\right)$, we have

$$TX=\frac{1}{2}X\pm \frac{2\varphi -1}{2}fX,NX=\pm \frac{2\varphi -1}{2}\omega X.$$

(19)

3. 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 $(\overline{M},\overline{g},J)$ is called pointwise slant if, at each point $x\in M$, the angle ${\theta }_x\left(X\right)$ between $JX$ and $T_{x}M$ (called the Wirtinger angle) is independent of the choice of the nonzero tangent vector $X\in T_{x}M\setminus \left\{0\right\}$, but it depends on $x\in M$. The Wirtinger angle is a real-valued function θ (called the Wirtinger function), verifying

$$cos{\theta }_x=\frac{\overline{g}(JX,TX)}{\parallel JX\parallel ·\parallel TX\parallel }=\frac{\parallel TX\parallel }{\parallel JX\parallel },$$

(20)

for any $x\in M$ and $X\in T_{x}M\setminus \left\{0\right\}$.

A pointwise slant submanifold of a Golden Riemannian manifold is called slant submanifold if its Wirtinger function $\theta$ is globally constant.

In a similar manner as in [23], we obtain

Proposition 1.

If M is an isometrically immersed submanifold in the Golden Riemannian manifold $(\overline{M},\overline{g},J)$, then M is a pointwise slant submanifold if and only if

$$T^2=\left({cos}^2{\theta }_x\right)(T+I),$$

(21)

for some real-valued function $x\mapsto {\theta }_x$, for $x\in M$.

From (8) and (21), we have

Proposition 2.

Let M be an isometrically immersed submanifold in the Golden Riemannian manifold $(\overline{M},\overline{g},J)$. If M is a pointwise slant submanifold with the Wirtinger angle ${\theta }_x$, then

$$\overline{g}(NX,NY)=\left({sin}^2{\theta }_x\right)[\overline{g}(TX,Y)+\overline{g}(X,Y)],$$

(22)

$$tNX=\left({sin}^2{\theta }_x\right)(TX+X),$$

(23)

for any X, $Y\in T_{x}M\setminus \left\{0\right\}$ and any $x\in M$.

From (21), by a direct computation, we obtain

Proposition 3.

Let M be an isometrically immersed submanifold in the Golden Riemannian manifold $(\overline{M},\overline{g},J)$. If M is a pointwise slant submanifold with the Wirtinger angle ${\theta }_x$, then

$$\left({\nabla }_X{T}^2\right)Y=\left({cos}^2{\theta }_x\right)\left({\nabla }_{X}T\right)Y-sin\left(2{\theta }_x\right)X\left({\theta }_x\right)(TY+Y),$$

(24)

for any X, $Y\in T_{x}M\setminus \left\{0\right\}$ and any $x\in M$.

4. 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.

Definition 2.

Let M be an immersed submanifold in the Golden Riemannian manifold $(\overline{M},\overline{g},J)$. We say that M is a pointwise bi-slant submanifold of $\overline{M}$ if there exists a pair of orthogonal distributions $D_1$ and $D_2$ on M such that

(i) $TM=D_1\oplus D_2$;

(ii) $J\left(D_1\right)\perp D_2$ and $J\left(D_2\right)\perp D_1$;

(iii) The distributions $D_1$ and $D_2$ are pointwise slant, with slant functions ${{\theta }_1}_x$ and ${{\theta }_2}_x$, for $x\in M$.

The pair $\{{\theta }_1,{\theta }_2\}$ of slant functions is called the bi-slant function.

A pointwise bi-slant submanifold M is called proper if its bi-slant functions ${\theta }_1$, ${\theta }_2\ne 0;\frac{\pi }{2}$ and both ${\theta }_1$ and ${\theta }_2$ are not constant on M.

In particular, if ${\theta }_1=0$ and ${\theta }_2\ne 0;\frac{\pi }{2}$, then M is called a pointwise semi-slant submanifold; if ${\theta }_1=\frac{\pi }{2}$ and ${\theta }_2\ne 0;\frac{\pi }{2}$, then M is called a pointwise hemi-slant submanifold.

If M is a pointwise bi-slant submanifold of $\overline{M}$, then the distributions $D_1$ and $D_2$ on M verify $T\left(D_1\right)\subseteq D_1$ and $T\left(D_2\right)\subseteq D_2$.

Now, we provide an example of a pointwise bi-slant submanifold in a Golden Riemannian manifold.

Example 1.

Let ${\mathbb{R}}^6$ be the Euclidean space endowed with the usual Euclidean metric $\langle ·,·\rangle$. Let $i:M\to {\mathbb{R}}^6$ be the immersion given by

$$i(u,v):=\left(cosucosv,cosusinv,sinucosv,sinusinv,sinv,cosv\right),$$

where $M:=\{(u,v)\mid u,v\in (0,\frac{\pi }{2})\}$.

We can find a local orthogonal frame on $TM$ given by

$$Z_1=-sinucosv\frac{\partial }{\partial x_1}-sinusinv\frac{\partial }{\partial x_2}+cosucosv\frac{\partial }{\partial x_3}+cosusinv\frac{\partial }{\partial x_4}$$

$$Z_2=-cosusinv\frac{\partial }{\partial x_1}+cosucosv\frac{\partial }{\partial x_2}-sinusinv\frac{\partial }{\partial x_3}+sinucosv\frac{\partial }{\partial x_4}$$

$$+cosv\frac{\partial }{\partial x_5}-sinv\frac{\partial }{\partial x_6}.$$

We define the Golden structure $J:{\mathbb{R}}^6\to {\mathbb{R}}^6$ by

$$J(X_1,X_2,X_3,X_4,X_5,X_6):=(\varphi X_1,\overline{\varphi }{X}_2,\varphi X_3,\overline{\varphi }{X}_4,\varphi X_5,\overline{\varphi }{X}_6),$$

where $\varphi :=\frac{1+\sqrt{5}}{2}$ is the Golden number and $\overline{\varphi }=1-\varphi$.

We remark that J verifies $J^2=J+I$ and $\langle JX,Y\rangle =\langle X,JY\rangle$, for any X, $Y\in {\mathbb{R}}^6$. Additionally, we have

$$J{Z}_1=-\varphi sinucosv\frac{\partial }{\partial x_1}-\overline{\varphi }sinusinv\frac{\partial }{\partial x_2}+\varphi cosucosv\frac{\partial }{\partial x_3}+\overline{\varphi }cosusinv\frac{\partial }{\partial x_4},$$

$$J{Z}_2=-\varphi cosusinv\frac{\partial }{\partial x_1}+\overline{\varphi }cosucosv\frac{\partial }{\partial x_2}-\varphi sinusinv\frac{\partial }{\partial x_3}+$$

$$+\overline{\varphi }sinucosv\frac{\partial }{\partial x_4}+\varphi cosv\frac{\partial }{\partial x_5}-\overline{\varphi }sinv\frac{\partial }{\partial x_6}.$$

We remark that $\langle J{Z}_1,Z_2\rangle =\langle J{Z}_2,Z_1\rangle =0$, $\langle J{Z}_1,Z_1\rangle =\varphi {cos}^{2}v+\overline{\varphi }{sin}^{2}v$ and $\langle J{Z}_2,Z_2\rangle =1$.

On the other hand, we get

$$\parallel Z_1{\parallel }^2=1,{\parallel Z_2\parallel }^2=2,$$

$$\parallel J{Z}_1{\parallel }^2={\varphi }^2{cos}^{2}v+{\overline{\varphi }}^2{sin}^{2}v=\varphi {cos}^{2}v+\overline{\varphi }{sin}^{2}v+1,{\parallel J{Z}_2\parallel }^2={\varphi }^2+{\overline{\varphi }}^2=3.$$

We denote by $D_1:=span\left\{Z_1\right\}$ the pointwise slant distribution with the slant angle ${\theta }_1$, where $cos{\theta }_1=\frac{f(u,v)}{\sqrt{f(u,v)+1}}$, for $f(u,v):=\varphi {cos}^{2}v+\overline{\varphi }{sin}^{2}v$ a real function on M. In addition, we denote by $D_2:=span\left\{Z_2\right\}$ the slant distribution with the slant angle ${\theta }_2$, where $cos{\theta }_2=\frac{1}{\sqrt{6}}$.

The distributions $D_1$ and $D_2$ satisfy the conditions from Definition 2.

If $M_1$ and $M_2$ are the integral manifolds of the distributions $D_1$ and $D_2$, respectively, then $M:=M_1{\times }_{\sqrt{2}}{M}_2$ with the Riemannian metric tensor

$$g:=d{u}^2+2d{v}^2$$

is a pointwise bi-slant submanifold in the Golden Riemannian manifold $({\mathbb{R}}^6,\langle ·,·\rangle ,J)$.

Example 2.

If, in Example 1, we consider that f is a Golden function (i.e., $f^2=f+1$), then $cos{\theta }_1=1$, and we remark that M is a semi-slant submanifold in the Golden Riemannian manifold $({\mathbb{R}}^6,\langle ·,·\rangle ,J)$, with the slant angle $\theta =arccos\frac{1}{\sqrt{6}}$.

Example 3.

On the other hand, if, in Example 2, we consider $f=0$ (i.e., $tanv=\pm \varphi$), then $cos{\theta }_1=0$, and we remark that M is a hemi-slant submanifold in the Golden Riemannian manifold $({\mathbb{R}}^6,\langle ·,·\rangle ,J)$, with the slant angle $\theta =arccos\frac{1}{\sqrt{6}}$.

If we denote by $P_i$ the projections from $TM$ onto $D_i$ for $i\in \{1,2\}$, then $X=P_{1}X+P_{2}X$ for any $X\in \Gamma \left(TM\right)$. In particular, if $X\in D_i$, then $X=P_{i}X$, for $i\in \{1,2\}$.

If we denote by $T_i=P_i\circ T$ for $i\in \{1,2\}$, then, from (5), we obtain

$$JX=T_{1}X+T_{2}X+NX.$$

(25)

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 $(\overline{M},\overline{g},J)$ with pointwise slant distributions $D_1$ and $D_2$ and slant functions ${\theta }_1$ and ${\theta }_2$, respectively. Then

(i) for any X, $Y\in D_1$ and $Z\in D_2$, we have

$$({sin}^2{\theta }_1-{sin}^2{\theta }_2)\overline{g}({\nabla }_{X}Y,T_{2}Z+Z)$$

(26)

$$=\overline{g}({\nabla }_{X}Y,T_{2}Z)+\overline{g}({\nabla }_{X}Z,T_{1}Y)+({cos}^2{\theta }_1+1)\overline{g}(A_{NZ}Y+A_{NY}Z,X)$$

$$-\overline{g}(A_{N{T}_{1}Y}Z+A_{N{T}_{2}Z}Y,X)-\overline{g}(A_{NZ}{T}_{1}Y+A_{NY}{T}_{2}Z,X);$$

(ii) for any $X\in D_1$ and Z, $W\in D_2$, we have

$$({sin}^2{\theta }_2-{sin}^2{\theta }_1)\overline{g}({\nabla }_{Z}W,T_{1}X+X)$$

(27)

$$=\overline{g}({\nabla }_{Z}W,T_{1}X)+\overline{g}({\nabla }_{Z}X,T_{2}W)+({cos}^2{\theta }_2+1)\overline{g}(A_{NX}W+A_{NW}X,Z)$$

$$-\overline{g}(A_{N{T}_{2}W}X+A_{N{T}_{1}X}W,Z)-\overline{g}(A_{NW}{T}_{1}X+A_{NX}{T}_{2}W,Z).$$

Proof. 

From (2), we have

$$\overline{g}({\nabla }_{X}Y,Z)=\overline{g}({\overline{\nabla }}_{X}Y,Z)=\overline{g}(J^2{\overline{\nabla }}_{X}Y,Z)-\overline{g}(J{\overline{\nabla }}_{X}Y,Z),$$

(28)

for any X, $Y\in D_1$ and $Z\in D_2$.

By using (3) and $\left({\overline{\nabla }}_{X}J\right)Y=0$, we obtain

$$\overline{g}({\nabla }_{X}Y,Z)=\overline{g}({\overline{\nabla }}_X{J}^{2}Y,Z)-\overline{g}({\overline{\nabla }}_{X}JY,Z).$$

(29)

From (25), we obtain $JX=T_{1}X+NX$, $JY=T_{1}Y+NY$ and $JZ=T_{2}Z+NZ$ for any X, $Y\in D_1$ and $Z\in D_2$ and, from here, we obtain

$$\overline{g}({\nabla }_{X}Y,Z)=\overline{g}({\overline{\nabla }}_{X}J{T}_{1}Y,Z)+\overline{g}({\overline{\nabla }}_{X}JNY,Z)-\overline{g}({\overline{\nabla }}_X(T_{1}Y+NY),Z)$$

$$=\overline{g}({\overline{\nabla }}_X{T}_1^{2}Y,Z)+\overline{g}({\overline{\nabla }}_{X}N{T}_{1}Y,Z)+\overline{g}({\overline{\nabla }}_{X}NY,JZ)-\overline{g}({\overline{\nabla }}_X{T}_{1}Y,Z)$$

$$+\overline{g}(A_{NY}X,Z)$$

$$=\overline{g}({\overline{\nabla }}_X\left({cos}^2{\theta }_1(T_{1}Y+Y)\right),Z)-\overline{g}(A_{N{T}_{1}Y}X,Z)+\overline{g}({\overline{\nabla }}_{X}NY,T_{2}Z+NZ)$$

$$+\overline{g}(T_{1}Y,{\overline{\nabla }}_{X}Z)+\overline{g}(A_{NY}X,Z).$$

Thus, we obtain

$$\overline{g}({\nabla }_{X}Y,Z)={cos}^2{\theta }_1\overline{g}({\overline{\nabla }}_X(T_{1}Y+Y),Z)-sin\left(2{\theta }_1\right)X\left({\theta }_1\right)\overline{g}(T_{1}Y+Y,Z)$$

$$-\overline{g}(A_{N{T}_{1}Y}X,Z)-\overline{g}(A_{NY}X,T_{2}Z)-\overline{g}({\overline{\nabla }}_{X}NZ,JY)+\overline{g}({\overline{\nabla }}_{X}NZ,T_{1}Y)$$

$$+\overline{g}(T_{1}Y,{\overline{\nabla }}_{X}Z)+\overline{g}(A_{NY}X,Z).$$

By using $\overline{g}(T_{1}Y+Y,Z)=0$, we obtain

$${sin}^2{\theta }_1\overline{g}({\nabla }_{X}Y,Z)={cos}^2{\theta }_1\overline{g}({\overline{\nabla }}_X{T}_{1}Y,Z)-\overline{g}(A_{N{T}_{1}Y}Z+A_{NY}{T}_{2}Z,X)$$

$$+\overline{g}(JNZ,{\overline{\nabla }}_{X}Y)-\overline{g}(A_{NZ}X,T_{1}Y)+\overline{g}(T_{1}Y,{\overline{\nabla }}_{X}Z)+\overline{g}(A_{NY}Z,X).$$

By using (8) and (23), we find

$$\overline{g}(JNZ,{\overline{\nabla }}_{X}Y)=\overline{g}(tNZ+nNZ,{\overline{\nabla }}_{X}Y)$$

$$={sin}^2{\theta }_2\overline{g}({\nabla }_{X}Y,Z+T_{2}Z)+\overline{g}(NZ-N{T}_{2}Z,{\overline{\nabla }}_{X}Y)$$

$$={sin}^2{\theta }_2\overline{g}({\nabla }_{X}Y,Z)+{sin}^2{\theta }_2\overline{g}({\nabla }_{X}Y,T_{2}Z)$$

$$-\overline{g}({\overline{\nabla }}_{X}NZ,Y)+\overline{g}({\overline{\nabla }}_{X}N{T}_{2}Z,Y)$$

and from

$$\overline{g}({\overline{\nabla }}_X{T}_{1}Y,Z)=-\overline{g}(JY-NY,{\overline{\nabla }}_{X}Z)=\overline{g}({\overline{\nabla }}_{X}Y,JZ)-\overline{g}({\overline{\nabla }}_{X}NY,Z)$$

$$=\overline{g}({\overline{\nabla }}_{X}Y,T_{2}Z)-\overline{g}(Y,{\overline{\nabla }}_{X}NZ)-\overline{g}({\overline{\nabla }}_{X}NY,Z)$$

$$=\overline{g}({\overline{\nabla }}_{X}Y,T_{2}Z)+\overline{g}(Y,A_{NZ}X)+\overline{g}(A_{NY}X,Z)$$

we have

$$({sin}^2{\theta }_1-{sin}^2{\theta }_2)\overline{g}({\nabla }_{X}Y,Z)=(1-{sin}^2{\theta }_1)\overline{g}({\nabla }_{X}Y,T_{2}Z)$$

$$+{cos}^2{\theta }_1\overline{g}(A_{NZ}Y+A_{NY}Z,X)+{sin}^2{\theta }_2\overline{g}({\nabla }_{X}Y,T_{2}Z)$$

$$-\overline{g}(A_{N{T}_{1}Y}Z+A_{NY}{T}_{2}Z+A_{NZ}{T}_{1}Y+A_{N{T}_{2}Z}Y,X)$$

$$-\overline{g}(Y,{\overline{\nabla }}_{X}NZ)+\overline{g}(T_{1}Y,{\nabla }_{X}Z)+\overline{g}(A_{NY}Z,X)$$

and from here, we obtain (26).

In the same manner, we find (27). □

Lemma 2.

Let M be a pointwise semi-slant submanifold in a locally Golden Riemannian manifold $(\overline{M},\overline{g},J)$, with pointwise slant distributions $D_1$ and $D_2$ having slant functions ${\theta }_1$ and ${\theta }_2$.

(i) If the slant functions are ${\theta }_1=0$ and ${\theta }_2=\theta$, we obtain

$${sin}^2\theta \overline{g}({\nabla }_{X}Y,T_{2}Z+Z)=-\overline{g}({\nabla }_{X}Y,T_{2}Z)-\overline{g}({\nabla }_{X}Z,T_{1}Y)$$

(30)

$$-2\overline{g}(A_{NZ}Y,X)+\overline{g}(A_{NZ}{T}_{1}Y+A_{N{T}_{2}Z}Y,X),$$

for any X, $Y\in D^T$ and $Z\in D^{\theta }$, and

$${sin}^2\theta \overline{g}({\nabla }_{Z}W,T_{1}X+X)=\overline{g}({\nabla }_{Z}W,T_{1}X)+\overline{g}({\nabla }_{Z}X,T_{2}W)$$

(31)

$$+({cos}^2\theta +1)\overline{g}(A_{NW}X,Z)-\overline{g}(A_{N{T}_{2}W}X+A_{NW}{T}_{1}X,Z),$$

for any $X\in D^T$ and Z, $W\in D^{\theta }$.

(ii) If the slant functions are ${\theta }_1=\theta$ and ${\theta }_2=0$, we obtain

$${sin}^2\theta \overline{g}({\nabla }_{X}Y,T_{2}Z+Z)=\overline{g}({\nabla }_{X}Y,T_{2}Z)-\overline{g}({\nabla }_X{T}_{1}Y,Z)$$

(32)

$$+({cos}^2\theta +1)\overline{g}(A_{NY}X,Z)-\overline{g}(A_{N{T}_{1}Y}Z+A_{N{T}_{2}Z}Y,X)-\overline{g}(A_{NY}{T}_{2}Z,X),$$

for any $X,Y\in D^{\theta }$ and $Z\in D^T$, and

$${sin}^2\theta \overline{g}({\nabla }_{Z}W,T_{1}X+X)=-\overline{g}({\nabla }_{Z}W,T_{1}X)+\overline{g}({\nabla }_Z{T}_{2}W,X)$$

(33)

$$-2\overline{g}(A_{NX}Z,W)+\overline{g}(A_{N{T}_{2}W}Z,X)+\overline{g}(A_{N{T}_{1}X}Z,W)+\overline{g}(A_{NX}{T}_{2}W,Z),$$

for any $X\in D^{\theta }$ and Z, $W\in D^T$.

Lemma 3.

Let M be a pointwise hemi-slant submanifold in a locally Golden Riemannian manifold $(\overline{M},\overline{g},J)$, with pointwise slant distributions $D_1$ and $D_2$ having slant functions ${\theta }_1$ and ${\theta }_2$.

(i) If the slant functions are ${\theta }_1=\frac{\pi }{2}$ and ${\theta }_2=\theta$, we obtain

$${cos}^2\theta \overline{g}({\nabla }_{X}Y,T_{2}Z+Z)=\overline{g}({\nabla }_{X}Y,T_{2}Z)$$

(34)

$$+\overline{g}(A_{NZ}Y+A_{NY}Z,X)-\overline{g}(A_{N{T}_{2}Z}Y+A_{NY}{T}_{2}Z,X),$$

for any X, $Y\in D^{\perp }$ and $Z\in D^{\theta }$, and

$${cos}^2\theta \overline{g}({\nabla }_{Z}W,X)=-\overline{g}({\nabla }_{Z}X,T_{2}W)$$

(35)

$$-({cos}^2\theta +1)\overline{g}(A_{NX}W+A_{NW}X,Z)+\overline{g}(A_{N{T}_{2}W}X+A_{NX}{T}_{2}W,Z),$$

for any $X\in D^{\perp }$ and Z, $W\in D^{\theta }$.

(ii) If the slant functions are ${\theta }_1=\theta$ and ${\theta }_2=\frac{\pi }{2}$, we obtain

$${cos}^2\theta \overline{g}({\nabla }_{X}Y,Z)=-\overline{g}({\nabla }_{X}Z,T_{1}Y)$$

(36)

$$-({cos}^2\theta +1)\overline{g}(A_{NZ}Y+A_{NY}Z,X)+\overline{g}(A_{N{T}_{1}Y}Z+A_{NZ}{T}_{1}Y,X),$$

for any $X,Y\in D^{\theta }$ and $Z\in D^{\perp }$, and

$${cos}^2\theta \overline{g}({\nabla }_{Z}W,T_{1}X+X)=\overline{g}({\nabla }_{Z}W,T_{1}X)$$

(37)

$$+\overline{g}(A_{NX}W+A_{NW}X,Z)-\overline{g}(A_{N{T}_{1}X}W+A_{NW}{T}_{1}X,Z),$$

for any $X\in D^{\theta }$ and Z, $W\in D^{\perp }$.

5. 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 ${\pi }_1$, ${\pi }_2$ be the projection maps from the product manifold $M_1\times M_2$ onto $M_1$ and $M_2$, respectively. We denote by $\tilde{\phi }:=\phi \circ {\pi }_1$ the lift to $M_1\times M_2$ of a smooth function $\phi$ on $M_1$. Then, $M_1$ is called the base, and $M_2$ is called the fiber of $M_1\times M_2$. The unique element $\tilde{X}$ of $\Gamma \left(T(M_1\times M_2)\right)$ that is ${\pi }_1$-related to $X\in \Gamma \left(T{M}_1\right)$ and to the zero vector field on $M_2$ will be called the horizontal lift of X, and the unique element $\tilde{V}$ of $\Gamma \left(T(M_1\times M_2)\right)$ that is ${\pi }_2$-related to $V\in \Gamma \left(T{M}_2\right)$ and to the zero vector field on $M_1$ will be called the vertical lift of V. We denote by $\mathcal{L}\left(M_1\right)$ the set of all horizontal lifts of vector fields on $M_1$ and by $\mathcal{L}\left(M_2\right)$ the set of all vertical lifts of vector fields on $M_2$.

For $f:M_1\to (0,\infty )$, a smooth function on $M_1$, we consider the Riemannian metric g on $M:=M_1\times M_2$:

$$g:={\pi }_1^{*}{g}_1+{(f\circ {\pi }_1)}^2{\pi }_2^{*}{g}_2.$$

(38)

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 $M:=M_1{\times }_f{M}_2$ is called trivial if the warping function f is constant. In this case, M is the Riemannian product $M_1\times M_2$, where the manifold $M_2$ is equipped with the metric $f^2{g}_2$ (which is homothetic to $g_2$) [4].

In the next considerations, we shall denote by ${(f\circ {\pi }_1)}^2=:f^2$, ${\pi }_1^{*}{g}_1=:g_1$ and ${\pi }_2^{*}{g}_2=:g_2$, respectively.

Lemma 4

([4]). For $X\in \Gamma \left(T{M}_1\right)$ and $Z\in \Gamma \left(T{M}_2\right)$, we have on $M:=M_1{\times }_f{M}_2$ that

$${\nabla }_{X}Z={\nabla }_{Z}X=X(lnf)Z,$$

(39)

where ∇ denotes the Levi-Civita connection on M.

The warped product $M_1{\times }_f{M}_2$ of two pointwise slant submanifolds $M_1$ and $M_2$ in a Golden Riemannian manifold $(\overline{M},\overline{g},J)$ is called a warped product pointwise bi-slant submanifold. Moreover, the pointwise bi-slant submanifold $M_1{\times }_f{M}_2$ is called proper if both of the submanifolds $M_1$ and $M_2$ are proper pointwise slant in $(\overline{M},\overline{g},J)$.

Now, we provide an example of a warped product pointwise bi-slant submanifold in a Golden Riemannian manifold.

Example 4.

Let ${\mathbb{R}}^6$ be the Euclidean space endowed with the usual Euclidean metric $\langle ·,·\rangle$. Let $i:M\to {\mathbb{R}}^6$ be the immersion given by

$$i(f,u):=\left(fsinu,fcosu,f,fcosu,fsinu,u\right),$$

where $M:=\{(f,u)\mid f>0,u\in (0,\frac{\pi }{2})\}$.

We can find a local orthogonal frame on $TM$ given by

$$Z_1=sinu\frac{\partial }{\partial x_1}+cosu\frac{\partial }{\partial x_2}+\frac{\partial }{\partial x_3}+cosu\frac{\partial }{\partial x_4}+sinu\frac{\partial }{\partial x_5}$$

$$Z_2=fcosu\frac{\partial }{\partial x_1}-fsinu\frac{\partial }{\partial x_2}-fsinu\frac{\partial }{\partial x_4}+fcosu\frac{\partial }{\partial x_5}+\frac{\partial }{\partial x_6}.$$

We define the Golden structure $J:{\mathbb{R}}^6\to {\mathbb{R}}^6$ by

$$J(X_1,X_2,X_3,X_4,X_5,X_6):=(\varphi X_1,\varphi X_2,\varphi X_3,\overline{\varphi }{X}_4,\overline{\varphi }{X}_5,\overline{\varphi }{X}_6),$$

where ϕ is the Golden number and $\overline{\varphi }=1-\varphi$.

We remark that J verifies $J^2=J+I$ and $\langle JX,Y\rangle =\langle X,JY\rangle$, for any X, $Y\in {\mathbb{R}}^6$. Additionally, we have

$$J{Z}_1=\varphi sinu\frac{\partial }{\partial x_1}+\varphi cosu\frac{\partial }{\partial x_2}+\varphi \frac{\partial }{\partial x_3}+\overline{\varphi }cosu\frac{\partial }{\partial x_4}+\overline{\varphi }sinu\frac{\partial }{\partial x_5}$$

$$J{Z}_2=\varphi fcosu\frac{\partial }{\partial x_1}-\varphi fsinu\frac{\partial }{\partial x_2}-\overline{\varphi }fsinu\frac{\partial }{\partial x_4}+\overline{\varphi }fcosu\frac{\partial }{\partial x_5}+\overline{\varphi }\frac{\partial }{\partial x_6}.$$

We remark that $\langle J{Z}_1,Z_2\rangle =\langle J{Z}_2,Z_1\rangle =0$, $\langle J{Z}_1,Z_1\rangle =2\varphi +\overline{\varphi }=\varphi +1$ and $\langle J{Z}_2,Z_2\rangle =f^2(\varphi +\overline{\varphi })+\overline{\varphi }=f^2+1-\varphi$.

On the other hand, we obtain

$$\parallel Z_1{\parallel }^2=3,{\parallel Z_2\parallel }^2=2f^2+1,$$

$$\parallel J{Z}_1{\parallel }^2=2{\varphi }^2+{\overline{\varphi }}^2=\varphi +4,{\parallel J{Z}_2\parallel }^2=f^2({\varphi }^2+{\overline{\varphi }}^2)+{\overline{\varphi }}^2=3f^2+2-\varphi .$$

We denote by $D_1:=span\left\{Z_1\right\}$ the slant distribution with the slant angle ${\theta }_1$ and $D_2:=span\left\{Z_2\right\}$ the pointwise slant distribution with the slant angle ${\theta }_2$, where

$$cos{\theta }_1=\frac{\varphi +1}{\sqrt{3(\varphi +4)}}$$

and

$$cos{\theta }_2=\frac{f^2+1-\varphi }{\sqrt{(2f^2+1)(3f^2+2-\varphi )}}.$$

The distributions $D_1$ and $D_2$ are integrable and, if $M_1$ and $M_2$ are the integral manifolds of the distributions $D_1$ and $D_2$, respectively, then $M:=M_1{\times }_{\sqrt{2f^2+1}}{M}_2$ with the Riemannian metric tensor

$$g:=3d{f}^2+(2f^2+1)d{u}^2$$

satisfy the conditions of the warped product of $M_1$ and $M_2$ by the warping function $\sqrt{2f^2+1}$. Thus, we obtain a warped product pointwise bi-slant submanifold in the Golden Riemannian manifold $({\mathbb{R}}^6,\langle ·,·\rangle ,J)$.

In a similar manner as in [25], we obtain

Proposition 4.

Let $M:=M_1{\times }_f{M}_2$ be a warped product submanifold in a locally Golden Riemannian manifold $(\overline{M},\overline{g},J)$ with warping function f. Then, for any X, $Y\in \Gamma \left(T{M}_1\right)$ and Z, $W\in \Gamma \left(T{M}_2\right)$, we have

$$\overline{g}(h(X,Y),NZ)=-\overline{g}(h(X,Z),NY),$$

(40)

$$\overline{g}(h(X,Z),NW)=0,$$

(41)

and

$$\overline{g}(h(Z,W),NX)=T_{1}X(lnf)\overline{g}(Z,W)-X(lnf)\overline{g}(Z,T_{2}W).$$

(42)

Proof. 

For any X, $Y\in \Gamma \left(T{M}_1\right)$ and $Z\in \Gamma \left(T{M}_2\right)$, by using (3), (5), (10), (39) and $\overline{\nabla }J=0$, we obtain

$$\overline{g}(h(X,Y),NZ)=\overline{g}({\overline{\nabla }}_{X}Y,JZ)-\overline{g}({\overline{\nabla }}_{X}Y,T_{2}Z)$$

$$=\overline{g}({\overline{\nabla }}_X{T}_{1}Y,Z)+\overline{g}({\overline{\nabla }}_{X}NY,Z)+\overline{g}({\overline{\nabla }}_X{T}_{2}Z,Y)$$

$$=-\overline{g}({\nabla }_{X}Z,T_{1}Y)-\overline{g}(A_{NY}X,Z)+\overline{g}(Y,{\nabla }_X{T}_{2}Z)$$

$$=-X(lnf)\overline{g}(T_{1}Y,Z)-\overline{g}(h(X,Z),NY)+X(lnf)\overline{g}(Y,T_{2}Z).$$

On the other hand, $\overline{g}(T_{1}Y,Z)=\overline{g}(JY,Z)=\overline{g}(Y,JZ)=\overline{g}(Y,T_{2}Z)$. Thus, we obtain (40).

For any $X\in \Gamma \left(T{M}_1\right)$ and Z, $W\in \Gamma \left(T{M}_2\right)$, by using (3), (5), (10), (39) and $\overline{\nabla }J=0$, we obtain

$$\overline{g}(h(X,Z),NW)=\overline{g}({\overline{\nabla }}_{X}Z,JW)-\overline{g}({\overline{\nabla }}_{X}Z,T_{2}W)$$

$$=\overline{g}({\nabla }_X{T}_{2}Z,W)-\overline{g}(A_{NZ}X,W)-\overline{g}({\nabla }_{X}Z,T_{2}W)$$

$$=X(lnf)[\overline{g}(T_{2}Z,W)-\overline{g}(Z,T_{2}W)]-\overline{g}(h(X,W),NZ)$$

and using

$$\overline{g}(T_{2}Z,W)-\overline{g}(Z,T_{2}W)=\overline{g}(JZ,W)-\overline{g}(Z,JW)=0,$$

we obtain

$$\overline{g}(h(X,Z),NW)=-\overline{g}(h(X,W),NZ).$$

(43)

On the other hand, after interchanging Z by X, we have

$$\overline{g}(h(Z,X),NW)=\overline{g}({\nabla }_Z{T}_{1}X,W)-\overline{g}(A_{NX}Z,W)-\overline{g}({\nabla }_{Z}X,T_{2}W)$$

$$=T_{1}X(lnf)\overline{g}(Z,W)-X(lnf)\overline{g}(Z,T_{2}W)-\overline{g}(h(Z,W),NX)=\overline{g}(h(X,W),NZ)$$

and using (43), we obtain (41).

For any $X\in \Gamma \left(T{M}_1\right)$ and Z, $W\in \Gamma \left(T{M}_2\right)$, by using (3), (5), (10), (39) and $\overline{\nabla }J=0$, we obtain

$$\overline{g}(h(Z,W),NX)=\overline{g}({\overline{\nabla }}_{Z}W,JX)-\overline{g}({\overline{\nabla }}_{Z}W,T_{1}X)$$

$$=\overline{g}({\nabla }_Z{T}_{2}W,X)+\overline{g}({\overline{\nabla }}_{Z}NW,X)-\overline{g}({\nabla }_{Z}W,T_{1}X)$$

$$=-\overline{g}(T_{2}W,{\nabla }_{Z}X)-\overline{g}(A_{NW}Z,X)+\overline{g}(W,{\nabla }_Z{T}_{1}X)$$

$$=-X(lnf)\overline{g}(Z,T_{2}W)+T_{1}X(lnf)\overline{g}(Z,W)$$

and we obtain (42). □

6. 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.

Definition 4.

Let $M:=M_1{\times }_f{M}_2$ be a warped product bi-slant submanifold in a Golden Riemannian manifold $(\overline{M},\overline{g},J)$ such that one of the components $M_i$ ($i\in \{1,2\}$) is an invariant submanifold (respectively, anti-invariant submanifold) in $\overline{M}$ and the other one is a pointwise slant submanifold in $\overline{M}$, with the Wirtinger angle ${\theta }_x\in [0,\frac{\pi }{2}]$. Then, we call the submanifold M warped product pointwise semi-slant submanifold (respectively, warped product pointwise hemi-slant submanifold) in the Golden Riemannian manifold $(\overline{M},\overline{g},J)$.

Now, we provide an example of a warped product semi-slant submanifold in a Golden Riemannian manifold.

Example 5.

Let ${\mathbb{R}}^7$ be the Euclidean space endowed with the usual Euclidean metric $\langle ·,·\rangle$ and consider the immersion $i:M\to {\mathbb{R}}^7$, given by

$$i(f,u,v):=(fcosu,fsinu,fcosv,fsinv,f,u,v),$$

where $M:=\{(f,u,v)\mid f>0,u,v\in [0,\frac{\pi }{2}]\}$.

The local orthogonal frame on $TM$ is given by

$$Z_1=cosu\frac{\partial }{\partial x_1}+sinu\frac{\partial }{\partial x_2}+cosv\frac{\partial }{\partial x_3}+sinv\frac{\partial }{\partial x_4}+\frac{\partial }{\partial y_1},$$

$$Z_2=-fsinu\frac{\partial }{\partial x_1}+fcosu\frac{\partial }{\partial x_2}+\frac{\partial }{\partial y_2},$$

$$Z_3=-fsinv\frac{\partial }{\partial x_3}+fcosv\frac{\partial }{\partial x_4}+\frac{\partial }{\partial y_3}.$$

We define the Golden structure $J:{\mathbb{R}}^7\to {\mathbb{R}}^7$ by

$$J\left(\frac{\partial }{\partial x_k},\frac{\partial }{\partial y_l}\right):=\left(\varphi \frac{\partial }{\partial x_1},\varphi \frac{\partial }{\partial x_2},\overline{\varphi }\frac{\partial }{\partial x_3},\overline{\varphi }\frac{\partial }{\partial x_4},\overline{\varphi }\frac{\partial }{\partial y_1},\varphi \frac{\partial }{\partial y_2},\overline{\varphi }\frac{\partial }{\partial y_3}\right),$$

for $k\in \{1,2,3,4\}$ and $l\in \{1,2,3\}$, where ϕ is the Golden number and $\overline{\varphi }=1-\varphi$. Since

$$J{Z}_1=\varphi cosu\frac{\partial }{\partial x_1}+\varphi sinu\frac{\partial }{\partial x_2}+\overline{\varphi }cosv\frac{\partial }{\partial x_3}+\overline{\varphi }sinv\frac{\partial }{\partial x_4}+\overline{\varphi }\frac{\partial }{\partial y_1},$$

$$J{Z}_2=\varphi Z_2,J{Z}_3=\overline{\varphi }{Z}_3,$$

we remark that $\langle J{Z}_k,Z_l\rangle =0,$ for any $k\ne l$, where $k,l\in \{1,2,3\}$, and $\langle J{Z}_1,Z_1\rangle =\varphi +2\overline{\varphi }=2-\varphi$.

We find that

$$\parallel Z_1{\parallel }^2=3,\parallel Z_2{\parallel }^2={\parallel Z_3\parallel }^2=f^2+1,$$

$$\parallel J{Z}_1{\parallel }^2={\varphi }^2+2{\overline{\varphi }}^2=5-\varphi ,\parallel J{Z}_2{\parallel }^2={\varphi }^2(f^2+1),{\parallel J{Z}_3\parallel }^2={\overline{\varphi }}^2(f^2+1).$$

Denote by $D_1:=span\left\{Z_1\right\}$ the slant distribution with the slant angle θ, where $cos\theta =\frac{2-\varphi }{\sqrt{3(5-\varphi )}}$ and by $D_2:=span\{Z_2,Z_3\}$ the invariant distribution (with respect to J).

If $M_{\theta }$ and $M_T$ are the integral manifolds of the distributions $D_1$ and $D_2$, respectively, then $M:=M_{\theta }{\times }_{\sqrt{f^2+1}}{M}_T$ with the metric

$$g:=3d{f}^2+(f^2+1)(d{u}^2+d{v}^2)=g_{M_{\theta }}+(f^2+1)g_{M_T}$$

is a warped product semi-slant submanifold in the Golden Riemannian manifold $({\mathbb{R}}^7,\langle ·,·\rangle ,J)$.

Now, we provide an example of a warped product hemi-slant submanifold in a Golden Riemannian manifold.

Example 6.

Let ${\mathbb{R}}^5$ be the Euclidean space endowed with the usual Euclidean metric $\langle ·,·\rangle$ and consider the immersion $i:M\to {\mathbb{R}}^5$, given by

$$i(f,u):=\left(fsinu,fcosu,\varphi fsinu,\varphi fcosu,-f\right),$$

where $M:=\{(f,u)\mid f>0,u\in (0,\frac{\pi }{2})\}$ and ϕ is the Golden number.

The local orthogonal frame on $TM$ is given by

$$Z_1=sinu\frac{\partial }{\partial x_1}+cosu\frac{\partial }{\partial x_2}+\varphi sinu\frac{\partial }{\partial x_3}+\varphi cosu\frac{\partial }{\partial x_4}-\frac{\partial }{\partial x_5},$$

$$Z_2=fcosu\frac{\partial }{\partial x_1}-fsinu\frac{\partial }{\partial x_2}+\varphi fcosu\frac{\partial }{\partial x_3}-\varphi fsinu\frac{\partial }{\partial x_4}.$$

We define the Golden structure $J:{\mathbb{R}}^5\to {\mathbb{R}}^5$ by

$$J(X_1,X_2,X_3,X_4,X_5):=(\varphi X_1,\varphi X_2,\overline{\varphi }{X}_3,\overline{\varphi }{X}_4,\varphi X_5),$$

where $\overline{\varphi }=1-\varphi$. Since

$$J{Z}_1=\varphi \left(sinu\frac{\partial }{\partial x_1}+cosu\frac{\partial }{\partial x_2}\right)-\left(sinu\frac{\partial }{\partial x_3}+cosu\frac{\partial }{\partial x_4}\right)-\varphi \frac{\partial }{\partial x_5},$$

$$J{Z}_2=\varphi f\left(cosu\frac{\partial }{\partial x_1}-sinu\frac{\partial }{\partial x_2}\right)-f\left(cosu\frac{\partial }{\partial x_3}-sinu\frac{\partial }{\partial x_4}\right),$$

we remark that $\langle J{Z}_2,Z_k\rangle =0$, for any $k\in \{1,2\}$, and $\langle J{Z}_1,Z_1\rangle =\varphi$.

We find that

$$\parallel Z_1{\parallel }^2={\varphi }^2+2=\varphi +3,{\parallel Z_2\parallel }^2=f^2({\varphi }^2+1)=f^2(\varphi +2),$$

$$\parallel J{Z}_1{\parallel }^2=2{\varphi }^2+1=2\varphi +3,{\parallel J{Z}_2\parallel }^2=f^2({\varphi }^2+1)=f^2(\varphi +2).$$

Denote by $D_1:=span\left\{Z_1\right\}$ the slant distribution with the slant angle θ, where $cos\theta =\frac{\varphi }{\sqrt{(2\varphi +3)(\varphi +3)}}$ and by $D_2:=span\left\{Z_2\right\}$ the anti-invariant distribution (with respect to J).

If $M_{\theta }$ and $M_{\perp }$ are the integral manifolds of the distributions $D_1$ and $D_2$, respectively, then $M:=M_{\theta }{\times }_{f\sqrt{\varphi +2}}{M}_{\perp }$ with the metric

$$g:=\left(\varphi +3\right)d{f}^2+f^2\left(\varphi +2\right)d{u}^2=g_{M_{\theta }}+f^2\left(\varphi +2\right)g_{M_{\perp }}$$

is a warped product hemi-slant submanifold in the Golden Riemannian manifold $({\mathbb{R}}^5,\langle ·,·\rangle ,J)$.

In a similar manner as in Theorem 2 from [25], we obtain

Theorem 1.

If $M:=M_T{\times }_f{M}_{\theta }$ is a warped product pointwise semi-slant submanifold in a locally Golden Riemannian manifold $(\overline{M},\overline{g},J)$ with the pointwise slant angle ${\theta }_x\in (0,\frac{\pi }{2})$, for $x\in M_{\theta }$, then the warping function f is constant on the connected components of $M_T$.

Proof. 

For any $X\in \Gamma \left(T{M}_T\right)$, $Z\in \Gamma \left(T{M}_{\theta }\right)\setminus \left\{0\right\}$, by using (10) in ${\overline{\nabla }}_{Z}JX=J{\overline{\nabla }}_{Z}X$ and (39), we obtain

$$TX(lnf)Z+h(TX,Z)=T{\nabla }_{Z}X+N{\nabla }_{Z}X+th(X,Z)+nh(X,Z).$$

From the equality of the normal components of the last equation, it follows

$$h(TX,Z)=X(lnf)NZ+nh(X,Z)$$

(44)

and replacing X with $TX=JX$ (for $X\in \Gamma \left(T{M}_T\right)$) in (44), we obtain

$$h(J^{2}X,Z)=TX(lnf)NZ+nh(TX,Z).$$

Thus, we obtain

$$TX(lnf)\overline{g}(NZ,NZ)=\overline{g}(h(J^{2}X,Z),NZ)-\overline{g}(nh(TX,Z),NZ)$$

$$=\overline{g}(h(TX,Z),NZ)+\overline{g}(h(X,Z),NZ)-\overline{g}(nh(TX,Z),NZ),$$

for any $X\in \Gamma \left(T{M}_T\right)$ and $Z\in \Gamma \left(T{M}_{\theta }\right)$.

From (41), we have $\overline{g}(h(TX,Z),NZ)=\overline{g}(h(X,Z),NZ)=0$, for any $X\in \Gamma \left(T{M}_T\right)$ and $Z\in \Gamma \left(T{M}_{\theta }\right)$ and by using (22), we obtain

$$TX(lnf){sin}^2\theta [\overline{g}(TZ,Z)+\overline{g}(Z,Z)]=-\overline{g}(nh(TX,Z),NZ).$$

(45)

On the other hand, for any $X\in \Gamma \left(T{M}_T\right)$ and $Z\in \Gamma \left(T{M}_{\theta }\right)$, we have $TX\in \Gamma \left(T{M}_T\right)$ and $TZ\in \Gamma \left(T{M}_{\theta }\right)$, and from (41), we obtain

$$\overline{g}(h(TX,Z),NZ)=\overline{g}(h(TX,Z),NTZ)=0.$$

Thus, by using (2) and (6), we have

$$\overline{g}(nh(TX,Z),NZ)=\overline{g}(h(TX,Z),nNZ)=\overline{g}(h(TX,Z),J^{2}Z-JTZ)$$

$$=\overline{g}(h(TX,Z),NZ)+\overline{g}(h(TX,Z),Z)-\overline{g}(h(TX,Z),NTZ)=0$$

and using (45), we obtain

$$TX(lnf){tan}^2{\theta }_x\overline{g}(TZ,TZ)=0,$$

for any $Z\in \Gamma \left(T{M}_{\theta }\right)$ and $x\in M_{\theta }$.

Since ${\theta }_x\in (0,\frac{\pi }{2})$ and $TZ\ne 0$, we get $TX(lnf)=0$, for any $X\in \Gamma \left(T{M}_T\right)$, which implies that the warping function f is constant on the connected components of $M_T$. □

Theorem 2.

Let $M:=M_{\theta }{\times }_f{M}_T$ be a warped product pointwise semi-slant submanifold in a locally Golden Riemannian manifold $(\overline{M},\overline{g},J)$ with the pointwise slant angle ${\theta }_x\in (0,\frac{\pi }{2})$, for $x\in M_{\theta }$. Then

$$(A_{N{T}_{1}Y}X-A_{N{T}_{1}X}Y)\in \Gamma \left(T{M}_{\theta }\right),$$

for any $X,Y\in \Gamma \left(T{M}_{\theta }\right)$.

Proof. 

For any $X,Y\in \Gamma \left(T{M}_{\theta }\right)$ and $Z\in \Gamma \left(T{M}_T\right)\setminus \left\{0\right\}$, from (32) and the symmetry of the shape operator, we have

$${sin}^2\theta \overline{g}([X,Y],T_2^{2}Z)=\overline{g}([X,Y],T_{2}Z)-\overline{g}({\nabla }_X{T}_{1}Y-{\nabla }_Y{T}_{1}X,Z)$$

$$+({cos}^2\theta +1)[\overline{g}(h(X,Z),NY)-\overline{g}(h(Y,Z),NX)]-\overline{g}(h(X,Z),N{T}_{1}Y)$$

$$+\overline{g}(h(Y,Z),N{T}_{1}X)+\overline{g}(h(X,Y),N{T}_{2}Z)-\overline{g}(h(Y,X),N{T}_{2}Z)$$

$$-\overline{g}(h(X,T_{2}Z),NY)+\overline{g}(h(Y,T_{2}Z),NX).$$

Using (3) and (40), we obtain

$$\overline{g}({\nabla }_X{T}_{1}Y-{\nabla }_Y{T}_{1}X,Z)=\overline{g}({\nabla }_{X}JY-{\nabla }_{Y}NX-{\nabla }_{Y}JX+{\nabla }_{Y}NX,Z)$$

$$=\overline{g}({\nabla }_{X}Y,JZ)-\overline{g}({\nabla }_{Y}X,JZ)+\overline{g}(A_{NY}X,Z)-\overline{g}(A_{NX}Y,Z)$$

$$=\overline{g}([X,Y],JZ)+\overline{g}(h(X,Z),NY)-\overline{g}(h(Z,Y),NX)=\overline{g}([X,Y],T_{2}Z).$$

From (40), we obtain

$$\overline{g}(h(X,Z),NY)=\overline{g}(h(Y,Z),NX)=-\overline{g}(h(X,Y),NZ).$$

Thus, using the symmetry of the shape operator, we have

$$\overline{g}(h(X,T_{2}Z),NY)-\overline{g}(h(Y,T_{2}Z),NX)=-\overline{g}\left(h(X,Y)N{T}_{2}Z\right)+\overline{g}(h(Y,X),N{T}_{2}Z)=0$$

and

$$\overline{g}(h(X,Z),N{T}_{1}Y)-\overline{g}(h(Y,Z),N{T}_{1}X)=\overline{g}(A_{N{T}_{1}Y}X-A_{N{T}_{1}X}Y,Z).$$

Thus, we obtain

$$0={sin}^2\theta \overline{g}([X,Y],T_2^{2}Z)=\overline{g}(A_{N{T}_{1}Y}X-A_{N{T}_{1}X}Y,Z).$$

□

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.

Theorem 3.

If $M:=M_{\perp }{\times }_f{M}_{\theta }$ (or $M:=M_{\theta }{\times }_f{M}_{\perp }$) is a warped product pointwise hemi-slant submanifold in a locally Golden Riemannian manifold $(\overline{M},\overline{g},J)$ with the pointwise slant angle ${\theta }_x\in (0,\frac{\pi }{2})$, for $x\in M_{\theta }$, then the warping function f is constant on the connected components of $M_{\perp }$ if and only if

$$A_{NZ}X=A_{NX}Z,$$

(46)

for any $X\in \Gamma \left(T{M}_{\perp }\right)$ and $Z\in \Gamma \left(T{M}_{\theta }\right)$ (or $X\in \Gamma \left(T{M}_{\theta }\right)$ and $Z\in \Gamma \left(T{M}_{\perp }\right)$, respectively).