A note on submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to certain connections

TL;DR

This paper investigates the properties of almost semi-invariant submanifolds within generalized Sasakian-space-forms under various specialized connections, extending previous results in differential geometry.

Contribution

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

Findings

Results on submanifold properties under semisymmetric metric connection

Findings on submanifold behavior with Schouten-van Kampen connection

Extensions of previous work on semi-invariant submanifolds

Abstract

The present paper deals with some results of almsot semi-invariant 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

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

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

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

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

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

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

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

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

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

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

$$\widetilde{\stackrel{ˉ}{R}}(X,Y)Z$$

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

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

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

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

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

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

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

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

$$\displaystyle+g\big{(}h(X,Z),h(Y,W)\big{)},$$

$$\widetilde{\stackrel{ˉ}{R}}(X,Y,Z,W)$$

$$\displaystyle+g\big{(}\tilde{h}(X,Z),\tilde{h}(Y,W)\big{)},$$

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

$$\displaystyle+g\big{(}h^{{}^{\prime}}(X,Z),h^{{}^{\prime}}(Y,W)\big{)},$$

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

$$\displaystyle+g\big{(}\hat{h}(X,Z),\hat{h}(Y,W)\big{)},$$

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

$$\displaystyle+g\big{(}\stackrel{{\scriptstyle*}}{{h}}(X,Z),\stackrel{{\scriptstyle*}}{{h}}(Y,W)\big{)},$$

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

$$T_{p}M=D_p^1\oplus D_p^0\oplus D_p^{{\lambda }_1}\oplus \cdots \oplus D_p^{{\lambda }_l}\oplus span\{{\xi }_p\},$$

$$g(TX,TY)=\sum _{\lambda }{\lambda }^{2}g(U^{\lambda }X,U^{\lambda }Y).$$

$$\tilde{S}(X,Y)$$

Full text

• corresponding author

1. Introduction

The notion of generalized Sasakian-space-form was introduced by Alegre et al. [2]. An almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$ whose curvature tensor $\bar{R}$ satisfies

[TABLE]

for all vector fields $X$, $Y$, $Z$ on $\bar{M}$ and $f_{1},f_{2},f_{3}$ are certain smooth functions on $\bar{M}$ is said to be generalized Sasakian-space-form [2]. 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, ([9], [10], [11], [12], [13], [14], [16], [17] and [22]). It reduces to Sasakian-space-form if $f_{1}=\frac{c+3}{4}$, $f_{2}=f_{3}=\frac{c-1}{4}$ [2]. We denote Sasakian-space-form of dimension $(2n+1)$ by $M^{2n+1}(c)$.

After introduced the semisymmetric linear connection by Friedman and Schouten [6], Hayden [8] gave the idea of metric connection with torsion on a Riemannian manifold. Later, Yano [29] and many others (see, [20], [21], [23] 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 ([19], [27]). In $2006$, Bejancu [5] studied Schouten-van Kampen connection on foliated manifolds. Recently Olszak [18] studied Schouten-van Kampen connection on almost(para) contact metric structure.

The Tanaka-Webster connection ([24], [28]) is the canonical affine connection defined on a non-degenerate pseudo-Hermitian CR-manifold. Tanno [25] defined the Tanaka-Webster connection for contact metric manifolds.

In [3], Alegre and Carriazo studied almost semi-invariant submanifolds of generalized Sasakian-space-form with respect to Levi-Civita connection. In this paper, we have studied the results of [3] with respect to certain connections, namely semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection, Tanaka-Webster connection.

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 $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$.

The semisymmetric metric connection $\tilde{\bar{\nabla}}$ 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 $\tilde{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$ 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 [18]

[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 [7]

[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]

Let $M$ be a $(m+1)$-dimensional 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 $g(h(X,Y),V)=g(A_{V}X,Y)$ [30].

Moreover, if $h(X,Y)=0$ for all $X,Y\in\Gamma(TM)$ then $M$ is said to be totally geodesic and if $H=0$ then $M$ is minimal in $\bar{M}$, where $H$ is the mean curvature tensor.

From (2.15), we have the Gauss equations as

[TABLE]

where $R$ is the curvature tensor of $M$. Let $\tilde{\nabla}$, $\nabla^{{}^{\prime}}$, $\hat{\nabla}$ and $\stackrel{{\scriptstyle*}}{{\nabla}}$ are the induced connection of $M$ from the connection $\tilde{\bar{\nabla}}$, $\bar{\nabla}^{{}^{\prime}}$, $\hat{\bar{\nabla}}$ and $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ respectively. Then Gauss equation with respect to $\tilde{\bar{\nabla}}$, $\bar{\nabla}^{{}^{\prime}}$, $\hat{\bar{\nabla}}$ and $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\tilde{h}$, $h^{{}^{\prime}}$, $\hat{h}$, $\stackrel{{\scriptstyle*}}{{h}}$ are the second fundamental forms with respect to $\tilde{\nabla}$, ${\nabla}^{{}^{\prime}}$, $\hat{\nabla}$ and $\stackrel{{\scriptstyle*}}{{\nabla}}$ respectively. Also $\tilde{H}$, $H^{{}^{\prime}}$, $\hat{H}$, $\stackrel{{\scriptstyle*}}{{H}}$ be the mean curvature of $M$ with respect to $\tilde{\nabla}$, ${\nabla}^{{}^{\prime}}$, $\hat{\nabla}$ and $\stackrel{{\scriptstyle*}}{{\nabla}}$ respectively.

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

[TABLE]

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

Definition 2.1**.**

([3], [26]) A submanifold $M$ of an almost contact metric manifold $\bar{M}$, $\xi$ tangent to $M$, is said to be an almost semi-invariant submanifold if their exist $l$ functions $\lambda_{1},\cdots,\lambda_{l}$, defined on $M$ with values in $(0,1)$, such that

1. (i)

$-\lambda_{1}^{2}(p),\cdots,-\lambda_{l}^{2}(p)$* are distinct eigenvalues of $T^{2}|_{D}$ at $p\in M$, with*

[TABLE]

where $D^{\lambda}_{p}$, $\lambda\in\{1,0,\lambda_{1}(p),\cdots,\lambda_{l}(p)\}$, denotes the eigenspace associated to the eigenvalue $-\lambda^{2}$. 2. (ii)

the dimension of $D^{1}_{p},D^{0}_{p},D^{\lambda_{1}}_{p},\cdots,D^{\lambda_{l}}_{p}$ are independent of $p\in M$.

Let the orthogonal projection from $TM$ on $D^{\lambda}$ be $U^{\lambda}$. Then we have

[TABLE]

Let us consider $\{E_{1},\cdots,E_{m},E_{m+1}=\xi\}$ and $\{F_{1},\cdots,F_{2n-m}\}$ local orthonormal basis of $TM$ and $T^{\bot}M$ respectively, and denote $A_{F_{k}}=A_{k}$.

3. Ricci tensor on $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with $\tilde{\bar{\nabla}}$

Lemma 3.1**.**

The Ricci tensor $\tilde{S}$ of submanifold $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2) and (2.17) we have the above Lemma. ∎

Lemma 3.2**.**

The Ricci tensor $\tilde{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2.22) and (3.1) we have the above Lemma. ∎

Corollary 3.1**.**

For an almost semi-invariant submanifolds $M$ of Sasakian-space-form $\bar{M}^{2n+1}(c)$ with respect to $\tilde{\bar{\nabla}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Putting $f_{1}=\frac{c+3}{4},\ f_{2}=f_{3}=\frac{c-1}{4}$ in (3.2) we obtain the result. ∎

Lemma 3.3**.**

The scalar curvature $\tilde{\tau}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$ is

[TABLE]

Proof.

Let us consider an orthonormal frame $\{E_{1},\cdots,E_{n(\lambda)}\}$ in $D^{\lambda}$. Then we have

[TABLE]

Using (2), (2.17) in (3.5) we get (3.3). ∎

Theorem 3.1**.**

If $M$ is an almost semi-invariant minimal submanifold of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$, then the following relation holds:

• (i)

$\tilde{S}(X,X)\leq\sum\limits_{\lambda}(mf_{1}+3f_{2}\lambda^{2}-f_{3}+1)g(U^{\lambda}X,U^{\lambda}X)$* $+m(f_{1}-f_{3}+1)\eta(X)\eta(X)+(f_{1}-f_{3})(m-1)g(TX,X)$,*

• (ii)

$\tilde{\tau}\leq f_{1}+\frac{1}{m(m+1)}\{3f_{2}\sum\limits_{\lambda}n(\lambda)\lambda^{2}-2m(f_{3}-1)\}$.

Proof.

Since $M$ is minimal submanifold with respect to $\tilde{\bar{\nabla}}$, then we have

[TABLE]

Using (3) in (3.2) we have

[TABLE]

which proves (i).

The second part (ii) comes directly from Lemma $3.3$. ∎

Remark 3.1**.**

The equality of (i) and (ii) in Theorem $3.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$.

Proof.

If $M$ is totally geodesic submanifold with respect to $\tilde{\bar{\nabla}}$, then $M$ is minimal submanifold with respect to $\tilde{\bar{\nabla}}$. Then by virtue of Lemma $3.2$ we have the equality case (i) and by virtue of Lemma $3.3$ we have equality case of (ii). ∎

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

Lemma 4.1**.**

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

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2) and (2.18) we have the above Lemma. ∎

Lemma 4.2**.**

The Ricci tensor $S^{{}^{\prime}}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\bar{\nabla}^{{}^{\prime}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2.22) and (4.1) we have the above Lemma. ∎

Corollary 4.1**.**

The Ricci tensor $S^{{}^{\prime}}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(c)$ with respect to $\bar{\nabla}^{{}^{\prime}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Putting $f_{1}=\frac{c+3}{4},\ f_{2}=f_{3}=\frac{c-1}{4}$ in (4.2) we obtain the result. ∎

Lemma 4.3**.**

The scalar curvature $\tau^{{}^{\prime}}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\bar{\nabla}^{{}^{\prime}}$ is

[TABLE]

Proof.

It is known that

[TABLE]

Using (2), (2.18) in (4.5) we get (4.3). ∎

Theorem 4.1**.**

If $M$ is an almost semi-invariant minimal submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\bar{\nabla}^{{}^{\prime}}$, then the following condition holds:

• (i)

$S^{{}^{\prime}}(X,X)\leq\sum\limits_{\lambda}(mf_{1}+3f_{2}\lambda^{2}-f_{3}+1)g(U^{\lambda}X,U^{\lambda}X)+m(f_{1}-f_{3})\eta(X)\eta(X)+(f_{1}-f_{3})g(TX,X)$,

• (ii)

$\tau^{{}^{\prime}}\leq f_{1}+\frac{1}{m(m+1)}\{3f_{2}\sum\limits_{\lambda}n(\lambda)\lambda^{2}-2mf_{3}+m\}$.

Proof.

Since $M$ is minimal submanifold with respect to $\bar{\nabla}^{{}^{\prime}}$, then we have

[TABLE]

Using (4.2) and (4) we have

[TABLE]

This proves (i).

The second part (ii) is comes directly from Lemma $4.3$. ∎

Remark 4.1**.**

The equality of (i) and (ii) in Theorem $4.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\bar{\nabla}^{{}^{\prime}}$.

Proof.

If $M$ is totally geodesic submanifold with respect to $\bar{\nabla}^{{}^{\prime}}$, then $M$ is minimal submanifold with respect to $\bar{\nabla}^{{}^{\prime}}$. Then by virtue of Lemma $4.2$ we have the equality case of (i) and by virtue of Lemma $4.3$ we have the equality case (ii). ∎

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

Lemma 5.1**.**

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

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2) and (2.19) we have the above Lemma. ∎

Lemma 5.2**.**

The Ricci tensor $\hat{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2.22) and (5.1) we have the above Lemma. ∎

Corollary 5.1**.**

The Ricci tensor $\hat{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(c)$ with respect to $\hat{\bar{\nabla}}$ is

[TABLE]

Proof.

Putting $f_{1}=\frac{c+3}{4},\ f_{2}=f_{3}=\frac{c-1}{4}$ in (5.2) we obtain the result. ∎

Lemma 5.3**.**

The scalar curvature $\hat{\tau}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is

[TABLE]

Proof.

It is known that

[TABLE]

Using (2), (2.19) in (5.5) we get (5.3). ∎

Theorem 5.1**.**

If $M$ is an almost semi-invariant minimal submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$, then the following condition holds:

• (i)

\hat{S}\leq\sum\limits_{\lambda}\big{[}mf_{1}+3f_{2}\lambda^{2}-f_{3}+(f_{1}-f_{3})^{2}(\lambda^{2}-1)\big{]}g(U^{\lambda}X,U^{\lambda}Y)+m\big{[}f_{1}-\{f_{3}+(f_{1}-f_{3})^{2}\}\big{]}\eta(X)\eta(Y),

• (ii)

\hat{\tau}\leq f_{1}+\frac{1}{m(m+1)}\big{[}\{3f_{2}+(f_{1}-f_{3})^{2}\}\sum\limits_{\lambda}n(\lambda)\lambda^{2}-2m\{f_{3}+(f_{1}-f_{3})^{2}\}\big{]}.

Proof.

Since $M$ is minimal submanifold with respect to with respect to $\hat{\bar{\nabla}}$, then we have

[TABLE]

Using (5.2) and (5) we have

[TABLE]

This complete the proves (i).

The second part (ii) is comes directly from Lemma $5.3$. ∎

Remark 5.1**.**

The equality of (i) and (ii) in Theorem $5.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$.

Proof.

If $M$ is totally geodesic submanifold with respect to $\hat{\bar{\nabla}}$, then $M$ is minimal submanifold with respect to $\hat{\bar{\nabla}}$. Then by virtue of Lemma $5.2$ we have the equality case (i) and by virtue of Lemma $5.3$ we have the equality case of (ii). ∎

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

Lemma 6.1**.**

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

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2) and (2.20) we have the above Lemma. ∎

Lemma 6.2**.**

The Ricci tensor $\stackrel{{\scriptstyle*}}{{S}}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Using (2.22) and (6.1) we have the above Lemma. ∎

Corollary 6.1**.**

The Ricci tensor $\stackrel{{\scriptstyle*}}{{S}}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(c)$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ is

[TABLE]

for any vector fields $X,Y$ on $M$.

Proof.

Putting $f_{1}=\frac{c+3}{4},\ f_{2}=f_{3}=\frac{c-1}{4}$ in (6.2) we obtain the result. ∎

Lemma 6.3**.**

The scalar curvature $\stackrel{{\scriptstyle*}}{{\tau}}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$ is

[TABLE]

Proof.

It is known that

[TABLE]

Using (2), (2.20) in (6.5) we get (6.3). ∎

Theorem 6.1**.**

Let $M$ be an almost semi-invariant minimal submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$, then the following condition holds:

• (i)

\stackrel{{\scriptstyle*}}{{S}}(X,Y)\leq\sum\limits_{\lambda}\big{[}mf_{1}+\{3f_{2}+2(f_{1}-f_{3})\}\lambda^{2}-f_{3}+(f_{1}-f_{3})^{2}(\lambda^{2}-1)\big{]}g(U^{\lambda}X,U^{\lambda}X)+m\big{[}f_{1}-\{f_{3}+(f_{1}-f_{3})^{2}\}\big{]}\eta(X)\eta(X),

• (ii)

\stackrel{{\scriptstyle*}}{{\tau}}\leq f_{1}+\frac{1}{m(m+1)}\big{[}\{3f_{2}+2(f_{1}-f_{3})+(f_{1}-f_{3})^{2}\}\sum_{\lambda}n(\lambda)\lambda^{2}-2m\{f_{3}+(f_{1}-f_{3})^{2}\}\big{]}.

Proof.

Since $M$ is minimal submanifold with respect to with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$, then we have

[TABLE]

Using (6.2) and (6) we have

[TABLE]

which proves (i).

The proof of (ii) comes directly from Lemma $6.3$. ∎

Remark 6.1**.**

The equality of (i) and (ii) in Theorem $6.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$.

Proof.

If $M$ is totally geodesic submanifold with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$, then $M$ is minimal submanifold with respect to $\stackrel{{\scriptstyle*}}{{\bar{\nabla}}}$. Then by virtue of Lemma $6.2$ we have the equality case (i) and by virtue of Lemma $6.3$ we have the equality case of (ii). ∎

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 second author (S. K. Hui) are thankful to University of Burdwan for providing administrative and technical support.