Gauss and Ricci Equations in Contact Manifolds with a Quarter-Symmetric Metric Connection

Abstract

In the present paper, we study the extrinsic and intrinsic geometry of submanifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection. We deduce Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and show some applications of these equations. Finally, we give an example verifying the results.

1 Introduction

The importance of the Gauss, Codazzi and Ricci equations in differential geometry is that if the ambient space has a constant sectional curvature, they play an analogous role to that of the compatibility equation in the local theory of surfaces. For a submanifold $M$ of a Riemannian manifold $\bar{M}$, if the Riemannian curvature tensors are denoted by $R$ and $\bar{R}$, respectively, then the usual Gauss, Codazzi and Ricci equations are given by the following:

$$\begin{aligned} g\big (\bar{R}(X,Y)Z,W\big )&= g\big (R(X,Y)Z,W\big )-g\big (h(X,W),h(Y,Z)\big ) \nonumber \\&+g\big (h(Y,W),h(X,Z)\big ), \end{aligned}$$

(1.1)

$$\begin{aligned} \big (\bar{R}(X,Y,Z)\big )^{\perp }&= (\nabla _{X}h)(Y,Z)-(\nabla _{Y}h)(X,Z), \end{aligned}$$

(1.2)

$$\begin{aligned} g\big (\bar{R}(X,Y)U,V\big )&= g\big (R^{\perp }(X,Y)U,V\big )+g\big ([A_{V},A_{U}]X,Y\big ), \end{aligned}$$

(1.3)

for $X,Y,Z,W$ tangent to $M$ and $U,V$ normal to $M$, where $h$ is the second fundamental form, $A$ is the associated shape operator of the immersion, and $R^{\perp }$ is the curvature tensor of the normal bundle. For an isometric immersion $i:M\rightarrow \bar{M}$ of Riemannian manifolds, the Gauss equation shows that the curvature tensor of $\bar{M}$, when evaluated on vector fields tangent to $M$, differs from the curvature tensor of $M$ by a tensor involving only the second fundamental form of the immersion. Gauss–Codazzi–Ricci equations are very important instruments for describing a submanifold in a Riemannian space. By nature, these equations appear in the Cauchy problem of general relativity [20].

On the other hand, in 1975, Golab [13] introduced the notion of quarter-symmetric connection on a differentiable manifold. A linear connection $\nabla$ is said to be quarter-symmetric if its torsion tensor $T$ defined by $T(X,Y)=\nabla _{X}Y-\nabla _{Y}X-[X,Y]$, is of the form:

$$\begin{aligned} T(X,Y)=u(Y)\psi X-u(X)\psi Y, \end{aligned}$$

(1.4)

where $u$ is a $1-$form and $\psi$ is a $(1,1)-$tensor field. When $T$ vanishes, the connection $\nabla$ is called symmetric; otherwise, it is called non-symmetric. $\nabla$ is called a metric connection if there is a Riemannian metric $g$ such that $\nabla g=0$; otherwise, it is called non-metric. It is well known that a linear connection is both symmetric and metric if and only if it is the Riemannian (or, the Levi-Civita) connection. If in (1.4), $\psi$ is an identity function, then it reduces to semi-symmetric metric connection. Hence, quarter-symmetric connection is a generalization of semi-symmetric connection. In the present paper, we deduce the Gauss, Codazzi and Ricci equations for submanifolds of an almost contact metric manifold admitting a quarter-symmetric metric connection.

In [8], De and Mondal proved the existence and uniqueness of quarter-symmetric metric connection in Riemannian manifolds. Many authors studied other geometric properties of almost Hermitian and almost contact manifolds with quarter-symmetric and semi-symmetric connections ([1, 7, 14, 16, 17, 19]). Ozgur [16] proved several results including the equations of Gauss Codazzi and Ricci for submanifolds of a Riemannian manifold admitting a particular type of semi-symmetric non-metric connection. Later on, hypersurfaces and submanifolds of different ambient manifolds admitting quarter-symmetric metric connection have been studied by several authors ([9, 11, 15]). In this paper, we generalize all the results obtained in these previous studies, by considering submanifolds of any codimension of an almost contact metric manifold admitting a quarter-symmetric metric connection.

The present paper has been organized as follows:After preliminaries, in sect. 3, we consider submanifolds of an almost contact metric manifold endowed with a quarter-symmetric metric connection, and we show that the connection induced on the submanifold is also a quarter-symmetric metric connection. Furthermore, we prove that the mean curvature with respect to both the connections coincides, and applying this result, we obtain a necessary condition of a submanifold to be invariant. In sect. 4, we deduce the Gauss, Codazzi and Ricci equations corresponding to the quarter-symmetric metric connection and obtain some results applying these equations. Finally, in sect. 5, we provide an example verifying the obtained results.

2 Preliminaries

Let $\bar{M}$ be an $(n+p)-$dimensional (where $n+p$ is odd) differentiable manifold endowed with an almost contact metric structure $(\phi ,\xi ,\eta ,g)$, where $\phi$, $\xi$, $\eta$ are tensor fields on $\bar{M}$ of types $(1,1)$, $(1,0)$, $(0,1)$, respectively, and $g$ is a compatible metric with the almost contact structure, such that [2, 3, 5, 21],

$$\begin{aligned} \phi ^{2}&= -I+\eta \otimes \xi ,\,\phi \xi =0,\,\eta (\xi )=1,\,\eta \circ \phi =0, \end{aligned}$$

(2.1)

$$\begin{aligned} g(\phi X,\phi Y)&= g(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

(2.2)

$$\begin{aligned} g(\phi X,Y)+g(X,\phi Y)&= 0, \end{aligned}$$

(2.3)

for all vector fields $X,Y\in T\bar{M},$ where $T\bar{M}$ is the Lie algebra of vector fields of the manifold $\bar{M}$. The fundamental $2-$form $\Phi$ is defined by $\Phi (X,Y)=g(X,\phi Y)$. If $[\phi ,\phi ]+d\eta \otimes \xi =0,$ where $[\phi ,\phi ](X,Y)=\phi ^{2}[X,Y]+[\phi X,\phi Y]-\phi [\phi X,Y]-\phi [X,\phi Y],$ then the almost contact structure is said to be normal [10]. If $\Phi =d\eta ,$ the almost contact structure becomes a contact structure. A normal contact metric manifold is called Sasakian. On a Sasakian manifold, we have the following [2, 12]:

$$\begin{aligned} \big (\bar{\nabla }_{X}\phi \big )Y&= g(X,Y)\xi -\eta (Y)X,\end{aligned}$$

(2.4)

$$\begin{aligned} \bar{R}(X,Y)\xi&= \eta (Y)X-\eta (X)Y. \end{aligned}$$

(2.5)

An almost contact metric structure $(\phi ,\xi ,\eta ,g)$ is cosymplectic if and only if $\phi$ is parallel. In a cosymplectic manifold, we have $\big (\bar{\nabla }_{X}\eta \big )Y=0.$

A Riemannian manifold of dimension >2 is said to be Einstein if its Ricci tensor S satisfies $S(X,Y)=\mu g(X,Y)$, where $\mu$ is a constant [6].

Let $M$ be a submanifold of an almost contact metric manifold $\bar{M}$ with a positive definite metric $g.$ Let the induced metric on $M$ also be denoted by $g.$ The usual Gauss and Weingarten formulae are given, respectively, by [4, 18]

$$\begin{aligned} \bar{\nabla }_{X}Y&= \nabla _{X}Y+h(X,Y),\,\,\,X,Y\in TM \end{aligned}$$

(2.6)

$$\begin{aligned} \bar{\nabla }_{X}N&= -A_{N}X+\nabla _{X}^{\bot }N,\,\,\,N\in T^{\bot }M \end{aligned}$$

(2.7)

where $\nabla$ is the induced Riemannian connection on $M,$ $h$ is the second fundamental form of the immersion, $A$ is the shape operator, and $\nabla ^{\bot }$ is the normal connection on $T^{\bot }M$, the normal bundle of $M.$ From (2.6) and (2.7), one gets

$$\begin{aligned} g(h(X,Y),N)=g(A_{N}X,Y). \end{aligned}$$

(2.8)

The submanifold $M$ of an almost contact manifold $\bar{M}$ is called invariant (resp. anti-invariant) if for each point $x\in M,$ $\phi T_{x}M\subset T_{x}M$ (resp. $\phi T_{x}M\subset T_{x}^{\bot }M.$ The submanifold is called totally umbilical if $h(X,Y)=g(X,Y)H$, for all $X,Y\in TM$, where $H$ is the mean curvature vector of the submanifold, defined by $H=\frac{1}{n}\sum \{h(e_{i},e_{i})\},\{e_{i}\},i=1,2,...,n$ being an orthonomal basis of $TM$ and $n$ the dimension of $M$. The submanifold is called totally geodesic if $h(X,Y)=0$ for all $X,Y\in TM$. Let the codimension of $M$ be $p$, and let $\{N_{\alpha }\},$ $\alpha =1,2,...,p$ be an orthonormal basis of $T^{\bot }M$.

3 Basic Results

On a submanifold $M$ of an almost contact metric manifold $\bar{M}$ with the quarter-symmetric metric connection $\nabla ^{*}$, we obtain the following results:

Theorem 3.1

The connection induced on a submanifold of an almost contact metric manifold with a quarter-symmetric metric connection is also a quarter-symmetric metric connection.

Proof

We define the quarter-symmetric metric connection $\nabla ^{*}$ on $\bar{M}$ by

$$\begin{aligned} \nabla ^{*}_{X}Y=\bar{\nabla }_{X}Y-\eta (X)\phi Y. \end{aligned}$$

(3.1)

If $\nabla ^{^{\prime }}$ is the induced connection on $M$ from the connection $\nabla ^{*}$, then we have

$$\begin{aligned} \nabla _{X}^{*}Y=\nabla _{X}^{^{\prime }}Y+m(X,Y), \end{aligned}$$

(3.2)

where $m$ is a tensor field of type $(1,2)$ in $T^{\perp }M$, the normal part of $M$. We term $m(X,Y)$ the second fundamental form with respect to the quarter-symmetric connection.

For $X\in TM$ and $N\in T^{\bot }M$, we put

$$\begin{aligned} \phi X&= PX+QX, \quad PX\in TM, \quad QX\in T^{\bot }M, \end{aligned}$$

(3.3)

$$\begin{aligned} \phi N&= tN+qN,\quad tN\in TM,\quad qN\in T^{\bot }M. \end{aligned}$$

(3.4)

Using (3.3), from (3.1) and (3.2), we have

$$\begin{aligned} \nabla _{X}^{^{\prime }} Y+m(X,Y)= \nabla _{X}Y +h(X,Y)-\eta (X)PY-\eta (X)QY. \end{aligned}$$

(3.5)

Now equating tangential and normal parts, we have

$$\begin{aligned} \nabla _{X}^{^{\prime }}Y=\nabla _{X}Y -\eta (X)PY, \end{aligned}$$

(3.6)

and

$$\begin{aligned} m(X,Y)=h(X,Y)-\eta (X)QY. \end{aligned}$$

(3.7)

From (3.6), the torsion tensor with respect to $\nabla ^{^{\prime }}$ is given by

$$\begin{aligned} T^{\prime }(X,Y)=\eta (Y)P{X}-\eta (X)P{Y}. \end{aligned}$$

Also using (3.2), we have

$$\begin{aligned} \big (\nabla _{X}^{^{\prime }}g\big )(Y,Z)=\big (\nabla _{X}^{*}g\big )(Y,Z). \end{aligned}$$

(3.8)

Hence, the result.$\square$

From (3.6), it follows that if the submanifold is anti-invariant, that is, $PX=0$, then we have the following:

Theorem 3.2

On an anti-invariant submanifold of an almost contact metric manifold with a quarter-symmetric metric connection, the induced quarter-symmetric connection and the induced Riemannian connection are equivalent.

So we concentrate mostly on invariant submanifolds. Equation (3.2) is the Gauss formula for the quarter-symmetric metric connection. Also, from (3.1), we have

$$\begin{aligned} \nabla _{X}^{*}N&= \bar{\nabla }_{X}N-\eta (X)\phi N \nonumber \\&= -A_{N}X-\eta (X)tN+\nabla _{X}^{\bot }N-\eta (X)qN \nonumber \\&= D_{X}N-A_{N}^{^{^{\prime }}}X, \end{aligned}$$

(3.9)

where $D_{X}N=\nabla _{X}^{\bot }N-\eta (X)qN$ is the normal connection, and $A_{N}^{^{\prime }}X=A_{N}X+\eta (X)tN$ is the shape operator corresponding to the quarter-symmetric metric connection. By simple calculations, we obtain

$$\begin{aligned} g(m(X,Y),N)=g(A_{N}^{\prime }X,Y). \end{aligned}$$

(3.10)

Equation (3.9) is the Weingarten formula with respect to the quarter-symmmetric metric connection.

Remark 3.1

Unlike the second fundamental form corresponding to the Levi-Civita connection, $m$ is neither symmetric nor skew-symmetric, in general, which is evident from (3.7). Thus, the shape operator $A^{\prime }$corresponding to the quarter-symmetric connection is also not symmetric. However, for invariant submanifolds both of them are symmetric.

We define the covariant derivative of $m$ and $\eta$ with respect to the quarter-symmetric metric connection as follows:

$$\begin{aligned} \big (\nabla _{X}^{*}m\big )(Y,Z)&= D_{X}\big (m(Y,Z)\big )-m\big ( \nabla _{X}^{^{\prime }}Y,Z\big )-m\big (Y,\nabla _{X}^{^{\prime }}Z\big ), \end{aligned}$$

(3.11)

$$\begin{aligned} \big (\nabla _{X}^{*}\eta \big )Y&= X\big (\eta (Y)\big )-\eta \big (\nabla _{X}^{\prime }Y\big ). \end{aligned}$$

(3.12)

Equation (3.11) may be called the van der Waerden–Bortolotti connection corresponding to the quarter-symmetric metric connection.

Now, we prove the following:

Theorem 3.3

The mean curvature of the submanifold $M$ with respect to the Riemannian connection coincides with that of $M$ with respect to the quarter-symmetric metric connection.

Proof

Let $\{e_{1},e_{2},...,e_{n}\}$ be an orthonormal basis of $TM$. We consider two cases:

Case I: $\xi \in TM$ and let $e_{n}=\xi$. Then from (3.7), we obtain

$$\begin{aligned} m(e_{i},e_{i})=h(e_{i},e_{i})-\eta (e_{i})Q(e_{i}). \end{aligned}$$

(3.13)

Since $\eta (e_{i})=0$, for $i=1,2,...,n-1$, and $\phi (e_{n})=0$, summing up for $i=1,2,...,n$ and dividing by $n$, we obtain the required result.

Case II: $\xi \notin TM$, then again from (3.7), we obtain

$$\begin{aligned} m(e_{i},e_{i})=h(e_{i},e_{i})-\eta (e_{i})Q(e_{i}) \end{aligned}$$

(3.14)

for each $i=1,2,...,n$. From (3.14), we obtain $m(e_{i},e_{i})=h(e_{i},e_{i})$, since $\eta (e_{i})=0$, for all the $i=1,2,...,n$ . Summing up for $i=1,2,...,n$ and dividing by $n$, we obtain the required result.$\square$

The following corrolaries are the direct consequence of the above theorem.

Corollary 3.1

Any submanifold of an almost contact manifold endowed with a quarter-symmetric metric connection is minimal with respect to the quarter-symmetric metric connection if and only if it is minimal with respect to the Riemannian connection.

Corollary 3.2

If a submanifold $M$ of an almost contact manifold endowed with a quarter-symmetric metric connection is tangent to $\xi$ and is totally umbilical with respect to both the connections, then $M$ is invariant. Conversely, if $M$ is invariant, then $M$ is totally umbilical with respect to quarter-symmetric connection if and only if $M$ is totally umbilical with respect to the Riemannian connection.

Proof

From (3.7), for all $X,Y\in TM$, we have,

$$\begin{aligned} \eta (X)QY=m(X,Y)-h(X,Y). \end{aligned}$$

(3.15)

If $M$ is totally umbilical with respect to both quarter-symmetric connection and Riemannian connection, then from Theorem 3.3, we have,

$$\begin{aligned} m(X,Y)=g(X,Y)H=h(X,Y). \end{aligned}$$

Thus, from (3.15), we get for all $X,Y\in TM$,

$$\begin{aligned} \eta (X)QY=0, \end{aligned}$$

(3.16)

for any $X,Y\in TM$. Putting $X=\xi$ in (3.16), we obtain, $QY=0$, for all $Y\in TM$, which implies that $M$ is an invariant submanifold.

The converse part follows directly from (3.7).$\square$

Remark 3.2

In this connection, we should note that, if any submanifold of a contact metric manifold is normal to $\xi$, then by the well-known result of Yano and Kon [22], the submanifold is always anti-invariant.

Theorem 3.4

The covariant derivative of the fundamental $2-$form $\Phi$ with respect to the quarter-symmetric connection is equal to the covariant derivative of $\Phi$ with respect to the Riemannian connection.

Proof

We have $\Phi (X,Y)=g(X,\phi Y)$.

Therefore,

$$\begin{aligned} \big (\nabla _{X}^{*}\Phi \big )(Y,Z)&= X\Phi (Y,Z)-\Phi \big (\nabla _{X}^{*}Y,Z\big )-\Phi \big (Y,\nabla _{X}^{*}Z\big ) \nonumber \\&= X\Phi (Y,Z)-\Phi \big (\bar{\nabla }_{X}Y,Z\big )+\eta (X)\Phi (Y,\phi Z) \nonumber \\&-\,\Phi \big (Y,\bar{\nabla }_{X}Z\big )+\eta (X)\Phi (Y,\phi Z) \nonumber \\&= \big (\bar{\nabla }_{X}\Phi \big )(Y,Z),\,\,\text {since,} \,\ \Phi (Y,\phi Z)=-\Phi (Y,\phi Z). \end{aligned}$$

(3.17)

Hence, the result.$\square$

4 The Gauss, Codazzi-Mainardi and Ricci Equations

In this section, we find the relations between the curvature tensors corresponding to the Levi-civita connection and the quarter symmetric metric connection. We denote the Riemannian curvature tensors corresponding to the Levi-Civita connection and the quarter-symmetric connection by $\bar{R}$ and $R^{*}$, respectively, and that corresponding to the induced connections $\nabla$ and $\nabla ^{^{\prime }}$ by $R$ and $R^{^{\prime }}$, respectively.

We have,

$$\begin{aligned} \nabla _{X}^{*}\nabla _{Y}^{*}Z&= \bar{ \nabla }_{X}\bar{\nabla }_{Y}Z-\eta (X)\phi \big (\bar{\nabla }_{Y}Z\big )-\eta (Y)\bar{ \nabla }_{X}\phi Z \nonumber \\&+\,\eta (X)\eta (Y)\phi ^{2}Z-X(\eta (Y))\phi Z,\end{aligned}$$

(4.1)

$$\begin{aligned} \nabla _{Y}^{*}\nabla _{X}^{*}Z&= \bar{ \nabla }_{Y}\bar{\nabla }_{X}Z-\eta (Y)\phi \big (\bar{\nabla }_{X}Z)-\eta (X)\bar{ \nabla }_{Y}\phi Z \nonumber \\&+\,\eta (X)\eta (Y)\phi ^{2}Z-Y\big (\eta (X)\big )\phi Z, \end{aligned}$$

(4.2)

and

$$\begin{aligned} \nabla _{[X,Y]}^{*}Z=\bar{\nabla }_{[X,Y]}Z-\eta \big ([X,Y]\big )\phi Z. \end{aligned}$$

(4.3)

Therefore, we have

$$\begin{aligned} R^{*}(X,Y)Z&= \bar{R}(X,Y)Z+\eta (X)\big (\bar{ \nabla }_{Y}\phi \big )Z-\eta (Y)\big (\bar{\nabla }_{X}\phi \big )Z \nonumber \\&-\,\big [(\bar{\nabla }_{X}\eta )Y-\big (\bar{\nabla }_{Y}\eta \big )X\big ]\phi Z. \end{aligned}$$

(4.4)

Hence, we derive

$$\begin{aligned} R^{*}(X,Y,Z,W)&= g(R^{*}(X,Y)Z,W) \nonumber \\&= g\big (\bar{R}(X,Y)Z,W\big )+\eta (X)g\big ((\bar{\nabla } _{Y}\phi )Z,W\big )-\,\eta (Y)g\big ((\bar{\nabla }_{X}\phi )Z,W\big ) \nonumber \\&-\,\big [\big (\bar{\nabla }_{X}\eta \big )Y-\big (\bar{\nabla }_{Y}\eta \big )X\big ]g(\phi Z,W) \nonumber \\&= \bar{R}(X,Y,Z,W)+\eta (X)g\big ((\bar{\nabla } _{Y}\phi )Z,W\big )-\,\eta (Y)g\big ((\bar{\nabla }_{X}\phi )Z,W\big ) \nonumber \\&-\,\big [\big (\bar{\nabla }_{X}\eta \big )Y-\big (\bar{\nabla }_{Y}\eta \big )X\big ]g(\phi Z,W). \end{aligned}$$

(4.5)

From (4.4) and (4.5), we can conclude the following:

Remark 4.1

• (i) $\ R^{*}(X,Y,Z,W)\ne R^{*}(Z,W,X,Y)$

• (ii) $R^{*}(X,Y,Z,W)\ne -R^{*}(X,Y,W,Z)$

• (iii) The first Bianchi identity with respect to the quarter-symmetric connection is given by

  $$\begin{aligned} R^{*}(X,Y)Z+R^{*}(Y,Z)X+R^{*}(Z,X)Y=k(X,Y)Z+k(Y,Z)X+k(Z,X)Y, \end{aligned}$$

where $k(X,Y)Z$ is a (1,3)-tensor defined by

$k(X,Y)Z=\eta (Z)\big [\big (\bar{\nabla }_{X}\phi \big )Y-\big (\bar{\nabla }_{Y}\phi \big )X\big ]-\big [\big (\bar{ \nabla }_{X}\eta \big )Y-\big (\bar{\nabla }_{Y}\eta \big )X\big ]\phi Z$.

Now, putting $Z=\xi$ in (4.4), we get

$$\begin{aligned} R^{*}(X,Y)\xi =\bar{R}(X,Y)\xi +\eta (X)\big (\bar{\nabla }_{Y}\phi \big )\xi -\eta (Y)\big (\bar{\nabla }_{X}\phi \big )\xi \end{aligned}$$

(4.6)

From (4.6), we obtain the following:

Remark 4.2

If the ambient manifold $\bar{M}$ is a Sasakian manifold, then we have

$$\begin{aligned} R^{*}(X,Y)\xi&= \bar{R}(X,Y)\xi +\eta (X)\big [\eta (Y)\xi -Y\big ]-\eta (Y)\big [\eta (X)\xi -X\big ] \\&= 2\bar{R}(X,Y)\xi . \end{aligned}$$

Remark 4.3

If the ambient manifold $\bar{M}$ is cosymplectic, then

$$\begin{aligned} R^{*}(X,Y)\xi =\bar{R}(X,Y)\xi . \end{aligned}$$

Again, we have

$$\begin{aligned} \nabla _{X}^{*}\nabla _{Y}^{*}Z&= \nabla _{X}^{^{\prime }}\nabla _{Y}^{^{\prime }}Z+m\big (X,\nabla _{Y}^{^{\prime }}Z\big ) \nonumber \\&+\,D_{X}\big (m(Y,Z)\big )-A_{m(Y,Z)}^{\prime }X.\end{aligned}$$

(4.7)

$$\begin{aligned} \nabla _{Y}^{*}\nabla _{X}^{*}Z&= \nabla _{Y}^{^{\prime }}\nabla _{X}^{^{\prime }}Z+m(Y,\nabla _{X}^{^{\prime }}Z) \nonumber \\&+\,D_{Y}\big (m(X,Z)\big )-A_{m(X,Z)}Y.\end{aligned}$$

(4.8)

$$\begin{aligned} \nabla _{[X,Y]}^{*}Z&= \nabla _{[X,Y]}^{^{\prime }}Z+m\big ([X,Y],Z\big ). \end{aligned}$$

(4.9)

By direct computations, we obtain

$$\begin{aligned} R^{*}(X,Y)Z&= R^{\prime }(X,Y)Z+\big (\nabla _{X}^{\prime }m\big )(Y,Z)-\big (\nabla _{Y}^{\prime }m\big )(X,Z) \nonumber \\&+\,A_{m(X,Z)}^{\prime }Y-A_{m(Y,Z)}^{\prime }X. \end{aligned}$$

(4.10)

Hence, the Gauss equation for the quarter symmetric metric connection $\nabla ^{*}$ is given by

$$\begin{aligned} R^{*}(X,Y,Z,W)&= g\big (R^{*}(X,Y)Z,W\big ) \nonumber \\&= g\big (R^{\prime }(X,Y)Z,W\big )+g\big (m(Y,W),m(X,Z)\big ) \nonumber \\&-\,g\big (m(Y,Z),m(X,W)\big ) \nonumber \\&= R^{\prime }(X,Y,Z,W)+g\big (m(Y,W),m(X,Z)\big ) \nonumber \\&-\,g\big (m(Y,Z),m(X,W)\big ). \end{aligned}$$

(4.11)

Putting $X=W=e_{i}$, $Y=Z=e_{j}$, in (4.11), we obtain

$$\begin{aligned} R^{*}(e_{i},e_{j},e_{j},e_{i})&= R^{\prime }(e_{i},e_{j},e_{j},e_{i})+g\big (m(e_{i},e_{j}),m(e_{j},e_{i})\big ) \nonumber \\&-\,g\big (m(e_{j},e_{j}),m(e_{i},e_{i})\big ). \end{aligned}$$

(4.12)

Summing over $i$ and $j$ and using Theorem 3.3, we get

$$\begin{aligned} \tau ^{*}=\tau ^{\prime }+||m||^{2}-n^{2}\Vert H\Vert ^{2}, \end{aligned}$$

(4.13)

where $\tau ^{*}$ and $\tau ^{\prime }$ are the scalar curvatures corresponding to the quarter symmetric metric connection defined on $\bar{M}$ and the induced quarter-symmetric metric connection on $M$, respectively, and $||m||^{2}$ denotes the squared norm of the second fundamental form with respect to the quarter-symmetric connection. From (4.13), we can also write

$$\begin{aligned} \Vert H\Vert ^{2}\ge \frac{1}{n^{2}}\big (\tau ^{\prime }-\tau ^{*}\big ). \end{aligned}$$

(4.14)

Hence, the following:

Theorem 4.1

On a minimal submanifold of an almost contact metric manifold admitting a quarter-symmetric metric connection, the scalar curvature corresponding to the quarter-symmetric connection is never less than that of the induced quarter-symmetric connecton.

Since $\{N_{\alpha }\},$ $\alpha =1,2,...,p$ is a basis of $T^{\bot }M$, we can express $m(X,Y)=\underset{\alpha =1}{\overset{p}{\sum }}m_{\alpha }(X,Y)N_{\alpha }$, where each $m_{\alpha }$ is a $(0,2)$ tensor.

Hence, the Gauss equation (4.11) can be rewritten in the following form:

$$\begin{aligned} R^{*}(X,Y,Z,W)&= R^{\prime }(X,Y,Z,W)+\sum _{\alpha =1}^{p}\big [m_{\alpha }(Y,W)m_{\alpha }(X,Z) \nonumber \\&-\,m_{\alpha }(Y,Z)m_{\alpha }(X,W)\big ]. \end{aligned}$$

(4.15)

From (3.2) and (3.9), we can easily deduce that

$$\begin{aligned} m_{\alpha }(X,Y)=g\big (A_{N_{\alpha }}^{^{\prime }}X,Y\big ). \end{aligned}$$

(4.16)

Hence, by (4.15), the Gauss equation can also be represented in terms of the shape operator as

$$\begin{aligned} R^{*}(X,Y,Z,W)&= R^{\prime }(X,Y,Z,W)+ \underset{\alpha =1}{\overset{p}{\sum }}\{g(A_{N_{\alpha }}^{^{\prime }}Y,W)g(A_{N_{\alpha }}^{^{\prime }}X,Z) \nonumber \\&-\,g(A_{N_{\alpha }}^{^{\prime }}Y,Z)g(A_{N_{\alpha }}^{^{\prime }}X,W)\}. \end{aligned}$$

(4.17)

From (4.17), we get

$$\begin{aligned} R^{*}(X,Y,X,Y)&= R^{\prime }(X,Y,X,Y)+ \underset{\alpha =1}{\overset{p}{\sum }}\{g(A_{N_{\alpha }}^{^{\prime }}Y,Y)g(A_{N_{\alpha }}^{^{\prime }}X,X) \nonumber \\&-\,g(A_{N_{\alpha }}^{^{\prime }}Y,X)g(A_{N_{\alpha }}^{^{\prime }}X,Y)\}. \end{aligned}$$

(4.18)

Combining with Remark 3.1, we can state the following:

Theorem 4.2

Let ${\mathcal {P}}$ be a $2-$dimensional invariant subspace of $T_{x}M$ and let $K^{*}({\mathcal {P}})$ and $K^{^{\prime }}({\mathcal {P}})$ be the sectional curvature of ${\mathcal {P}}$ in $\bar{M}$ and $M,$ respectively, with respect to the quarter-symmetric metric connection. If $X$and $Y$ form an orthonormal basis of ${\mathcal {P}}$, then

$$\begin{aligned} K^{*}({\mathcal {P}})=K^{^{\prime }}({\mathcal {P}})+\underset{\alpha =1}{ \overset{p}{\sum }}\{g(A_{N_{\alpha }}^{^{\prime }}Y,Y)g(A_{N_{\alpha }}^{^{\prime }}X,X)-g(A_{N_{\alpha }}^{^{\prime }}X,Y)^{2}\}. \end{aligned}$$

Now, contracting equation (4.15)  we have the expression of Ricci tensor corresponding to the quarter-symmetric connection as

$$\begin{aligned} S^{*}(Y,Z)&= S^{\prime }(Y,Z)+\sum _{\alpha =1}^{p}R^{*}(N_{\alpha },Y,Z,N_{\alpha }) \nonumber \\&\!+\!\sum _{\alpha =1}^{p}\left[ \sum _{i=1}^{n}g(A_{N_{\alpha }}^{^{\prime }}Y,e_{i})g(A_{N_{\alpha }}^{^{\prime }}Z,e_{i})\!-\!f_{\alpha }m_{\alpha }(Y,Z) \right] \! \text { (if }A\text {is symmetric)} \nonumber \\&= S^{\prime }(Y,Z)+\sum _{\alpha =1}^{p}R^{*}(N_{\alpha },Y,Z,N_{\alpha }) \nonumber \\&+\,\sum _{\alpha =1}^{p}\left[ g(A_{N_{\alpha }}^{^{\prime }}A_{N_{\alpha }}^{^{\prime }}Y,Z)-f_{\alpha }m_{\alpha }(Y,Z)\right] \text { (if }A\text {is symmetric)} \nonumber \\&= S^{\prime }(Y,Z)+\sum _{\alpha =1}^{p}R^{*}(N_{\alpha },Y,Z,N_{\alpha }) \nonumber \\&+\sum _{\alpha =1}^{p}\left[ m_{\alpha }(A_{N_{\alpha }}^{^{\prime }}Y,Z)-f_{\alpha }m_{\alpha }(Y,Z)\right] . \end{aligned}$$

(4.19)

where $f_{\alpha }$ denote the trace of $A_{N_{\alpha }}^{\prime }$.

Suppose that the quarter-symmetric metric connection $\nabla ^{*}$ is of constant sectional curvature. Then

$$\begin{aligned} R^{*}(X,Y,Z,W)=\lambda (g(Y,Z)g(X,W)-g(X,Z)g(Y,W)). \end{aligned}$$

(4.20)

Therefore, from equation (4.19), we have

$$\begin{aligned} S^{\prime }(Y,Z)&= \sum _{\alpha =1}^{p}\left\{ f_{\alpha }m_{\alpha }(Y,Z)-m_{\alpha }(A_{N_{\alpha }}^{\prime }Y,Z)\right\} \\&+\sum _{\alpha =1}^{p}R^{*}(N_{\alpha },Y,Z,N_{\alpha }). \end{aligned}$$

Therefore,

$$\begin{aligned} S^{\prime }(Y,Z)=\lambda (n-1)g(Y,Z)+\sum _{\alpha =1}^{p}\left\{ f_{\alpha }m_{\alpha }(Y,Z)-m_{\alpha }(A_{N_{\alpha }}^{\prime }Y,Z)\right\} . \end{aligned}$$

(4.21)

Thus, we can state the following:

Theorem 4.3

Let $M$ be an invariant submanifold of an almost contact metric manifold $\bar{M}$ of constant sectional curvature with a quarter-symmetric metric connection. Then

• (i) the Ricci tensor of $M$ induced from the quarter-symmetric connection is symmetric.

• (ii) The Ricci tensor of $M$ induced from the quarter-symmetric connection is not parallel.

From Eq. (4.21), we can also conclude the following:

Theorem 4.4

Let $M$ be a totally umbilical submanifold of an almost contact metric manifold $\bar{M}$ of constant sectional curvature with a quarter-symmetric metric connection. Then the submanifold $M$ is an Einstein manifold with respect to the quarter-symmetric metric connection.

From (4.10), we obtain the normal part of $R^{*}(X,Y)Z$ as

$$\begin{aligned} R^{\perp }(X,Y)Z=(\nabla _{X}^{\prime }m)(Y,Z)-(\nabla _{Y}^{\prime }m)(X,Z), \end{aligned}$$

(4.22)

which is the Codazzi equation corresponding to the quarter-symmetric metric connection $\nabla ^{*}$.

Let $\zeta _{1},\zeta _{2}\in T^{\perp }M$, then we have

$$\begin{aligned} R^{*}(X,Y)\zeta _{1}&= R^{\prime }(X,Y)\zeta _{1}-A_{D_{Y}\zeta _{1}}^{\prime }X+A_{D_{X}\zeta _{1}}^{\prime }Y+\nabla _{Y}^{\prime }(A_{\zeta _{1}}^{\prime }X) \nonumber \\&+\,\nabla _{X}^{\prime }(A_{\zeta _{1}}^{\prime }Y)+m(Y,A_{\zeta _{1}}^{\prime }X)-m(X,A_{\zeta _{1}}^{\prime }Y)-A_{\zeta {_{1}}}^{\prime }\big ([X,Y]\big ).\nonumber \\ \end{aligned}$$

(4.23)

So, we have

$$\begin{aligned} R^{*}(X,Y,\zeta _{1},\zeta _{2})&= g(R^{*}(X,Y)\zeta _{1},\zeta _{2}) \nonumber \\&= g(R^{\prime }(X,Y)\zeta _{1},\zeta _{2})+g(A_{\zeta _{1}}^{\prime }X,A_{\zeta _{2}}^{\prime }Y)-g(A_{\zeta _{1}}^{\prime }Y,A_{\zeta _{2}}^{\prime }X) \nonumber \\&= R^{\prime }(X,Y,\zeta _{1},\zeta _{2})+g(A_{\zeta _{1}}^{\prime }X,A_{\zeta _{2}}^{\prime }Y) -g(A_{\zeta _{1}}^{\prime }Y,A_{\zeta _{2}}^{\prime }X),\quad \end{aligned}$$

(4.24)

which is the Ricci equation corresponding to the quarter-symmetric metric connection.

Remark 4.4

If we consider the submanifold M to be invariant, then from Lemma 3.1, we have the shape operator to be symmetric. Thus, we can express the Ricci equation in the following form:

$$\begin{aligned} R^{*}(X,Y,\zeta _{1},\zeta _{2})=R^{\prime }(X,Y,\zeta _{1},\zeta _{2})+g\big ([A_{\zeta _{1}}^{\prime },A_{\zeta _{2}}^{\prime }]X,Y\big ). \end{aligned}$$

(4.25)

5 Example

Example 5.1

Let us consider the $5$-dimensional manifold $\bar{M}=\{(x,y,z,u,v)\in {\mathbb {R}}^{5},(x,y,z,u,v)\not =(0,0,0,0,0)\},$ where $(x,y,z,u,v)$ are the standard coordinates in ${\mathbb {R}}^{5}.$ The vector fields

$$\begin{aligned} e_{1}\!=\!2\left( -\frac{\partial }{\partial x}+y\frac{\partial }{\partial z}\right) ,\,\, e_{2}\!=\!2\frac{\partial }{\partial y},\,\,e_{3}=2\frac{\partial }{\partial z},\,\,e_{4}=2\left( -\frac{\partial }{\partial u}+v\frac{ \partial }{\partial z}\right) ,\,\,e_{5}=2\frac{\partial }{\partial v} \end{aligned}$$

(5.1)

are linearly independent at each point of $\bar{M}$. Let $g$ be the metirc defined by

$$\begin{aligned} g(e_{i},e_{j})=\left\{ \begin{array}{lll} 1,~~{\text{ when }}~i=j &{} &{} \\ &{} &{} \\ 0,~~{\text{ when }}~i\ne j. &{} &{} \\ &{} &{} \end{array} \right. \end{aligned}$$

Let $\eta$ be the $1$-form defined by $\eta (Z)=g(Z,e_{3})$, for all $Z\in T \bar{M}$, and let $\phi$ be the $(1,1)$ tensor field defined by

$$\begin{aligned} \phi e_{1}=e_{2},~~\phi e_{2}=-e_{1},~~\phi e_{3}=0,~~\phi e_{4}=e_{5},~~\phi e_{5}=-e_{4}. \end{aligned}$$

(5.2)

Then, using the linearity of $\phi$ and $g$, we have

$$\begin{aligned} \eta (e_{3})=1,~~\phi ^{2}Z=-Z+\eta (Z)e_{3} \end{aligned}$$

and

$$\begin{aligned} g(\phi Z,\phi W)=g(Z,W)-\eta (Z)\eta (W),~~\forall ~~Z,W\in T\bar{M}. \end{aligned}$$

Thus, for $e_{3}=\xi ,~~(\bar{M},\phi ,\xi ,\eta ,g)$ be an almost contact metric manifold. Let $\bar{\nabla }$ be the Levi-Civita connection with respect to the metric $g$. Then, we have

$$\begin{aligned}{}[e_{1},e_{2}]=-2e_{3}=[e_{4},e_{5}]~~{\text{ and }}~~[e_{i},e_{j}]=0,~~{ \text{ for } \text{ all } \text{ other }}~i,j. \end{aligned}$$

Taking $e_{3}=\xi$ and using Koszul’s formula for the metric $g$, it can be easily calculated that

$$\begin{aligned} \bar{\nabla }_{e_{1}}e_{2}&= -e_{3},~~\bar{\nabla } _{e_{1}}e_{3}=e_{2},~~\bar{\nabla }_{e_{2}}e_{1}=-e_{3},~~\bar{\nabla } _{e_{2}}e_{3}=-e_{1},~~\bar{\nabla }_{e_{3}}e_{1}=e_{2}, \nonumber \\ \bar{\nabla }_{e_{3}}e_{2}&= -e_{1},~~\bar{\nabla } _{e_{3}}e_{4}=e_{5},~~\bar{\nabla }_{e_{3}}e_{5}=-e_{4},~~\bar{\nabla } _{e_{4}}e_{3}=e_{5},~~\bar{\nabla }_{e_{5}}e_{3}=-e_{4}, \end{aligned}$$

(5.3)

and the rest of the terms are $0$.

Since $\{e_{1},e_{2},e_{3},e_{4},e_{5}\}$ is a frame filed, then any vector field $X,Y\in T\bar{M}$ can be wrriten as

$$\begin{aligned} X=a_{1}e_{1}+b_{1}e_{2}+c_{1}e_{3}+d_{1}e_{4}+f_{1}e_{5}, ~~Y=a_{2}e_{1}+b_{2}e_{2}+c_{c}e_{3}+d_{2}e_{4}+f_{2}e_{5} \end{aligned}$$

where $a_{i},b_{i},c_{i},d_{i},f_{i}\in {{\mathbb {R}}},~~i=1,2$ such that

$$\begin{aligned} a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}+f_{1}f_{2}\ne 0. \end{aligned}$$

Hence, we derive

$$\begin{aligned} g(X,Y)=a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}+d_{1}d_{2}+f_{1}f_{2}. \end{aligned}$$

(5.4)

Now, using (5.3), we get

$$\begin{aligned} \bar{\nabla }_{X}Y&=-(b_{1}c_{2}+b_{2}c_{1})e_{1}+(a_{1}c_{2} +a_{2}c_{1})e_{2}-(a_{2}b_{1}a_{1}b_{2})e_{3}\\&\quad -(c_{1}f_{2}+c_{2}f_{1})e_{4}+(c_{1}d_{2}+c_{2}d_{1})e_{5}. \end{aligned}$$

Thus, from (3.1), we obtain

$$\begin{aligned} \nabla _{X}^{*}Y=a_{1}c_{2}e_{2}-b_{1}c_{2}e_{1}-(a_{2}b_{1}+a_{1}b_{2})e_{3} -c_{2}f_{1}e_{4}+c_{2}d_{1}e_{5}. \end{aligned}$$

(5.5)

Also from (5.4), it follows that $\nabla ^{*}g=0.$ Thus, in an almost contact metric manifold, the quarter-symmetric metric connection is given by (5.5).

Now, let $M$ be a subset of $\bar{M}$ and consider an isometric immersion $f:M\longrightarrow$ $\bar{M}$ by

$$\begin{aligned} f(x,y,z)=(x,y,z,0,0). \end{aligned}$$

It can be easily seen that $M$ is a $3$-dimensional submanifold of the $5$ -dimensional almost contact metric manifold $\bar{M}$. Now, $M=\{(x,y,z)\in {\mathbb {R}}^{3},~(x,y,z)\ne 0\}$, where $(x,y,z)$ are the standard coordinates in ${\mathbb {R}}^{3}$. The vector fields

$$\begin{aligned} e_{1}=2\left( -\frac{\partial }{\partial x}+y\frac{\partial }{\partial z}\right) ,\,\,e_{2}=2\frac{\partial }{\partial y},\,\,e_{3}=2\frac{\partial }{\partial z} \end{aligned}$$

are linearly independent at each point of $M$. Let us denote the induced metric by the same symbol $g$ such that

$$\begin{aligned} g(e_{1},e_{2})=g(e_{1},e_{3})=g(e_{2},e_{3})=0 \end{aligned}$$

and

$$\begin{aligned} g(e_{1},e_{1})=g(e_{2},e_{2})=g(e_{3},e_{3})=1. \end{aligned}$$

Let $\nabla$ be the Levi-Civita connection with respect to the metric $g$ on $M$. Then, we have

$$\begin{aligned}{}[e_{1},e_{2}]=-2e_{3}~~{\text{ and }}~~[e_{1},e_{3}]=0=[e_{2},e_{3}]. \end{aligned}$$

Taking $e_{3}=\xi$ and using Koszul’s formula for the induced metric $g$, it can be easily calculated that

$$\begin{aligned} \nabla _{e_{1}}e_{2}=-e_{3},~~\nabla _{e_{1}}e_{3}=e_{2},~~\nabla _{e_{1}}e_{1}=0,~~\nabla _{e_{2}}e_{3}=-e_{1},~~\nabla _{e_{2}}e_{2}=0, \end{aligned}$$

$$\begin{aligned} \nabla _{e_{2}}e_{1}=-e_{3},~~\nabla _{e_{3}}e_{3}=0,~~\nabla _{e_{3}}e_{2}=-e_{1},~~\nabla _{e_{3}}e_{1}=e_{2}. \end{aligned}$$

Clearly, $\{e_{4},e_{5}\}$ is the frame for the normal bundle $T^{\perp }M$. If we take $X,Y\in TM$, then we can express them as

$$\begin{aligned} X=a_{1}e_{1}=b_{1}e_{2}+c_{1}e_{3},~~~~Y=a_{2}e_{1}=b_{2}e_{2}+c_{2}e_{3} \end{aligned}$$

and therefore

$$\begin{aligned} \nabla _{X}Y=-(b_{1}c_{2}+b_{2}c_{1})e_{1}+(a_{1}c_{2}+a_{2}c_{1})e_{2}-(a_{1}b_{2}+a_{2}b_{1})e_{3} \end{aligned}$$

(5.6)

which is the tangential part of $\bar{\nabla }_{X}Y$. The second fundamental form is given by

$$\begin{aligned} h(X,Y)=-(c_{1}f_{2}+c_{2}f_{1})e_{4}+(c_{1}d_{2}+c_{2}d_{1})e_{5}. \end{aligned}$$

(5.7)

Now, the tangential part of $\nabla _{X}^{*}Y$ is given by

$$\begin{aligned} \nabla _{X}^{\prime }Y=-b_{1}c_{2}e_{1}+a_{1}c_{2}e_{2}-(a_{1}b_{2}+a_{2}b_{1})e_{3}=\nabla _{X}Y-\eta (X)PY. \end{aligned}$$

(5.8)

And, the normal part of $\nabla _{X}^{*}Y$ will be

$$\begin{aligned} m(X,Y)=-c_{2}f_{1}e_{4}+c_{2}d_{1}e_{5}=h(X,Y)-\eta (X)QY. \end{aligned}$$

(5.9)

It is easy to check that $\nabla _{X}^{\prime }g=\nabla _{X}^{*}g$, for any $X\in TM$.

We can easily check that the submanifold $M$ is minimal with respect to both the connections, the Levi-Civita as well as the quarter-symmetric connection.