On the Geometry of Semi-Invariant Submanifolds in (α, p)-Golden Riemannian Manifolds

Abstract

The main aim of this paper is to study some properties of submanifolds in a Riemannian manifold equipped with a new structure of golden type, called the ($\alpha , p$)-golden structure, which generalizes the almost golden structure (for $\alpha =1$) and the almost complex golden structure (for $\alpha =-1$). We present some characterizations of isometrically immersed submanifolds in an ($\alpha , p$)-golden Riemannian manifold, especially in the case of the semi-invariant submanifolds, and we find some conditions for the integrability of the distributions.

1. Introduction

A polynomial structure on a differentiable manifold M, defined in [1], arises as a $C^{\infty }$-tensor field f of type $(1,1)$, which satisfies the algebraic equation

$$Q\left(f\right):=f^n+a_n{f}^{n-1}+\cdots +a_{2}f+a_{1}Id=0,$$

where $Id$ is the identity map on the Lie algebra of vector fields on M and

$$f^{n-1}\left(x\right),f^{n-2}\left(x\right),\cdots ,f\left(x\right),Id$$

are linearly independent for every $x\in M$. The polynomial $Q\left(f\right)$ is called the structure polynomial.

The almost complex structure and the almost product structure have the structure polynomial $Q\left(f\right):=f^2+Id$ and $Q\left(f\right):=f^2-Id$, respectively.

An ($\alpha ,p$)-golden structure on a differentiable manifold has the structure polynomial of the form

$$Q\left(f\right):=f^2-pf-{\displaystyle \frac{5\alpha -1}{4}}{p}^2·Id,$$

where p is a nonzero real number and $\alpha \in \{-1,1\}$ [2].

By adding a compatible Riemannian metric, we focus on the study of the structure induced on submanifolds in this setting and on its properties, and we discuss the case of semi-invariant submanifolds in an ($\alpha ,p$)-golden Riemannian manifold.

This structure is a generalization of the golden structure, determined by an endomorphism $\Phi$ which satisfies the polynomial equation ${\Phi }^2=\Phi +Id$ (introduced in [3]). Also, it is related to the metallic structure, which is a solution of the polynomial equation ${\Phi }^2=p\Phi +qId$, where p and q are positive integers (see the references [4,5]). These structures can be obtained from the almost product structures on a differentiable manifold.

On the other hand, an almost complex metallic structure is defined as an endomorphism J which satisfies the relation $J^2-pJ+{\displaystyle \frac{3}{2}}qId=0$ (see [6]). For $p=q=1$, the almost complex metallic structure becomes a complex golden structure (defined in [3]). These structures are related to the almost complex structures on an even dimensional differentiable manifold.

The almost product and the almost complex structures can be unified under the notion of $\alpha$-structure, denoted by $J_{\alpha }$ (with the property $J_{\alpha }^2=\alpha Id$, where $\alpha \in \{-1,1\}$), defined in [7]. The structure ${\Phi }_{\alpha ,p}$ studied in this paper is related to the $\alpha$-structure on an even dimensional differentiable manifold.

Semi-invariant submanifolds in different kinds of ambient manifolds have been defined and studied by many geometers. Firstly, semi-invariant submanifolds in locally Riemannian product manifolds were introduced in 1960 by S. Tachibana [8] and then studied by A. Bejancu and N. Papaghiuc [9,10].

Semi-invariant submanifolds in Riemannian manifolds correspond to CR-submanifolds in complex manifolds. D. Blair and B.Y. Chen studied the properties of CR-submanifolds of Hermitian manifolds in [11]. Moreover, CR-submanifolds of Kaehler manifolds were studied by B.Y. Chen in [12,13] and A. Bejancu in [14,15].

In the paper [16], the authors studied semi-invariant submanifolds of a $(g,F,\mu )$-manifold, which are extensions of CR-submanifolds to this general class of manifolds.

The properties of semi-invariant submanifolds in golden (or metallic) Riemannian manifolds were treated in [17,18,19].

In the present paper, we present some properties of submanifolds of the ($\alpha , p$)-golden Riemannian manifold induced by the ($\alpha , p$)-golden structure of the ambient manifold, especially in the case of semi-invariant submanifolds.

The paper is organized as follows. In Section 2, we present some basic facts regarding a Riemannian manifold endowed with an $(\alpha , p)$-golden structure and a compatible Riemannian metric g, called an almost $(\alpha , p)$-golden Riemannian manifold.

In Section 3, we deal with the investigation of the structure induced on submanifolds by the almost $(\alpha , p)$-golden structure of a Riemannian manifold. Some properties of the projection operators are given when the condition $\overline{\nabla }{\Phi }_{\alpha ,p}=0$ is satisfied, where $\overline{\nabla }$ is the Levi-Civita connection on the ambient almost $(\alpha , p)$-golden Riemannian manifold.

In the last section we focus on the characterization of semi-invariant submanifolds in an $(\alpha , p)$-golden Riemannian manifold. We study the integrability conditions of both invariant and anti-invariant distributions. Finally, we obtain a characterization of the mixed totally geodesic semi-invariant submanifolds in an $(\alpha , p)$-golden Riemannian manifold.

2. Characterization of the ($\mathbf{\alpha }$, $\mathit{p}$)-Golden Riemannian Manifold

In this section, we consider several frameworks useful for our study. Let $\overline{M}$ be an even dimensional manifold $\overline{M}$ and let $\Gamma \left(T\overline{M}\right)$ be the set of smooth sections of $T\overline{M}={\bigcup }_{x\in M}{T}_x\overline{M}$.

It is known that a manifold has an ($\alpha ,\epsilon$)-structure if $\overline{M}$ is endowed with a tensor field $J_{\alpha }$ of type $(1,1)$, which, according to [7], satisfies the equations:

$$J_{\alpha }^2=\alpha ·Id$$

(1)

and

$$\overline{g}(J_{\alpha }X,J_{\alpha }Y)=\epsilon \overline{g}(X,Y),$$

(2)

for any vector fields $X,Y\in \Gamma \left(T\overline{M}\right)$, where $\alpha ;\epsilon \in \{-1,1\}$. The tensor field $J_{\alpha }$ of type $(1,1)$ is an isometry, for $\epsilon =1$, or an anti-isometry, for $\epsilon =-1$.

In particular, for $\epsilon$ = 1 in the equality (2), one obtains the following:

• If $\alpha =-1$, then $J_{\alpha }$ is an almost complex structure and $(\overline{M},J_{-1},\overline{g})$ is an almost Hermitian manifold ([20] p. 124);
• If $\alpha =1$, then $J_{\alpha }$ is an almost product structure and $(\overline{M},J_1,\overline{g})$ is an almost product manifold ([20] p. 423).

In our paper, we will consider an $\alpha$-structure $J_{\alpha }$ (having $\alpha \in \{-1,1\}$), which is an isometry ($\epsilon$ = 1).

If $\overline{M}$ is an even dimensional manifold, then the almost golden structure and the almost complex golden structure are related to the $\alpha$-structure $J_{\alpha }$, which is an endomorphism of the total space $T\overline{M}$ of the tangent bundle that satisfies relation (1). Moreover, if we fix a Riemannian metric $\overline{g}$ such that $\epsilon =1$, we obtain

$$\overline{g}(J_{\alpha }X,J_{\alpha }Y)=\overline{g}(X,Y),$$

(3)

which is equivalent to

$$\overline{g}(J_{\alpha }X,Y)=\alpha \overline{g}(X,J_{\alpha }Y),$$

(4)

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

For $\alpha =1$, one obtains that $\overline{g}$ is a pure metric [21] and the structure $J_1$ is an almost product structure.

Definition 1

([2], Definition 2). An endomorphism ${\Phi }_{\alpha ,p}$ of the total space $T\overline{M}$ of the tangent bundle is called an $(\alpha , p$)-golden structure on $\overline{M}$ if it satisfies the equality

$${\Phi }_{\alpha ,p}^2=p{\Phi }_{\alpha ,p}+{\displaystyle \frac{5\alpha -1}{4}}{p}^2·Id,$$

(5)

where p is a nonzero real number and $\alpha \in \{-1,1\}$.

Definition 2

([2]). An almost $(\alpha , p)$-golden Riemannian manifold is a triple $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$, where $\overline{g}$ is a Riemannian metric on an even dimensional manifold $\overline{M}$, which verifies the equality

$$\overline{g}({\Phi }_{\alpha ,p}X,Y)=\alpha \overline{g}(X,{\Phi }_{\alpha ,p}Y)+{\displaystyle \frac{p}{2}}(1-\alpha )\overline{g}(X,Y),$$

(6)

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

Remark 1

([2]). If $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ is an almost $(\alpha , p)$-golden Riemannian manifold, then the Riemannian metric $\overline{g}$ on $\overline{M}$ verifies

$$\overline{g}({\Phi }_{\alpha ,p}X,{\Phi }_{\alpha ,p}Y)={\displaystyle \frac{p}{2}}(\overline{g}({\Phi }_{\alpha ,p}X,Y)+\overline{g}(X,{\Phi }_{\alpha ,p}Y))+p^2\overline{g}(X,Y),$$

(7)

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

In particular, for $(\alpha ,p)$ = (1, 1), the almost ($1,1$)-golden structure becomes an almost golden structure $\Phi$ and $(\overline{M},\Phi )$ turns into an almost golden manifold, which was studied in [3].

On the other hand, if $(\alpha ,p)$ = (−1, 1), then one obtains an almost complex golden structure determined by the endomorphism ${\Phi }_c$, which satisfies the equation ${\Phi }_c^2={\Phi }_c-{\displaystyle \frac{3}{2}}Id$. In this case, $(\overline{M},{\Phi }_c)$ is called an almost complex golden manifold and it was studied in [22,23].

Remark 2.

Let ${\Phi }_{\alpha ,p}$ be an almost $(\alpha , p)$-golden structure. The structure ${\overline{\Phi }}_{\alpha ,p}=pId-{\Phi }_{\alpha ,p}$ is an almost $(\alpha , p)$-golden structure, too ([2]).

Proposition 1

([2], Proposition 1). Every α-structure $J_{\alpha }$ on $\overline{M}$ defines two almost $(\alpha , p)$-golden structures, given by the equality

$${\Phi }_{\alpha ,p}^{\pm }={\displaystyle \frac{p}{2}}(Id\pm \sqrt{5}{J}_{\alpha }).$$

(8)

Conversely, two α-structures $J_{\alpha }^{\pm }$ can be associated to a given almost $(\alpha , p)$-golden structure as follows:

$$J_{\alpha }^{\pm }=\pm {\displaystyle \frac{2}{p\sqrt{5}}}\left({\Phi }_{\alpha ,p}-{\displaystyle \frac{p}{2}}Id\right).$$

Example 1.

Let us assume that $(\overline{M},\overline{g})$ is a Riemannian manifold of dimension 2 m. We can define an α-structure $J_{\alpha }$, given by

$$J_{\alpha }(X^1,...,X^m,Y^1,...,Y^m)=(Y^1,...,Y^m,\alpha X^1,...,\alpha X^m),$$

(9)

for any $(X^1,...,X^m,Y^1,...,Y^m)\in \Gamma \left(TM\right)$ and any integer number $m\ge 1$. The metric g is given by

$$\overline{g}((X^i,Y^i),(Z^i,W^i))=\sum _{i=1}^m{X}^i{Z}^i+\sum _{i=1}^m{Y}^i{W}^i,$$

(10)

for any $(X^i,Y^i):=(X^1,...,X^m,Y^1,...,Y^m)$, $(Z^i,W^i):=(Z^1,...,Z^m,W^1,...,W^m)\in \Gamma (T\left(\overline{M}\right)$. We can verify that

$$\overline{g}(J_{\alpha }(X^i,Y^i),(Z^i,W^i))=\alpha \overline{g}((X^i,Y^i),J_{\alpha }(Z^i,W^i)),$$

for any $(X^i,Y^i),(Z^i,W^m)\in \Gamma (T\left(\overline{M}\right)$.

Using the identity (8) in (9), we obtain an $(\alpha , p)$-golden structure ${\Phi }_{\alpha ,p}$, given by the equality

$${\Phi }_{\alpha ,p}(X^i,Y^i)={\displaystyle \frac{p}{2}}(X^1+\sqrt{5}{Y}^1,...,X^m+\sqrt{5}{Y}^m,Y^1+\alpha \sqrt{5}{X}^1,...,Y^m+\alpha \sqrt{5}{X}^m).$$

Moreover, by using (8) in (10), we can verify that the metric g satisfies the equality

$$\overline{g}({\Phi }_{\alpha ,p}(X^i,Y^i),(Z^i,W^i))=\alpha \overline{g}((X^i,Y^i),{\Phi }_{\alpha ,p}(W^i,Z^i))+{\displaystyle \frac{p}{2}}(1-\alpha )\overline{g}((X^i,Y^i),(Z^i,W^i)),$$

for any $(X^i,Y^i),(Z^i,W^i)\in \Gamma \left(TM\right)$.

Thus, $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ is an almost $(\alpha , p)$-golden Riemannian manifold.

Now, let $\overline{\nabla }$ be the Levi-Civita connection on $(\overline{M},\overline{g})$. The covariant derivative $\overline{\nabla }{J}_{\alpha }$ is a tensor field of the type (1, 2), defined by

$$\left({\overline{\nabla }}_X{J}_{\alpha }\right)Y:={\overline{\nabla }}_X\left(J_{\alpha }Y\right)-J_{\alpha }\left({\overline{\nabla }}_{X}Y\right),$$

(11)

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

Using the compatibility formula ${\Phi }_{\alpha ,p}={\displaystyle \frac{p}{2}}(Id+\sqrt{5}{J}_{\alpha })$, we obtain ([2])

$$\left({\overline{\nabla }}_X{\Phi }_{\alpha ,p}\right)Y={\displaystyle \frac{p\sqrt{5}}{2}}\left({\overline{\nabla }}_X{J}_{\alpha }\right)Y,$$

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

Let us consider the Nijenhuis tensor field of an $\alpha$-structure $J_{\alpha }$, defined by the relation

$$N_{J_{\alpha }}(X,Y)=J_{\alpha }^2[X,Y]+[J_{\alpha }X,J_{\alpha }Y]-J_{\alpha }[J_{\alpha }X,Y]-J_{\alpha }[X,J_{\alpha }Y],$$

for any $X,Y\in \Gamma \left(T\overline{M}\right)$. The Nijenhuis tensor field corresponding to the $(\alpha , p)$-golden structure $\Phi :={\Phi }_{\alpha ,p}$ is given by the equality ([2])

$$N_{\Phi }\left(X,Y\right)=({\overline{\nabla }}_{\Phi X}\Phi )Y-({\overline{\nabla }}_{\Phi Y}\Phi )X-\Phi ({\overline{\nabla }}_X\Phi )Y+\Phi ({\overline{\nabla }}_Y\Phi )X,$$

and it verifies

$$N_{\Phi }\left(X,Y\right)={\displaystyle \frac{5p^2}{4}}{N}_{J_{\alpha }}(X,Y),$$

(12)

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

An $\alpha$-structure $J_{\alpha }$ on a differentiable manifold is integrable if the Nijenhuis tensor field $N_{J_{\alpha }}$, corresponding to $J_{\alpha }$, vanishes identically (i.e., $N_{J_{\alpha }}=0$).

Remark 3.

${\Phi }_{\alpha ,p}$ is integrable if and only if the associated almost α-structure $J_{\alpha }$ is integrable.

Remark 4.

${\Phi }_{\alpha ,p}$ is integrable (i.e., $N_{{\Phi }_{\alpha ,p}}=0$) if $\overline{\nabla }{\Phi }_{\alpha ,p}=0$.

From ([20], Theorem 3.1, p. 125), we remark that if $\alpha =-1$, then $J_{-1}$ is an almost complex structure on the manifold $(\overline{M},\overline{g}$) and $\overline{M}$ is a complex manifold (i.e., $N_{J_{-1}}=0$) if and only if it admits a linear connection $\overline{\nabla }$ such that $\overline{\nabla }J=0$ and $T=0$, where T denotes the torsion of $\overline{\nabla }$. Thus, from (11) and (12), we obtain the following property:

Proposition 2.

The structure ${\Phi }_{-1,p}:=\Phi$ on an almost $(-1,p)$-golden Riemannian manifold $(\overline{M},\Phi ,\overline{g})$ is integrable (i.e., $N_{\Phi }=0$) if and only if it admits a linear connection $\overline{\nabla }$, having the torsion $T=0$, such that $\overline{\nabla }\Phi =0$.

From ([20], Theorem 2.3, p. 420), it is known that an integrable almost product Riemannian manifold, with structure tensor J, is a locally product Riemannian manifold. Sufficient conditions for the integrability of almost product structures on Riemannian manifolds were presented in [21], where it was shown that the condition $\overline{\nabla }J=0$ is equivalent to decomposability of the pure metric $\overline{g}$. In [24], the authors studied an integrability condition for the locally decomposable metallic Riemannian structures.

Definition 3.

The structure ${\Phi }_{1,p}$ on an almost $(1,p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{1,p},\overline{g})$ is called locally decomposable if $\overline{\nabla }{\Phi }_{1,p}=0$, where $\overline{\nabla }$ is the Levi-Civita connection corresponding to the metric $\overline{g}$.

Taking into account these observations, we may consider an almost $(\alpha , p)$-golden Riemannian manifold which is covariant constant (i.e., $\overline{\nabla }{\Phi }_{\alpha ,p}=0$, where $\overline{\nabla }$ is the Levi-Civita connection corresponding to the metric $\overline{g}$) and we introduce the following definition:

Definition 4.

A locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ is an almost $(\alpha , p)$-golden Riemannian manifold whose $(\alpha , p)$-golden structure is parallel with respect to the Levi-Civita connection $\overline{\nabla }$ (i.e., $\overline{\nabla }{\Phi }_{\alpha ,p}=0$).

3. Submanifolds in ($\mathbf{\alpha }$, $\mathit{p}$)-Golden Riemannian Manifold

In this section, we assume that M is an isometrically immersed submanifold in an even dimensional almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. If $\Gamma \left(TM\right)$ is the Lie algebra of vector fields on M and $T_{x}M$ (respectively $T_x^{\perp }M$) is the tangent space (respectively, the normal space) of M at a given point $x\in M$, one obtains the direct sum

$$T_x\overline{M}=T_{x}M\oplus T_x^{\perp }M.$$

Let g be the induced Riemannian metric on M, given by $g(X,Y):=\overline{g}(i_{*}X,i_{*}Y)$ for any $X,Y\in \Gamma \left(TM\right)$, where $i_{*}$ is the differential of the immersion $i:M\to \overline{M}$. We shall assume that all immersions are injective.

In the rest of the paper, one uses the simple notation $X:=i_{*}X$, for any $X\in \Gamma \left(TM\right)$.

By using (1), (3), and (4), we obtain that the induced metric on the submanifold M and the $\alpha$-structure $J_{\alpha }$ verify the equalities:

$$g(J_{\alpha }X,Y)=\alpha g(X,J_{\alpha }Y)\iff g(J_{\alpha }X,J_{\alpha }Y)=g(X,Y),$$

for any $X,Y\in \Gamma \left(TM\right)$. Moreover, using relations (6) and (7), one obtains the equality

$$g({\Phi }_{\alpha ,p}X,Y)=\alpha g(X,{\Phi }_{\alpha ,p}Y)+{\displaystyle \frac{p}{2}}(1-\alpha )g(X,Y),$$

which is equivalent to the equality

$$g({\Phi }_{\alpha ,p}X,{\Phi }_{\alpha ,p}Y)={\displaystyle \frac{p}{2}}(g({\Phi }_{\alpha ,p}X,Y)+g(X,{\Phi }_{\alpha ,p}Y))+p^{2}g(X,Y),$$

for any $X,Y\in \Gamma \left(TM\right)$.

Let $h:\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(T^{\perp }M\right)$ be the second fundamental form of M in $\overline{M}$ and let $A_U$ be the shape operator of M with respect to $U\in \Gamma \left(T^{\perp }M\right)$. One denotes by ${\nabla }^{\perp }$ the normal connection on the normal bundle $\Gamma \left(T^{\perp }M\right)$.

The Gauss and Weingarten formulas are

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y)$$

(13)

and

$${\overline{\nabla }}_{X}U=-A_{U}X+{\nabla }_X^{\perp }U,$$

(14)

respectively, for any tangent vector fields $X,Y\in \Gamma \left(TM\right)$ and for any normal vector field $U\in \Gamma \left(T^{\perp }M\right)$, where $\overline{\nabla }$ and ∇ are the Levi-Civita connections on $\overline{M}$ and on the submanifold M, respectively. Moreover, the second fundamental form h and the shape operator $A_U$ are related by

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

(15)

First of all, we consider the endomorphisms given by the relations

$$\mathcal{T}:\Gamma \left(TM\right)\to \Gamma \left(TM\right),\mathcal{T}X={\left({\Phi }_{\alpha ,p}X\right)}^{\top }$$

(16)

and

$$\mathfrak{n}:\Gamma \left(T^{\perp }M\right)\to \Gamma \left(T^{\perp }M\right),\mathfrak{n}U={\left({\Phi }_{\alpha ,p}U\right)}^{\perp }$$

(17)

for any tangent vector field $X\in \Gamma \left(TM\right)$ and any normal vector field $U\in \Gamma \left(T^{\perp }M\right)$.

On the other hand, we consider the operators (bundle-valued 2-forms) given by

$$\mathcal{N}:\Gamma \left(TM\right)\to \Gamma \left(T^{\perp }M\right),\mathcal{N}X={\left({\Phi }_{\alpha ,p}X\right)}^{\perp }$$

(18)

and

$$\mathfrak{t}:\Gamma \left(T^{\perp }M\right)\to \Gamma \left(TM\right),\mathfrak{t}U={\left({\Phi }_{\alpha ,p}U\right)}^{\top },$$

(19)

for any tangent vector field $X\in \Gamma \left(TM\right)$ and any normal vector field $U\in \Gamma \left(T^{\perp }M\right)$.

For any vector field $X\in \Gamma \left(TM\right)$, we have the decomposition into the tangential and normal parts of ${\Phi }_{\alpha ,p}X$ given by the equality

$${\Phi }_{\alpha ,p}X=\mathcal{T}X+\mathcal{N}X.$$

(20)

Similarly, for any vector field $U\in \Gamma \left(T^{\perp }M\right)$, the decomposition into the tangential and normal parts of ${\Phi }_{\alpha ,p}V$ is given by the equality

$${\Phi }_{\alpha ,p}U=\mathfrak{t}U+\mathfrak{n}U.$$

(21)

Proposition 3

([2], Proposition 8). Let $(\overline{M},\overline{g})$ be a Riemannian manifold endowed with an almost $(\alpha , p)$-golden structure ${\Phi }_{\alpha ,p}$. Thus, for any $X,Y\in \Gamma \left(TM\right)$, the maps $\mathcal{T}$ and $\mathfrak{n}$ satisfy the equalities

$$\overline{g}(\mathcal{T}X,Y)=\alpha \overline{g}(X,\mathcal{T}Y)+{\displaystyle \frac{p(1-\alpha )}{2}}\overline{g}(X,Y)$$

(22)

and

$$\overline{g}(\mathfrak{n}U,V)=\alpha \overline{g}(U,\mathfrak{n}V)+{\displaystyle \frac{p(1-\alpha )}{2}}\overline{g}(U,V),$$

(23)

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

$$\overline{g}(\mathcal{N}X,U)=\alpha \overline{g}(X,\mathfrak{t}U),$$

(24)

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

Remark 5.

If $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ is an almost $(\alpha , p)$-golden Riemannian manifold, then we obtain the equalities

$$\overline{g}({\Phi }_{\alpha ,p}X,{\Phi }_{\alpha ,p}Y)=\overline{g}(\mathcal{T}X,\mathcal{T}Y)+\overline{g}(\mathcal{N}X,\mathcal{N}Y),$$

for any $X,Y\in \Gamma \left(TM\right)$, and

$$\overline{g}({\Phi }_{\alpha ,p}U,{\Phi }_{\alpha ,p}V)=\overline{g}(\mathfrak{t}U,\mathfrak{t}V)+\overline{g}(\mathfrak{n}U,\mathfrak{n}V),$$

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

Proposition 4.

The structure induced on a submanifold M by an almost $(\alpha , p)$-golden structure ${\Phi }_{\alpha ,p}$ on $\overline{M}$ satisfies the following equalities:

$${\mathcal{T}}^{2}X=p\mathcal{T}X+{\displaystyle \frac{5\alpha -1}{4}}{p}^{2}X-\mathfrak{t}\left(\mathcal{N}X\right)$$

(25)

and

$$\mathcal{N}\left(\mathcal{T}X\right)=p\mathcal{N}X-\mathfrak{n}\left(\mathcal{N}X\right),$$

(26)

for any $X\in \Gamma \left(TM\right)$. Moreover, for any $U\in \Gamma \left(T^{\perp }M\right)$, one obtains the equalities

$${\mathfrak{n}}^{2}U=p\mathfrak{n}U+{\displaystyle \frac{5\alpha -1}{4}}{p}^{2}U-\mathcal{N}\left(\mathfrak{t}U\right)$$

(27)

and

$$\mathfrak{t}\left(\mathfrak{n}U\right)=p\mathfrak{t}U-\mathcal{T}\left(\mathfrak{t}U\right),$$

(28)

where the operators $\mathcal{T},\mathcal{N},\mathfrak{t}$, and $\mathfrak{n}$ are defined in (16)–(19).

Proof. 

By using (5) and the decomposition (20) for ${\Phi }_{\alpha ,p}X$ in the equality ${\Phi }_{\alpha ,p}^{2}X={\Phi }_{\alpha ,p}\left({\Phi }_{\alpha ,p}X\right)$, one obtains

$$p\mathcal{T}X+p\mathcal{N}X+{\displaystyle \frac{5\alpha -1}{4}}{p}^{2}X={\mathcal{T}}^{2}X+\mathcal{N}\left(\mathcal{T}X\right)+\mathfrak{t}\left(\mathcal{N}X\right)+\mathfrak{n}\left(\mathcal{N}X\right),$$

for any $X\in \Gamma \left(TM\right)$. Equalizing the tangential and the normal parts from both members of the last equality, we obtain the relations (25) and (26), respectively.

In the same manner, the decomposition Formula (21) for ${\Phi }_{\alpha ,p}U$ applied in ${\Phi }_{\alpha ,p}^{2}U={\Phi }_{\alpha ,p}\left({\Phi }_{\alpha ,p}U\right)$ leads to the equality

$$p\mathfrak{t}U+p\mathfrak{n}U+{\displaystyle \frac{5\alpha -1}{4}}{p}^{2}U=\mathcal{T}\left(\mathfrak{t}U\right)+\mathcal{N}\left(\mathfrak{t}U\right)+\mathfrak{t}\left(\mathfrak{n}U\right)+{\mathfrak{n}}^{2}U,$$

for any $U\in \Gamma \left(T^{\perp }M\right)$, which implies the relations (27) and (28), respectively. □

We remark that, if we consider $\mathfrak{t}\mathcal{N}=0$ in (25), then the endomorphism $\mathcal{T}$ verifies the equation of the $(\alpha , p)$-golden structure (5). Since the induced metric g on the submanifold M verifies the compatibility relation (22), we obtain the following property regarding the induced structure on a submanifold:

Proposition 5.

If M is a submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold ($\overline{M},{\Phi }_{\alpha ,p},\overline{g}$) and the operators $\mathcal{N}$ and $\mathfrak{t}$, defined in (18) and (19), satisfy $\mathfrak{t}\mathcal{N}=0$, then the submanifold $(M,\mathcal{T},g)$ is also an almost $(\alpha , p)$-golden Riemannian manifold.

Now, we recall a property proved in ([2], Theorem 2), similar to one in the case of the metallic Riemannian manifolds (see ([4], Proposition 4.3)).

Proposition 6

([2]). A necessary and sufficient condition for the invariance of a submanifold M in an even dimensional Riemannian manifold $(\overline{M},\overline{g})$, endowed with an almost $(\alpha , p)$-golden structure ${\Phi }_{\alpha ,p}$, is that the structure $\mathcal{T}$, induced on M by ${\Phi }_{\alpha ,p}$, is also an almost $(\alpha , p)$-golden structure.

The covariant derivatives of the tangential and normal parts of ${\Phi }_{\alpha ,p}$ are given, for any $X,Y\in \Gamma \left(TM\right)$, by the equalities

$$\left({\nabla }_X\mathcal{T}\right)Y={\nabla }_X\left(\mathcal{T}Y\right)-\mathcal{T}\left({\nabla }_{X}Y\right)$$

(29)

and

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

(30)

Moreover, the covariant derivatives of the tangential and normal parts of ${\Phi }_{\alpha ,p}$ are given, for any $X\in \Gamma \left(TM\right)$ and $U\in \Gamma \left(T^{\perp }M\right)$, by the equalities

$$\left({\nabla }_X\mathfrak{t}\right)U={\nabla }_X\left(\mathfrak{t}U\right)-\mathfrak{t}\left({\nabla }_X^{\perp }U\right)$$

(31)

and

$$\left({\overline{\nabla }}_X\mathfrak{n}\right)U={\nabla }_X^{\perp }\left(\mathfrak{n}U\right)-\mathfrak{n}\left({\nabla }_X^{\perp }U\right).$$

(32)

Proposition 7.

If $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ is a locally $(\alpha , p)$-golden Riemannian manifold, then the operators $\mathcal{T},\mathcal{N},\mathfrak{t}$, and $\mathfrak{n}$, defined in (16)–(19), verify the equalities

$$\left({\nabla }_X\mathcal{T}\right)Y=A_{\mathcal{N}Y}X+\mathfrak{t}h(X,Y),$$

(33)

$$\left({\overline{\nabla }}_X\mathcal{N}\right)Y=-h(X,\mathcal{T}Y)+\mathfrak{n}h(X,Y),$$

(34)

$$\left({\nabla }_X\mathfrak{t}\right)U=A_{\mathfrak{n}U}X-\mathcal{T}\left(A_{U}X\right),$$

(35)

and

$$\left({\overline{\nabla }}_X\mathfrak{n}\right)U=-h(X,\mathfrak{t}U)-\mathcal{N}\left(A_{U}X\right),$$

(36)

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

Proof. 

From ${\overline{\nabla }}_X\left({\Phi }_{\alpha ,p}Y\right)={\Phi }_{\alpha ,p}\left({\overline{\nabla }}_{X}Y\right)$, using the Gauss and Weingarten Equations (13) and (14), for any X, $Y\in \Gamma \left(TM\right)$, one obtains the equality

$$\begin{array}{c}\hfill \begin{array}{c}{\nabla }_X\left(\mathcal{T}Y\right)+h(X,\mathcal{T}Y)-A_{\mathcal{N}Y}X+{\nabla }_X^{\perp }\left(\mathcal{N}Y\right)\hfill \\ =\mathcal{T}\left({\nabla }_{X}Y\right)+\mathcal{N}\left({\nabla }_{X}Y\right)+\mathfrak{t}h(X,Y)+\mathfrak{n}h(X,Y).\hfill \end{array}\end{array}$$

(37)

Equalizing the tangential and the normal components from both members of equality (37) and using (29) and (30), we obtain (33) and (34), respectively.

From ${\overline{\nabla }}_X\left({\Phi }_{\alpha ,p}U\right)={\Phi }_{\alpha ,p}\left({\overline{\nabla }}_{X}U\right)$, using the Gauss and Weingarten Equations (13) and (14), for any $X\in \Gamma \left(TM\right)$ and $U\in \Gamma \left(T^{\perp }M\right)$, one obtains the equality

$$\begin{array}{c}\hfill \begin{array}{c}{\nabla }_X\left(\mathfrak{t}U\right)+h(X,\mathfrak{t}U)-A_{\mathfrak{n}U}X+{\nabla }_X^{\perp }\left(\mathfrak{n}U\right)\hfill \\ =-\mathcal{T}\left(A_{U}X\right)-\mathcal{N}\left(A_{U}X\right)+\mathfrak{t}\left({\nabla }_X^{\perp }U\right)+\mathfrak{n}\left({\nabla }_X^{\perp }U\right).\hfill \end{array}\end{array}$$

(38)

Equalizing the tangential and the normal components from both members of equality (38) and using (31) and (32), we obtain (35) and (36), respectively. □

Proposition 8.

If M is an isometrically immersed submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ (in the sense of the Definition 4), then the operators $\mathcal{T},\mathcal{N},\mathfrak{t}$, and $\mathfrak{n}$, defined in (16)–(19), verify the equalities

$$g(\left({\nabla }_X\mathcal{T}\right)Y,Z)=\alpha g(Y,\left({\nabla }_X\mathcal{T}\right)Z),$$

(39)

$$\overline{g}(\left({\overline{\nabla }}_X\mathfrak{n}\right)V,U)=\alpha \overline{g}(V,\left({\overline{\nabla }}_X\mathfrak{n}\right)U),$$

(40)

and

$$\overline{g}(\left({\overline{\nabla }}_X\mathcal{N}\right)Y,U)=\alpha \overline{g}(Y,\left({\nabla }_X\mathfrak{t}\right)U),$$

(41)

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

Proof. 

First of all, using the relations (15), (24), and (33), one obtains

$$g(\left({\nabla }_X\mathcal{T}\right)Y,Z)=\overline{g}(h(X,Z),\mathcal{N}Y)+\alpha \overline{g}(h(X,Y),\mathcal{N}Z),$$

(42)

for any $X,Y,Z\in \Gamma \left(TM\right)$. On the other hand, interchanging Y and Z in (42), we have the equality

$$g(Y,\left({\nabla }_X\mathcal{T}\right)Z)=\overline{g}(h(X,Y),\mathcal{N}Z)+\alpha \overline{g}(h(X,Z),\mathcal{N}Y).$$

(43)

Thus, from (42) and (43) multiplied by $\alpha$, we obtain Equation (39).

Using (24) and (36), we obtain the equality

$$\begin{array}{c}\hfill \begin{array}{c}\overline{g}(\left({\overline{\nabla }}_X\mathfrak{n}\right)V,U)=-\overline{g}(h(X,\mathfrak{t}V),U)-\overline{g}(\mathcal{N}\left(A_{V}X\right),U)\hfill \\ =-\alpha \overline{g}(\mathcal{N}\left(A_{U}X\right)+h(X,\mathfrak{t}U),V)=\alpha \overline{g}(V,\left({\overline{\nabla }}_X\mathfrak{n}\right)U),\hfill \end{array}\end{array}$$

for any $X\in \Gamma \left(TM\right)$ and any $U,V\in \Gamma \left(T^{\perp }M\right)$, which implies Equation (40).

From the relations (15), (22)–(24), (34) and (35) we obtain the equalities

$$\begin{array}{c}\hfill \begin{array}{c}\overline{g}(\left({\overline{\nabla }}_X\mathcal{N}\right)Y,U)=-\overline{g}(\mathcal{T}Y,A_{U}X)+\alpha \overline{g}(h(X,Y),\mathfrak{n}U)+{\displaystyle \frac{p(1-\alpha )}{2}}\overline{g}(h(X,Y),U)\hfill \\ =-\alpha \overline{g}(Y,\mathcal{T}\left(A_{U}X\right))+\alpha \overline{g}(h(X,Y),\mathfrak{n}U)=\alpha \overline{g}(Y,-\mathcal{T}\left(A_{U}X\right)+A_{\mathfrak{n}U}X)=\alpha \overline{g}(Y,\left({\overline{\nabla }}_X\mathfrak{t}\right)U),\hfill \end{array}\end{array}$$

for any $X,Y\in \Gamma \left(TM\right)$ and any $U\in \Gamma \left(T^{\perp }M\right)$, which imply (41). □

By using Equation (41), we obtain the following property:

Corollary 1.

Let M be an isometrically immersed submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Then, $\nabla \mathcal{N}=0$ if and only if $\nabla \mathfrak{t}=0$.

Proposition 9.

Let M be an isometrically immersed submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Then, $\nabla \mathcal{T}=0$ if and only if $A_{\mathcal{N}X}Y=-\alpha A_{\mathcal{N}Y}X$, for any $X,Y\in \Gamma \left(TM\right)$.

Proof. 

By using Equation (33), we have $\nabla \mathcal{T}=0$ if and only if

$$0=\overline{g}(A_{\mathcal{N}Y}X,Z)+\overline{g}(\mathfrak{t}h(X,Y),Z)$$

for any $X,Y,Z\in \Gamma \left(TM\right)$ and from relations (15) and (24), we get

$$0=\overline{g}(h(X,Z),\mathcal{N}Y)+\alpha \overline{g}(h(X,Y),\mathcal{N}Z).$$

Thus, we obtain the equality

$$\overline{g}(A_{\mathcal{N}Y}Z+\alpha A_{\mathcal{N}Z}Y,X)=0,$$

for any $X,Y,Z\in \Gamma \left(TM\right)$, which implies the conclusion. □

By using (35) and Corollary 1, one obtains the following property.

Corollary 2.

Let M be an isometrically immersed submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Then, $\nabla \mathfrak{t}=0$ (or equivalently, $\nabla \mathcal{N}=0$) if and only if $A_{\mathfrak{n}U}X=\mathcal{T}{A}_{U}X$, for any $X\in \Gamma \left(TM\right)$ and any $U\in \Gamma \left(T{M}^{\perp }\right)$.

Proposition 10.

Let M be an isometrically immersed submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Then, $\overline{\nabla }\mathfrak{n}=0$ if and only if $A_V\left(\mathfrak{t}U\right)=-\alpha A_U\left(\mathfrak{t}V\right)$ for any $U,V\in \Gamma \left(T{M}^{\perp }\right)$.

Proof. 

From the Equation (36) we get $\nabla \mathfrak{n}=0$ if and only if

$$0=\overline{g}(h(X,\mathfrak{t}U),V)+\overline{g}(\mathcal{N}\left(A_{U}X\right),V),$$

for any $X\in \Gamma \left(TM\right)$ and $U,V\in \Gamma \left(T{M}^{\perp }\right)$. By using (15) and (24) in the last equality, we obtain

$$0=\overline{g}(A_V\left(\mathfrak{t}U\right),X)+\alpha \overline{g}(A_{U}X,\mathfrak{t}V)=\overline{g}(A_V\left(\mathfrak{t}U\right)+\alpha A_U\mathfrak{t}V,X),$$

for any $X\in \Gamma \left(TM\right)$ and $U,V\in \Gamma \left(T{M}^{\perp }\right)$, which implies the conclusion. □

4. Semi-Invariant Submanifolds in ($\mathbf{\alpha }$, $\mathit{p}$)-Golden Riemannian Manifold

4.1. Characterization of Semi-Invariant Submanifolds

Let M be an isometrically immersed submanifold in an even dimensional almost $(\alpha , p)$-golden Riemannian manifold ($\overline{M},\overline{g},{\Phi }_{\alpha ,p})$. In the rest of this paper, we suppose that M is a semi-invariant submanifold in an almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},\overline{g},{\Phi }_{\alpha ,p})$.

Definition 5.

A submanifold M is called a semi-invariant submanifold in $\overline{M}$ if $TM$ admits two orthogonally complementary distributions D and $D^{\perp }$ (i.e., $TM=D\oplus D^{\perp }$), where D is an invariant distribution with respect to ${\Phi }_{\alpha ,p}$ (i.e., ${\Phi }_{\alpha ,p}\left(D\right)=D$) and $D^{\perp }$ is an anti-invariant distribution with respect to ${\Phi }_{\alpha ,p}$ (i.e., ${\Phi }_{\alpha ,p}\left(D^{\perp }\right)\subset T^{\perp }M$).

The orthogonally complementary distributions D and $D^{\perp }$ are called the horizontal and the vertical distribution on M, respectively.

We denote the dimension of the invariant distribution D by $dim\left(D\right)$ and of the anti-invariant distribution $D^{\perp }$ by $dim\left(D^{\perp }\right)$. Thus, we can have the following situations:

• For $dim\left(D^{\perp }\right)=0$, the semi-invariant submanifold becomes an invariant submanifold;
• For $dim\left(D\right)=0$, the semi-invariant submanifold becomes an anti-invariant submanifold;
• If $dim\left(D\right)·dim\left(D^{\perp }\right)\ne 0$, the semi-invariant submanifold is called a proper semi-invariant submanifold.

Now, we denote the orthogonal complement of ${\Phi }_{\alpha ,p}\left(D^{\perp }\right)$ in $T{M}^{\perp }$ by $\tilde{D}$. Then, we have the direct sum

$$T{M}^{\perp }={\Phi }_{\alpha ,p}\left(D^{\perp }\right)\oplus \tilde{D}.$$

Proposition 11.

If M is a semi-invariant submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold $\overline{M}$, then the orthogonal complementary distributions $\tilde{D}$ of ${\Phi }_{\alpha ,p}\left(D^{\perp }\right)$ are invariant with respect to the structure ${\Phi }_{\alpha ,p}$ and ${\Phi }_{\alpha ,p}\left(\tilde{D}\right)=\tilde{D}.$

Proof. 

If $V\in \Gamma \left(\tilde{D}\right)$ and $W\in \Gamma \left({\Phi }_{\alpha ,p}\left(D^{\perp }\right)\right)$, then $\overline{g}(V,W)=0$. If $W={\Phi }_{\alpha ,p}Z$, for $Z\in \Gamma \left(D^{\perp }\right)$, from (5) and (6) we get

$$\begin{array}{c}\hfill \begin{array}{c}\overline{g}({\Phi }_{\alpha ,p}V,W)=\alpha \overline{g}(V,{\Phi }_{\alpha ,p}W)=\alpha \overline{g}(V,{\Phi }_{\alpha ,p}^{2}Z)\hfill \\ =\alpha p\overline{g}(V,W)+{\displaystyle \frac{\alpha (5\alpha -1)p^2}{4}}\overline{g}(V,Z)=0.\hfill \end{array}\end{array}$$

Therefore, ${\Phi }_{\alpha ,p}\left(\tilde{D}\right)\cap {\Phi }_{\alpha ,p}\left(D^{\perp }\right)=\left\{0\right\}$.

If $X\in \Gamma \left(D\right)$ and $V\in \Gamma \left(\tilde{D}\right)$, then $\overline{g}(X,V)=0$ and one obtains

$$\overline{g}({\Phi }_{\alpha ,p}V,X)=\alpha \overline{g}(V,{\Phi }_{\alpha ,p}X)=0,$$

and it implies ${\Phi }_{\alpha ,p}\left(\tilde{D}\right)\cap D=\left\{0\right\}$.

Moreover if $Y\in \Gamma \left(D^{\perp }\right)$ and $V\in \Gamma \left(\tilde{D}\right)$, then $\overline{g}(Y,V)=0$ and it follows that

$$\overline{g}({\Phi }_{\alpha ,p}V,Y)=\alpha \overline{g}(V,{\Phi }_{\alpha ,p}Y)=0,$$

which implies ${\Phi }_{\alpha ,p}\left(\tilde{D}\right)\cap D^{\perp }=\left\{0\right\}$.

Thus, ${\Phi }_{\alpha ,p}\left(\tilde{D}\right)\subseteq \tilde{D}$.

On the other hand, if $V\in \Gamma \left(\tilde{D}\right)$, then, using (5), we have

$$V={\displaystyle \frac{4}{p^2(5\alpha -1)}}{\Phi }_{\alpha ,p}({\Phi }_{\alpha ,p}V-pV)\in {\Phi }_{\alpha ,p}\left(\tilde{D}\right),$$

which leads to $\tilde{D}\subseteq {\Phi }_{\alpha ,p}\left(\tilde{D}\right)$. Thus, we have ${\Phi }_{\alpha ,p}\left(\tilde{D}\right)=\tilde{D}.$ □

Let us denote by P and Q the projection morphism of $TM$ to the orthogonally complementary distributions D and $D^{\perp }$, respectively. Thus, for any $X\in \Gamma \left(TM\right)$, one obtains

$$X=PX+QX.$$

(44)

Proposition 12.

If M is a semi-invariant submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ and D and $D^{\perp }$ are the horizontal and vertical distributions on M, respectively, then we obtain

$$\left(i\right)\mathcal{T}\left(D\right)=D,\left(ii\right)\mathcal{N}\left(D\right)=\left\{0\right\}$$

(45)

and

$$\left(i\right),\mathcal{T}\left(D^{\perp }\right)=\left\{0\right\}\left(ii\right)\mathcal{N}\left(D^{\perp }\right)={\Phi }_{\alpha ,p}\left(D^{\perp }\right)\subset T^{\perp }M,$$

(46)

where $\mathcal{T}$ and $\mathcal{N}$ were defined in (16) and (20).

Proof. 

From (20) and (44), we have

$${\Phi }_{\alpha ,p}X={\Phi }_{\alpha ,p}PX+{\Phi }_{\alpha ,p}QX=\mathcal{T}\left(PX\right)+\mathcal{N}\left(PX\right)+\mathcal{T}\left(QX\right)+\mathcal{N}\left(QX\right)$$

for any $X\in \Gamma \left(TM\right)$. By using Definition 5, it follows that

$${\Phi }_{\alpha ,p}X=\mathcal{T}\left(PX\right)+\mathcal{N}\left(QX\right)$$

and

$$\left(i\right)\mathcal{N}\left(PX\right)=0,\left(ii\right)\mathcal{T}\left(QX\right)=0,$$

for any $X\in \Gamma \left(TM\right)$. Thus, we obtain ${\Phi }_{\alpha ,p}X=\mathcal{T}X$ and $\mathcal{N}X=0$, for any $X\in \Gamma \left(D\right)$. Moreover, for any $Y\in \Gamma \left(D^{\perp }\right)$, we have ${\Phi }_{\alpha ,p}Y=\mathcal{N}Y$ and $\mathcal{T}Y=0$, which leads to (45) and (46). □

If we replace X by $\mathcal{T}X$ in (25), then we have the following property.

Proposition 13.

If M is a semi-invariant submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ such that $\mathcal{N}\left(\mathcal{T}X\right)=0$ for any $X\in \Gamma \left(TM\right)$, then we obtain the equalities

$${\mathcal{T}}^{3}X=p{\mathcal{T}}^{2}X+{\displaystyle \frac{5\alpha -1}{4}}{p}^2\mathcal{T}X$$

(47)

and

$$\mathfrak{n}\left(\mathcal{N}X\right)=p\mathcal{N}X,$$

for any $X\in \Gamma \left(TM\right)$.

Proposition 14.

Let M be a semi-invariant submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. If D is the horizontal distribution of M and P is the projector operator on D, then one obtains

$${\left(\mathcal{T}P\right)}^{2}X=p\mathcal{T}\left(PX\right)+{\displaystyle \frac{5\alpha -1}{4}}{p}^{2}PX$$

(48)

and

$$\mathfrak{n}\left(\mathcal{N}\right(PX\left)\right)=p\mathcal{N}\left(PX\right),$$

for any $X\in \Gamma \left(TM\right)$.

From the relation (48), we obtain the following property:

Remark 6.

If M is a semi-invariant submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold, then the structure $\mathcal{T}P$ is an $(\alpha , p)$-golden structure on D.

Theorem 1.

Let M be a submanifold isometrically immersed in an almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$, such that $\mathcal{N}\left(\mathcal{T}X\right)=0$, for any $X\in \Gamma \left(TM\right)$. If l and m are two operators on $TM$, defined by the equalities

$$\left(i\right)l={\displaystyle \frac{4}{(5\alpha -1)p^2}}({\mathcal{T}}^2-p\mathcal{T}),\left(ii\right)m={\displaystyle \frac{4}{(5\alpha -1)p^2}}(-{\mathcal{T}}^2+p\mathcal{T})+Id,$$

(49)

then l and m are orthogonal complementary projection operators on $TM$. Moreover, we obtain the following equalities

$$\left(i\right)\mathcal{T}\left(lX\right)=\mathcal{T}X,\left(ii\right)\mathcal{T}\left(mX\right)=0$$

(50)

and

$$\left(i\right)\mathcal{N}\left(lX\right)=0,\left(ii\right)\mathcal{N}\left(mX\right)=\mathcal{N}X,$$

(51)

for any $X\in \Gamma \left(TM\right)$.

Proof. 

First of all, applying $\mathcal{T}$ in (49) (i), we have

$$\mathcal{T}\left(lX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}({\mathcal{T}}^{3}X-p{\mathcal{T}}^{2}X),$$

and using the relation (47), we obtain (50) (i).

Similarly, applying $\mathcal{T}$ in (49) (ii), we have

$$\mathcal{T}\left(mX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}(-{\mathcal{T}}^{3}X+p{\mathcal{T}}^{2}X+{\displaystyle \frac{(5\alpha -1)p^2}{4}}\mathcal{T}X),$$

and from (47), we obtain (50) (ii).

Now, applying $\mathcal{N}$ in (49) (i), we get

$$\mathcal{N}\left(lX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}(\mathcal{N}\left({\mathcal{T}}^{2}X\right)-p\mathcal{N}\left(\mathcal{T}X\right)),$$

and using $\mathcal{N}\mathcal{T}=0,$ we obtain (51) (i). Moreover, applying $\mathcal{N}$ in (49) (ii), we have

$$\mathcal{N}\left(mX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}(-\mathcal{N}\left({\mathcal{T}}^{2}X\right)+p\mathcal{N}\left(\mathcal{T}X\right))+\mathcal{N}X,$$

and using $\mathcal{N}\mathcal{T}=0,$ (51) (ii) is proved.

On the other hand, we prove that l and m are orthogonal complementary projection operators, which means that they satisfy the equalities:

$$l+m=Id,l^2=l,m^2=m,l·m=m·l=0$$

By using (49) it follows that $l+m=Id$. Moreover, applying (49) (i) to $lX$, it leads to

$$l^{2}X=l\left(lX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}({\mathcal{T}}^2\left(lX\right)-p\mathcal{T}\left(lX\right)),$$

for any $X\in \Gamma \left(TM\right)$ and from relations (50) (i) and (49) (i), we obtain $l^{2}X=lX.$

Similarly, applying (49) (ii) to $mX$, we obtain

$$m^{2}X=m\left(mX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}(-{\mathcal{T}}^2\left(mX\right)+p\mathcal{T}\left(mX\right))+mX,$$

for any $X\in \Gamma \left(TM\right)$. From the relations (50) (ii) and (49) (ii), we have $m^{2}X=mX$.

From the relations (49) (i) and (50) (ii), we have

$$l\left(mX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}({\mathcal{T}}^{2}mX-p\mathcal{T}mX)=0,$$

for any $X\in \Gamma \left(TM\right)$. By using (49) (ii), (50), and (49) (i), we get

$$m\left(lX\right)={\displaystyle \frac{4}{(5\alpha -1)p^2}}(-{\mathcal{T}}^{2}lX+p\mathcal{T}lX)+lX=-lX+lX=0,$$

for any $X\in \Gamma \left(TM\right)$. □

Theorem 2.

Let M be an isometrically immersed submanifold in an almost $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Then M is a semi-invariant submanifold in $\overline{M}$ if and only if the operators $\mathcal{T}$ and $\mathcal{N}$ verify the identity $\mathcal{N}\mathcal{T}=0$.

Proof. 

If we suppose that M is a semi-invariant submanifold in $\overline{M}$ and using the decomposition of X given in (44), then, from $\mathcal{T}\left(PX\right)\in \Gamma \left(D\right)$ and $\mathcal{N}\left(D\right)=0$, we obtain

$$\mathcal{N}\left(\mathcal{T}X\right)=\mathcal{N}\left(\mathcal{T}\right(PX+QX\left)\right)=\mathcal{N}\left(\mathcal{T}\right(PX\left)\right)=0,$$

for any $X\in \Gamma \left(TM\right)$.

Conversely, if we consider that $\mathcal{N}\left(\mathcal{T}X\right)=0$ for any $X\in \Gamma \left(TM\right)$, then the operators l and m defined in (49) are orthogonal complementary projection operators and they define two complementary distributions $D_l$ and $D_m$ on $TM$. Also, from any $X\in \Gamma \left(TM\right)$, we obtain $X=lX+mX$, and using (50) and (51), we have

$${\Phi }_{\alpha ,p}X={\Phi }_{\alpha ,p}lX+{\Phi }_{\alpha ,p}mX=\mathcal{T}lX+\mathcal{N}mX.$$

(52)

By using (52), we have ${\Phi }_{\alpha ,p}X=\mathcal{T}X$ for any $X\in \Gamma \left(D_l\right)$, and ${\Phi }_{\alpha ,p}Z=\mathcal{N}Z$ for any $Z\in \Gamma \left(D_m\right)$. Thus, it follows that the distribution $D_l:=D$ is an invariant distribution and the distribution $D_m:=D^{\perp }$ is an anti-invariant distribution with respect to ${\Phi }_{\alpha ,p}$. Hence, M is a semi-invariant submanifold in $\overline{M}$. □

In particular, for $\alpha =1$ and $p=1$, we obtain the following property.

Corollary 3

([18], Theorem 1). Let M be any submanifold of a golden Riemannian manifold $(\overline{M},\overline{g},\overline{\Phi })$. Then, a necessary and sufficient condition for the submanifold M to be semi-invariant is that $\mathcal{N}\mathcal{T}=0$.

A similar property to that given in Corollary 3 is obtained in the case of an isometrically immersed submanifold in a metallic Riemannian manifold.

Corollary 4

([19], Proposition 3.2). If M is an isometrically immersed submanifold in a metallic Riemannian manifold $(\overline{M},\overline{g},\overline{J})$, then the submanifold M is semi-invariant if and only if $\mathcal{N}\mathcal{T}=0$.

4.2. On the Integrability of the Distributions of Semi-Invariant Submanifolds

In order to study the integrability of the invariant distribution D and of the anti-invariant distribution $D^{\perp }$, we calculate the tangential and the normal components of ${\Phi }_{\alpha ,p}\left([X,Y]\right)$ respectively, given in the next lemma.

Lemma 1.

If M is an isometrically immersed submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$, then the normal and the tangential part of ${\Phi }_{\alpha ,p}\left([X,Y]\right)$ are given by the equalities

$$\mathcal{T}\left([X,Y]\right)={\nabla }_X\left(\mathcal{T}Y\right)-{\nabla }_Y\left(\mathcal{T}X\right)+A_{\mathcal{N}X}Y-A_{\mathcal{N}Y}X$$

(53)

and

$$\mathcal{N}\left([X,Y]\right)=h(X,\mathcal{T}Y)-h(\mathcal{T}X,Y)+{\nabla }_X^{\perp }\left(\mathcal{N}Y\right)-{\nabla }_Y^{\perp }\left(\mathcal{N}X\right),$$

(54)

respectively, for any $X,Y\in \Gamma \left(TM\right)$, where ∇ is the Levi-Civita connection on M.

Proof. 

Using $[X,Y]={\nabla }_{X}Y-{\nabla }_{Y}X$ and (33), we have the equality

$$\mathcal{T}\left({\nabla }_{X}Y\right)={\nabla }_X\left(\mathcal{T}Y\right)-A_{\mathcal{N}Y}X-\mathfrak{t}h(X,Y),$$

for any $X,Y\in \Gamma \left(D\right)$. Interchanging X by Y and subtracting these two equalities, it leads to the relation (53).

By using the relation (34), it follows that

$$\mathcal{N}\left({\nabla }_{X}Y\right)={\nabla }_X^{\perp }\left(\mathcal{N}Y\right)+h(X,\mathcal{T}Y)-\mathfrak{n}h(X,Y),$$

for any $X,Y\in \Gamma \left(D\right)$. Interchanging X by Y and subtracting these two equalities, we obtain (54). □

Theorem 3.

Let M be a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Thus, we obtain the following:

• The invariant distribution D is integrable if and only if we have the equality

  $$h({\Phi }_{\alpha ,p}X,Y)=h(X,{\Phi }_{\alpha ,p}Y),$$

  (55)

  for any $X,Y\in \Gamma \left(D\right)$;
• The anti-invariant distribution $D^{\perp }$ is integrable if and only if we have the equality

  $$A_{{\Phi }_{\alpha ,p}Z}W=A_{{\Phi }_{\alpha ,p}W}Z,$$

  (56)

  for any $Z,W\in \Gamma \left(D^{\perp }\right)$.

Proof. 

Let us consider $X,Y\in \Gamma \left(D\right),$ which implies $\mathcal{N}X=\mathcal{N}Y=0$. The distribution D is integrable if and only if $\mathcal{N}\left(\right[X,Y\left]\right)=0$ for any $X,Y\in \Gamma \left(D\right)$. By using ${\Phi }_{\alpha ,p}X=\mathcal{T}X$, for any $X\in \Gamma \left(D\right)$ and (54), we obtain that the distribution D is integrable if and only if (55) holds.

Now, let us consider $Z,W\in \Gamma \left(D^{\perp }\right),$ which implies $\mathcal{T}Z=\mathcal{T}W=0$. The distribution $D^{\perp }$ is integrable if and only if $\mathcal{T}\left(\right[X,Y\left]\right)=0$, for any $Z,W\in \Gamma \left(D^{\perp }\right)$. By using ${\Phi }_{\alpha ,p}Z=\mathcal{N}Z$, for any $Z\in \Gamma \left(D^{\perp }\right)$ and (53), we obtain that the distribution $D^{\perp }$ is integrable if and only if (56) holds. □

In particular, for $\alpha =p=1$, one obtains the following:

Corollary 5

([17], Theorem 2.1). Let M be a semi-invariant submanifold in a golden Riemannian manifold $(\overline{M},P)$. Then, the distribution D is integrable if and only if $h(PX,Y)=h(X,PY)$, for any $X,Y\in \Gamma \left(D\right)$.

Using (33) and (34) in Theorem 3, we obtain the following necessary and sufficient conditions for the integrability of the invariant and anti-invariant distributions, respectively.

Corollary 6.

If M is a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ and D and $D^{\perp }$ are the invariant and anti-invariant distributions on M, respectively, then the following equivalences hold:

• The distribution D is integrable if and only if we have the equality

  $$\left({\overline{\nabla }}_X\mathcal{N}\right)Y=\left({\overline{\nabla }}_Y\mathcal{N}\right)X$$

  for any $X,Y\in \Gamma \left(D\right)$;
• The distribution $D^{\perp }$ is integrable if and only if we have the equality

  $$\left({\nabla }_Z\mathcal{T}\right)W=\left({\nabla }_W\mathcal{T}\right)Z$$

  for any $Z,W\in \Gamma \left(D^{\perp }\right)$.

Corollary 7.

Let M be a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. The invariant distribution D is integrable if and only if the following equality holds

$$h({\Phi }_{\alpha ,p}X,{\Phi }_{\alpha ,p}Y)=ph({\Phi }_{\alpha ,p}X,Y)+{\displaystyle \frac{5\alpha -1}{4}}{p}^{2}h(X,Y),$$

(57)

for any $X,Y\in \Gamma \left(D\right)$.

Proof. 

If we consider that the horizontal distribution D is integrable, then by using (55) from Theorem 3, we obtain the relation

$$h({\Phi }_{\alpha ,p}X,{\Phi }_{\alpha ,p}Y)=h({\Phi }_{\alpha ,p}^{2}X,Y),$$

for any $X,Y\in \Gamma \left(D\right)$. Thus, from Equation (5) of ${\Phi }_{\alpha ,p}$, we obtain (57).

Conversely, if (57) holds, then interchanging X and Y and subtracting these two equations, we obtain $h({\Phi }_{\alpha ,p}X,Y)=h(X,{\Phi }_{\alpha ,p}Y)$. Thus, by using Theorem 3, we have that the horizontal distribution D of M is integrable. □

Theorem 4.

Let M be a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. Let D and $D^{\perp }$ be the invariant and anti-invariant distributions on M, respectively. The distribution D on M is integrable if and only if

$$A_V\mathcal{T}X-\alpha \mathcal{T}{A}_{V}X-{\displaystyle \frac{p(1-\alpha )}{2}}{A}_{V}X\in D^{\perp }$$

(58)

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

Proof. 

From Theorem 3, the horizontal distribution D of M is integrable if and only if relation (55) holds. This implies the equality

$$\overline{g}(h(X,\mathcal{T}Y),V)=\overline{g}(h(Y,\mathcal{T}X),V),$$

(59)

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

From (15) and (59), we obtain

$$g(\mathcal{T}Y,A_{V}X)=g(Y,A_V\mathcal{T}X),$$

(60)

for any $X,Y\in \Gamma \left(D\right)$ and any $V\in \Gamma \left(T^{\perp }M\right)$. Now, using (22), we have

$$g(\mathcal{T}Y,A_{V}X)=\alpha g(Y,\mathcal{T}{A}_{V}X)+{\displaystyle \frac{p(1-\alpha )}{2}}g(Y,A_{V}X),$$

(61)

for any $X,Y\in \Gamma \left(D\right)$ and any $V\in \Gamma \left(T^{\perp }M\right)$. By using (60) and (61), it follows that

$$g\left(A_V\mathcal{T}X-\alpha \mathcal{T}{A}_{V}X-{\displaystyle \frac{p(1-\alpha )}{2}}{A}_{V}X,Y\right)=0,$$

for any $X,Y\in \Gamma \left(D\right)$ and any $V\in \Gamma \left(T^{\perp }M\right)$. Thus, the distribution D on M is integrable if and only if relation (58) holds. □

If we consider $\alpha =1$ (and $\alpha$ = −1, respectively) in (58), we obtain the next two corollaries.

Corollary 8.

If M is a semi-invariant submanifold isometrically immersed in a locally $(1,p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{1,p},\overline{g})$, then the horizontal distribution D on M is integrable if and only if

$$\mathcal{T}{A}_{V}X-A_V\mathcal{T}X\in D^{\perp },$$

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

Corollary 9.

If M is a semi-invariant submanifold isometrically immersed in a locally $(-1,p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{-1,p},\overline{g})$, then the horizontal distribution D on M is integrable if and only if

$$\mathcal{T}{A}_{V}X+A_V\mathcal{T}X-p{A}_{V}X\in D^{\perp },$$

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

Theorem 5.

Let M be a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$ and let D and $D^{\perp }$ be the horizontal and vertical distributions on M, respectively.

• For $\alpha =1$, the distribution $D^{\perp }$ is integrable if and only if

  $$A_{{\Phi }_{1,p}W}Z=0,$$

  (62)

  for any $Z,W\in \Gamma \left(D^{\perp }\right)$;
• For $\alpha =-1$, the distribution $D^{\perp }$ is integrable if and only if

  $${\nabla }_{X}W\in \Gamma \left(D\right),$$

  (63)

  for any $X\in \Gamma \left(TM\right)$ and any $W\in \Gamma \left(D^{\perp }\right)$.

Proof. 

We denote ${\Phi }_{\alpha ,p}:=\Phi$ and we consider $Z,W\in \Gamma \left(D^{\perp }\right)$ and $X\in \Gamma \left(TM\right)$. Thus, we have $\Phi Z=\mathcal{N}Z\in \Gamma \left(T^{\perp }M\right)$. By using (13) and (15), one obtains the equalities

$$\overline{g}(A_{\Phi Z}W,X)=\overline{g}(h(X,W),\Phi Z)=\overline{g}({\overline{\nabla }}_{X}W-{\nabla }_{X}W,\mathcal{N}Z)=\overline{g}({\overline{\nabla }}_{X}W,\Phi Z),$$

for any $Z,W\in \Gamma \left(D^{\perp }\right)$ and any $X\in \Gamma \left(TM\right)$. Now, using (6) and (14) and $\overline{\nabla }\Phi =0$ (which implies ${\overline{\nabla }}_X\Phi W=\Phi {\overline{\nabla }}_{X}W$), we have

$$\begin{array}{cc}\hfill \overline{g}(A_{\Phi Z}W,X)& =\overline{g}(\Phi Z,{\overline{\nabla }}_{X}W)=\alpha \overline{g}(Z,{\overline{\nabla }}_X\Phi W)+{\displaystyle \frac{p(1-\alpha )}{2}}\overline{g}({\overline{\nabla }}_{X}W,Z)\hfill \\ \hfill & =-\alpha \overline{g}(Z,A_{\Phi W}X)+{\displaystyle \frac{p(1-\alpha )}{2}}\overline{g}({\nabla }_{X}W,Z).\hfill \end{array}$$

Moreover, from $\Phi W\in \Gamma \left(T{M}^{\perp }\right)$, we get

$$\overline{g}(A_{\Phi Z}W+\alpha A_{\Phi W}Z,X)={\displaystyle \frac{p(1-\alpha )}{2}}\overline{g}({\nabla }_{X}W,Z),$$

(64)

for any $Z,W\in \Gamma \left(D^{\perp }\right)$ and any $X\in \Gamma \left(TM\right)$.

For $\alpha =1$, the equality (64) leads to $\overline{g}(A_{\Phi Z}W+A_{\Phi W}Z,X)=0,$ for any $X\in \Gamma \left(TM\right)$; thus, $A_{\Phi Z}W=-A_{\Phi W}Z$, for any $Z,W\in \Gamma \left(D^{\perp }\right)$. On the other hand, from (56), the distribution $D^{\perp }$ is integrable if and only $A_{\Phi Z}W=A_{\Phi W}Z$. Thus, we obtain (62).

For $\alpha =-1$, from (64), we have

$$\overline{g}(A_{\Phi Z}W-A_{\Phi W}Z,X)=p\overline{g}({\nabla }_{X}W,Z),$$

(65)

for any $Z,W\in \Gamma \left(D^{\perp }\right)$ and any $X\in \Gamma \left(TM\right)$. Thus, by taking (56) and (65) into account, we obtain that the distribution $D^{\perp }$ is integrable if and only $\overline{g}({\nabla }_{X}W,Z)=0$, for any $Z,W\in \Gamma \left(D^{\perp }\right)$ and any $X\in \Gamma \left(TM\right)$, which implies (63). □

4.3. Mixed Totally Geodesic Semi-Invariant Submanifolds

In this subsection we study the conditions which imply that the submanifold M is a $D-D^{\perp }$ mixed totally geodesic submanifold in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$.

Definition 6.

Let us consider that M is a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$. If $h(X,Z)=0$, for any $X\in \Gamma \left(D\right)$ and $Z\in \Gamma \left(D^{\perp }\right)$, where D and $D^{\perp }$ are the horizontal and vertical distributions on M, respectively, then M is called a $D-D^{\perp }$ mixed totally geodesic submanifold in $\overline{M}$.

Theorem 6.

If M is a semi-invariant submanifold isometrically immersed in a locally $(\alpha , p)$-golden Riemannian manifold $(\overline{M},{\Phi }_{\alpha ,p},\overline{g})$, then the submanifold M is a $D-D^{\perp }$ mixed totally geodesic submanifold if and only if $A_{V}X\in \Gamma \left(D\right)$ or $A_{V}Z\in \Gamma \left(D^{\perp }\right)$, for any $X\in \Gamma \left(D\right)$, $Z\in \Gamma \left(D^{\perp }\right)$, and $V\in \Gamma \left(T^{\perp }M\right)$.

Proof. 

The submanifold M is a $D-D^{\perp }$ mixed totally geodesic submanifold if and only if we have

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

for any $X\in \Gamma \left(D\right)$, $Z\in \Gamma \left(D^{\perp }\right)$, and $V\in \Gamma \left(T{M}^{\perp }\right)$, and these imply $A_{V}X\in \Gamma \left(D\right)$ and $A_{V}Z\in \Gamma \left(D^{\perp }\right)$. □