Submanifolds of Generalized Sasakian-space-forms with respect to certain connections

TL;DR

This paper investigates the properties of submanifolds within generalized Sasakian-space-forms under various specialized connections, extending existing geometric frameworks and exploring their structural implications.

Contribution

It introduces new results on submanifolds of generalized Sasakian-space-forms with respect to multiple connections, expanding the understanding of their geometric behavior.

Findings

Results on submanifold properties under semisymmetric metric connection

Findings related to semisymmetric non-metric connection

Insights into Schouten-van Kampen and Tanaka-webster connections

Abstract

The present paper deals with some results of submanifolds of generalized Sasakian-space-forms in \cite{ALEGRE3} with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection.

Equations

$$\bar{R}(X,Y)Z$$

$$\displaystyle-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi Z\big{\}}+f_{3}\big{\{}\eta(X)\eta(Z)Y$$

$$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\}}$$

$${\varphi }^2(X)=-X+\eta (X)\xi ,\varphi \xi =0,$$

$$\eta (\xi )=1,g(X,\xi )=\eta (X),\eta (\varphi X)=0,$$

$$g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y),$$

$$g(\varphi X,Y)=-g(X,\varphi Y).$$

$$({\bar{\nabla }}_X\varphi )(Y)=(f_1-f_3)[g(X,Y)\xi -\eta (Y)X],$$

$${\bar{\nabla }}_X\xi =-(f_1-f_3)\varphi X,$$

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

$$\varphi X=TX+FX,$$

$${\tilde{\bar{\nabla }}}_{X}Y={\bar{\nabla }}_{X}Y+\eta (Y)X-g(X,Y)\xi .$$

$$\tilde{\bar{R}}(X,Y)Z$$

$$\displaystyle=(f_{1}-1)\big{\{}g(Y,Z)X-g(X,Z)Y\big{\}}+f_{2}\big{\{}g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X$$

$$\displaystyle+2g(X,\phi Y)\phi Z\big{\}}+(f_{3}-1)\big{\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi$$

$$\displaystyle-g(Y,Z)\eta(X)\xi\big{\}}+(f_{1}-f_{3})\{g(X,\phi Z)Y-g(Y,\phi Z)X+g(Y,Z)\phi X$$

$$-g(X,Z)\varphi Y\}.$$

$${\bar{\nabla }}_X^{{}^{\prime }}Y={\bar{\nabla }}_{X}Y+\eta (Y)X.$$

$${\bar{R}}^{{}^{\prime }}(X,Y)Z$$

$${\hat{\bar{\nabla }}}_{X}Y={\bar{\nabla }}_{X}Y+(f_1-f_3)\eta (Y)\varphi X-(f_1-f_3)g(\varphi X,Y)\xi .$$

$$\hat{\bar{R}}(X,Y)Z$$

$$\displaystyle=f_{1}\big{\{}g(Y,Z)X-g(X,Z)Y\big{\}}+f_{2}\big{\{}g(X,\phi Z)\phi Y$$

$$\displaystyle-g(Y,\phi Z)\phi X+2g(X,\phi Y)\phi Z\big{\}}+\{f_{3}+(f_{1}-f_{3})^{2}\}\big{\{}\eta(X)\eta(Z)Y$$

$$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\}}$$

$$\displaystyle+(f_{1}-f_{3})^{2}\big{[}g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X\big{]}.$$

$${\stackrel{\ast }{\bar{\nabla }}}_{X}Y={\bar{\nabla }}_{X}Y+\eta (X)\varphi Y+(f_1-f_3)\eta (Y)\varphi X-(f_1-f_3)g(\varphi X,Y)\xi .$$

$$\stackrel{\ast }{\bar{R}}(X,Y)Z$$

$$\displaystyle=f_{1}\big{\{}g(Y,Z)X-g(X,Z)Y\big{\}}+f_{2}\big{\{}g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X$$

$$\displaystyle+2g(X,\phi Y)\phi Z\big{\}}+\{f_{3}+{(f_{1}-f_{3})^{2}}\}\big{\{}\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X$$

$$\displaystyle+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\}}+{(f_{1}-f_{3})^{2}}\big{[}g(X,\phi Z)\phi Y$$

$$\displaystyle-g(Y,\phi Z)\phi X\big{]}+2(f_{1}-f_{3})g(X,\phi Y)\phi Z.$$

$$tan(\tilde{\bar{R}}(X,Y)Z)$$

$$\displaystyle=(f_{1}-1)\big{\{}g(Y,Z)X-g(X,Z)Y\big{\}}+(f_{3}-1)\big{\{}\eta(X)\eta(Z)Y$$

$$\displaystyle-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big{\}},$$

$$nor(\tilde{\bar{R}}(X,Y)Z)$$

$$\tilde{\bar{R}}(X,Y)X$$

$$\tilde{\bar{R}}(U,V)X$$

$$g(X,\varphi U)=-g(\varphi X,U)=g(\varphi V,X)=0$$

Full text

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1. Introduction

As a generalization of Sasakian-space-form, Alegre et al. [2] introduced the notion of generalized Sasakian-space-form as that an almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$ whose curvature tensor $\bar{R}$ of $\bar{M}$ satisfies

[TABLE]

for all vector fields $X$, $Y$, $Z$ on $\bar{M}$ and $f_{1},f_{2},f_{3}$ are certain smooth functions on $\bar{M}$. Such a manifold of dimension $(2n+1)$, $n>1$ (the condition $n>1$ is assumed throughout the paper), is denoted by $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ [2]. Many authors studied this space form with different aspects. For this, we may refer ([11], [12], [13], [14], [15], [17], [18] and [23]). It reduces to Sasakian-space-form if $f_{1}=\frac{c+3}{4}$, $f_{2}=f_{3}=\frac{c-1}{4}$ [2].

After introducing the semisymmetric linear connection by Friedman and Schouten [7], Hayden [9] gave the idea of metric connection with torsion on a Riemannian manifold. Later, Yano [29] and many others (see, [21], [22], [24] and references therein) studied semisymmetric metric connection in different context. The idea of semisymmetric non-metric connection was introduced by Agashe and Chafle [1].

The Schouten-van Kampen connection introduced for the study of non-holomorphic manifolds ([20], [27]). In $2006$, Bejancu [6] studied Schouten-van Kampen connection on foliated manifolds. Recently Olszak [19] studied Schouten-van Kampen connection on almost(para) contact metric structure.

The Tanaka-Webster connection ([25], [28]) is the canonical affine connection defined on a non-degenerate pseudo-Hermitian CR-manifold. Tanno [26] defined the Tanaka-Webster connection for contact metric manifolds.

The submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ are studied in ([3], [10], [16]). In [3], Alegre and Carriazo studied submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to Levi-Civita connection $\bar{\nabla}$. The present paper deals with study of such submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection respectively.

2. preliminaries

In an almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$, we have [4]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$, we have [2]

[TABLE]

[TABLE]

where $\bar{\nabla}$ is the Levi-Civita connection of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$.

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$. If $\nabla$ and $\nabla^{\perp}$ are the induced connections on the tangent bundle $TM$ and the normal bundle $T^{\perp}{M}$ of $M$, respectively then the Gauss and Weingarten formulae are given by [30]

[TABLE]

for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and $A_{V}$ are second fundamental form and shape operator (corresponding to the normal vector field V), respectively and they are related by [30] $g(h(X,Y),V)=g(A_{V}X,Y)$.

For any $X\in\Gamma(TM)$, we may write

[TABLE]

where $TX$ is the tangential component and $FX$ is the normal component of $\phi X$.

In particular, if $F=0$ then $M$ is invariant [5] and here $\phi(TM)\subset TM$. Also if $T=0$ then $M$ is anti-invariant [5] and here $\phi(TM)\subset T^{\bot}M$. Also here we assume that $\xi$ is tangent to $M$.

The semisymmetric metric connection $\widetilde{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [29]

[TABLE]

The Riemannian curvature tensor $\widetilde{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$ is

[TABLE]

The semisymmetric non-metric connection ${\bar{\nabla}}^{{}^{\prime}}$ and the Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [1]

[TABLE]

The Riemannian curvature tensor ${\bar{R}}^{{}^{\prime}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{{}^{\prime}}$ is

[TABLE]

The Schouten-van Kampen connection $\hat{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$ of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [19]

[TABLE]

The Riemannian curvature tensor $\hat{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is

[TABLE]

The Tanaka-Webster connection $\stackrel{{\scriptstyle\ast}}{{\bar{\nabla}}}$ and the Riemannian connection $\bar{\nabla}$ of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [8]

[TABLE]

The Riemannian curvature tensor $\stackrel{{\scriptstyle*}}{{\bar{R}}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ is

[TABLE]

3. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with $\widetilde{\bar{\nabla}}$

Lemma 3.1**.**

If $M$ is invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$, for any $X,Y,Z\in\Gamma(TM)$.

Proof.

If $M$ is invariant then from (2.10) we say that $\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$. This proves the lemma. ∎

Lemma 3.2**.**

If $M$ is anti-invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$, then

[TABLE]

[TABLE]

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

Proof.

Since $M$ is anti-invariant, we have $\phi X,\phi Y\in\Gamma(T^{\bot}M)$. Then equating tangent and normal component of (2.10) we get the result. ∎

Lemma 3.3**.**

If $f_{1}(p)=f_{3}(p)$ and $M$ is either invariant or anti-invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$ for any $X,Y,Z\in\Gamma(TM)$.

Proof.

Using Lemma $3.1$ and Lemma $3.2$ we get the result. ∎

Lemma 3.4**.**

If $M$ is invariant or anti-invariant submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$, then $\widetilde{\bar{R}}(X,Y)V$ is normal to $M$, for any $X,Y,\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

Proof.

If $M$ is invariant from (2.10) we have $\widetilde{\bar{R}}(X,Y)V$ normal to $M$, and if $M$ is anti-invariant then $\widetilde{\bar{R}}(X,Y)V=0$ i.e. $\widetilde{\bar{R}}(X,Y)V$ normal to $M$ for any $X,Y,\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. ∎

Lemma 3.5**.**

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$, $f_{1}(p)=f_{3}(p)$ and $TM$ is invariant under the action of $\widetilde{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

Proof.

For $X,Y\in\Gamma(TM)$, we have from (2.10) that

[TABLE]

Note that $\widetilde{\bar{R}}(X,Y)X$ should be tangent if $[-3f_{2}g(Y,\phi X)\phi X+(f_{1}-f_{3})\{g(Y,X)\phi X-g(X,X)\phi Y\}]$ is tangent. Since $f_{2}(p)\neq 0$, $f_{1}(p)=f_{3}(p)$ at any point $p$ then by similar way of proof of Lemma $3.2$ of [3], we can prove that either $M$ is invariant or anti-invariant. This proves the Lemma. ∎

Remark 3.1*.*

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$. If $f_{1}(p)\neq f_{3}(p)$ and $TM$ is invariant under the action of $\widetilde{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is invariant.

From Lemma $3.3$ and Lemma $3.5$, we have

Theorem 3.1**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$, $f_{1}(p)=f_{3}(p)$ then $M$ is either invariant or anti-invariant if and only if $TM$ is invariant under the action of $\widetilde{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.

Proposition 3.1**.**

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant under the action of $\widetilde{\bar{R}}(U,V)$ for any $U,V\in\Gamma(T^{\bot}M)$.

Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2.10), we get

[TABLE]

As $M$ is invariant, $U,V\in\Gamma(T^{\bot}M)$, we have

[TABLE]

for any $X\in\Gamma(TM)$. Using (3.5) in (3), we have

[TABLE]

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

Proposition 3.2**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$. If $f_{2}(p)\neq 0$, $f_{1}(p)=f_{3}(p)$ for each $p\in M$ and $T^{\bot}M$ is invariant under the action of $\widetilde{\bar{R}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then $M$ is either invariant or anti-invariant.

Proof.

The proof is similar as it is an Lemma $3.4$, just assuming that $\widetilde{\bar{R}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎

4. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with ${\bar{\nabla}}^{{}^{\prime}}$

Lemma 4.1**.**

If $M$ is either invariant or anti-invarint submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{{}^{\prime}}$, then ${\bar{R}}^{{}^{\prime}}(X,Y)Z$ is tangent to $M$ and ${\bar{R}}^{{}^{\prime}}(X,Y)V$ normal to $M$ for any $X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

Proof.

If $M$ is invariant then from (2) we say that ${\bar{R}}^{{}^{\prime}}(X,Y)Z$ is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.

If $M$ is anti-invariant then

[TABLE]

From (2) and (4.1) we have

[TABLE]

which is tangent.

If $M$ is invariant then from (2), it follows that ${\bar{R}}^{{}^{\prime}}(X,Y)V$ is normal to $M$, and if $M$ is anti-invariant then ${\bar{R}}^{{}^{\prime}}(X,Y)V=0$ i.e. ${\bar{R}}^{{}^{\prime}}(X,Y)V$ is normal to $M$ for any $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎

Lemma 4.2**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in M$ and $TM$ is invariant under the action of $\bar{R}^{{}^{\prime}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

Proof.

For $X,Y\in\Gamma(TM)$, we have from (2) that

[TABLE]

Note that ${\bar{R}}^{{}^{\prime}}(X,Y)X$ should be tangent if $3f_{2}(p)g(Y,\phi X)\phi X$ is tangent. Since $f_{2}(p)\neq 0$ for each $p\in M$, as similar as proof of Lemma $3.2$ of [3], we may conclude that either $M$ is invariant or anti-invariant. This proves the Lemma. ∎

From Lemma $4.1$ and Lemma $4.2$, we have

Theorem 4.1**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in M$, then $M$ is either invariant or anti-invariant if and only if $TM$ is invariant under the action of ${\bar{R}}^{{}^{\prime}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.

Proposition 4.1**.**

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $M$ is invariant, then $TM$ is invariant under the action of ${\bar{R}}^{{}^{\prime}}(U,V)$ for any $U,V\in\Gamma(T^{\bot}M)$.

Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2), we get

[TABLE]

As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have

[TABLE]

for any $X\in\Gamma(TM)$. Using (4.5) in (4), we have

[TABLE]

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

Proposition 4.2**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{{}^{\prime}}$. If $f_{2}(p)\neq 0$ for each $p\in M$ and $T^{\bot}M$ is invariant under the action of ${\bar{R}}(U,V)$, $U,V\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

Proof.

The proof is similar as the proof of Lemma $4.2$, just imposing that ${\bar{R}}^{{}^{\prime}}(U,V)U$ is normal for any $U,V\in\Gamma(TM)$. ∎

5. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with $\hat{\bar{\nabla}}$

Lemma 5.1**.**

If $M$ is either invariant or anti-invarint submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$, then $\hat{\bar{R}}(X,Y)Z$ is tangent to $M$ and $\hat{\bar{R}}(X,Y)V$ is normal to $M$ for any $X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

Proof.

If $M$ is invariant then from (2.14) we say that $\hat{\bar{R}}(X,Y)Z$ is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.

If $M$ is anti-invariant then

[TABLE]

From (2.14) and (5.1) we have

[TABLE]

which is tangent.

If $M$ is invariant from (2.14) we have $\hat{\bar{R}}(X,Y)V$ is normal to $M$, and if $M$ is anti-invariant then $\hat{\bar{R}}(X,Y)V=0$ i.e. $\hat{\bar{R}}(X,Y)V$ is normal to $M$ for any $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎

Lemma 5.2**.**

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq(f_{1}-f_{3})^{2}$ on $M$ and $TM$ is invariant under the action of $\hat{\bar{R}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

Proof.

For $X,Y\in\Gamma(TM)$, we have from (2.14) that

[TABLE]

Now, we see that $\hat{\bar{R}}(X,Y)X$ should be tangent if $\{3f_{2}+(f_{1}-f_{3})^{2}\}g(Y,\phi X)\phi X$ is tangent. Since $3f_{2}\neq-(f_{1}-f_{3})^{2}$ then in similar way of proof of Lemma $3.2$ of [3] we may conclude that either $M$ is invariant or anti-invariant. This proves the Lemma. ∎

From Lemma $5.1$ and Lemma $5.2$, we can state the following:

Theorem 5.1**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq-(f_{1}-f_{3})^{2}$, then $M$ is either invariant or anti-invariant if and only if $TM$ is invariant under the action of $\hat{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.

Proposition 5.1**.**

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant under the action of $\hat{\bar{R}}(U,V)$ for any $U,V\in\Gamma(T^{\bot}M)$.

Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2.14), we get

[TABLE]

As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have

[TABLE]

for any $X\in\Gamma(TM)$. Using (5.5) in (5), we have

[TABLE]

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

Proposition 5.2**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$. If $3f_{2}\neq-(f_{1}-f_{3})^{2}$ on $M$ and $T^{\bot}M$ is invariant under the action of $\hat{\bar{R}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then $M$ is either invariant or anti-invariant.

Proof.

The proof is similar as the proof of Lemma $5.2$, just imposing that $\hat{\bar{R}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎

6. Submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$

Lemma 6.1**.**

If $M$ is either invariant or anti-invarint submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$, then $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ is tangent to $M$ and $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ is normal to $M$ for any $X,Y,Z\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$.

Proof.

If $M$ is invariant then from (2.16) we say that $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)Z$ is tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.

If $M$ is anti-invariant then

[TABLE]

From (2.16) and (6.1) we have

[TABLE]

which is tangent.

If $M$ is invariant from (2.16) we have $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ normal to $M$ and if $M$ is anti-invariant then $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V=0$ i.e. $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)V$ normal to $M$ for any $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$. This proves the Lemma. ∎

Lemma 6.2**.**

let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If $\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\}(p)\neq 0$ for each $p\in M$ and $TM$ is invariant under the action of $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)$, $X,Y\in\Gamma(TM)$, then $M$ is either invariant or anti-invariant.

Proof.

For $X,Y\in\Gamma(TM)$, we have from (2.16) that

[TABLE]

Now we see that $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)X$ should be tangent if $\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\}(p)g(Y,\phi X)\phi X$ is tangent. Since $\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\}(p)\neq 0$ then by similar way of proof of Lemma $3.2$ of [3] we can proved that either $M$ is invariant or anti-invariant. This proves the Lemma. ∎

From Lemma $6.1$ and Lemma $6.2$, we have

Theorem 6.1**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If $\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\}(p)\neq 0$, then $M$ is either invariant or anti-invariant if and only if $TM$ is invariant under the action of $\stackrel{{\scriptstyle*}}{{\bar{R}}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.

Proposition 6.1**.**

Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If $M$ is invariant, then $TM$ is invariant under the action of $\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)$ for any $U,V\in\Gamma(T^{\bot}M)$.

Proof.

Replacing $X,Y,Z$ by $U,V,X$ in (2.16), we get

[TABLE]

As $M$ is invariant, $U\in\Gamma(T^{\bot}M)$, we have

[TABLE]

for any $X\in\Gamma(TM)$. Using (6.5) in (6), we have

[TABLE]

which is tangent as $\phi X$ is tangent. This proves the proposition. ∎

Proposition 6.2**.**

Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. If $\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\}(p)\neq 0$ for each $p\in M$ and $T^{\bot}M$ is invariant under the action of $\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)$, $U,V\in\Gamma(T^{\bot}M)$, then $M$ is either invariant or anti-invariant.

Proof.

The proof is similar as the proof of Lemma $6.2$, just considering that $\stackrel{{\scriptstyle*}}{{\bar{R}}}(U,V)U$ is normal for any $U,V\in\Gamma(T^{\bot}M)$. ∎

Acknowledgement: The first author (P. Mandal) gratefully acknowledges to the CSIR(File No.:09/025(0221)/2017-EMR-I), Govt. of India for financial assistance. The Third author (S. K. Hui) are thankful to University of Burdwan for providing administrative and technical support.