Quarter-Symmetric Non-Metric Connection of Non-Integrable Distributions

Abstract

In this paper, we focus on non-integrable distributions with a quarter-symmetric non-metric connection (QSNMC) in generalized Riemannian manifold. First, by studying a quarter-symmetric connection on the generalized Riemannian manifold, we obtain the condition that the connection is non-metric. Then, the Gauss, Codazzi and Ricci equations are proved for non-integrable distributions with respect to a quarter-symmetric non-metric connection in generalized Riemannian manifold. Furthermore, we deduce Chen’s inequalities for non-integrable distributions of real space forms with a quarter-symmetric non-metric connection in generalized Riemannian manifold as applications. After that, we give some examples of non-integrable distributions in Riemannian manifold with quarter-symmetric non-metric connection.

1. Introduction

In 1924, Friedmann and Schouten introduced the concept of semi-symmetric connection [1], while in 1932, Hayden defined metric connections with torsion tensor [2]. Wang Yong discussed non-integrable distribution in a Riemannian manifold with a semi-symmetric metric connection and a semi-symmetric non-metric connection in [3]. And, he deduced the Gauss, Codazzi and Ricci equations for a non-integrable distribution corresponding to a semi-symmetric metric connection and a semi-symmetric non-metric connection. A linear connection ∇ is said to be semi-symmetric on an n-dimensional Riamannian manifold ($M^n,g$), if its torsion tensor T, defined by $T(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$, ($X,Y\in \Gamma (M)$), is of the form $T(X,Y)=\omega (Y)X-\omega (X)Y$, where $\omega$ is a 1-form. If we change X to AX, then the connection is called a quarter-symmetric connection when A is a (1,1)-tensor field [4].

At first, the idea of quarter-symmetric linear connections in differentiable manifold was introduced by Golab in 1975 [5]. Recently, more and more scholars pay attention to the quarter-symmetric connection on various manifolds. For example, in 2008, Sular, Özgür and De investigated the curvature tensor and the Ricci tensor of a Kenmotsu manifold with respect to the quarter-symmetric metric connection [6]. In 2010, De and Mondal studied a quarter-symmetric metric connection on a three-dimensional quasi-Sasakian manifold. They deduced the relation between the Riemannian connection and the quarter-symmetric metric connection on a three-dimensional quasi-Sasakian manifold. And, they investigated the curvature tensor, the Ricci tensor, scalar curvature and the first Bianchi identity of a three-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection [7]. In 2012, Mondal and De studied the above with a quarter-symmetric non-metric connection on a P-Sasakian manifold again [8]. In 2019, He and Zhao studied submanifolds of an almost contact metric manifold admitting a quarter-symmetric non-metric connection. And, they proved the induced connection on a submanifold is also quarter-symmetric non-metric connection. Furthermore, they obtain the Gauss, Cadazzi and Ricci equations for submanifolds with respect to the quarter-symmetric non-metric connection, and show some applications of these equations [9]. Furthermore, Yang Yuchen and Wang Yong studied non-integrable distributions with a quarter-symmetric metric connection and a quarter-symmetric non-metric connection in Riemannian manifold in [10]. In 2019, Wang Yong obtained Chen’s inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connections [11]. In 2020, Chen’s inequalities for submanifolds in $(\kappa ,\mu )$-contact space form with two kinds of generalized semi-symmetric non-metric connections were obtained by Wang Yong [12]. In 2021, He, Zhang and Zhao, in [13], derived a first Chen-type inequality for a statistical submanifold of a statistical manifold M of a constant curvature, such that M admits a non-integrable distribution on M with a constant rank. In the same year, Al-Khaldi, Aquib, Aslam et al. obtained improved Chen–Ricci inequalities for submanifolds of generalized space forms with quarter-symmetric metric connection, with the help of which they completely characterized the Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form [14]. Respectively, Li, Khatri, Singh et al. obtained Chen’s inequalities for submanifolds of generalized Sasakian-space-form, admitting a quarter-symmetric connection in [15]. However, the problems related to quarter-symmetric non-metric connections on non-integrable distributions remain to be solved. Some new results will be presented in this paper and the system work of non-integrable distribution will be further improved.

In Section 2, we introduce some basic concepts about a quarter-symmetric connection on the generalized Riemannian manifold. And, we explain the conditions for becoming a non-metric connection. In Section 3, by assigning coefficients a and b, the Gauss formula and the Weingarten formula are established, and the Gauss, Codazzi and Ricci equations for non integrable distributions with respect to a quarter-symmetric non-metric connection are obtained. In Section 4, we prove Chen’s inequalities, in this case as applications. Finally, in Section 5, some examples based on non-integrable distributions in a Riemannian manifold with quarter-symmetric non-metric connections are presented.

2. Quarter-Symmetric Non-Metric Connection (QSNMC)

In this section, we will give some important information that will be used in this paper. The pair $(M,G=g+F)$ is defined a generalized Riemannian manifold, where M denotes a differentiable manifold (dim$M=m$) and G denotes a non-symmetric (0,2) tensor [16]. For its non-symmetry, the tensor G (i.e., the generalized Riemannian metric) can be expressed as

$$G(X,Y)=g(X,Y)+F(X,Y),$$

where g represents its symmetric part and F represents its skew-symmetric part. Specifically, $g(X,Y)$ is defined as

$$g(X,Y)=\frac{1}{2}(G(X,Y)+G(Y,X)),$$

while $F(X,Y)$ is defined as

$$F(X,Y)=\frac{1}{2}(G(X,Y)-G(Y,X)).$$

Suppose that the symmetric part g is a non-degenerate part with arbitrary characteristics. Furthermore, there exists a relationship between the symmetric part g and the skew-symmetric part F given by

$$F(X,Y)=g(AX,Y),$$

where A denotes the (1,1) tensor field associated with skew-symmetric tensor F. So, the quarter-symmetric connection on the generalized Riemannian manifold is determined by the torsion tensor, given in the following form:

$${\nabla }_X^{Q}Y={\nabla }_{X}Y+a\omega (Y)AX+b\omega (X)AY,$$

(1)

whose torsion tensor given with

$$T^Q(X,Y)=(a-b)(\omega (Y)AX-\omega (X)AY),$$

where a and b denotes different real numbers (i.e., $a,b\in R,a\ne b$) and $\omega$ denotes a 1-form associated with vector field P, i.e., $\omega (X)=g(X,P)$.

If the linear connection ${\nabla }^Q$ preserves the generalized Riemannian metric G, it is referred to as a generalized metric (G-metric) connection. The preservation of G implies that both its symmetric part g and its skew-symmetric part F are preserved by ${\nabla }^Q$, i.e.,

$${\nabla }^{Q}G=0⟺{\nabla }^{Q}g={\nabla }^{Q}F=0⟺{\nabla }^{Q}g={\nabla }^{Q}A=0.$$

The connection ${\nabla }^Q$ is called a non-metric connection if it does not preserve the symmetric metric g, ${\nabla }^{Q}g\ne 0$; otherwise, it is considered metric.

In the following article, we will only focus on the quarter-symmetric non-metric connection in generalized Riemannian manifold $(M,G=g+F)$.

3. The Gauss, Codazzi and Ricci Equations for Non-Integrable Distribution with QSNMC

Let $(M,G=g+F)$ be a m-dimensional smooth generalized Riemannian manifold, and ∇ be the Levi-Civita connection of g. We denote $\Gamma (M)$ as the $C^{\infty }(M)$-module of vector fields on M and define ${\nabla }_{X}Y$ as the covariant derivative of Y with respect to X when $X,Y\in \Gamma (M)$. Let $D\subseteq TM$ be a non-integrable distribution, i.e., a subbundle of the tangent bundle $TM$ with constant rank n, and there exist $X^D,Y^D\in \Gamma (D)$, such that $[X^D,Y^D]$ is not in $\Gamma (D)$, where $\Gamma (D)$ is the space of sections of D. The distribution D inherits a metric tensor field $g^D$ from the original g in M. Let $D^{\perp }\subseteq TM$ be the orthogonal distribution to D, which inherits a metric tensor field $g^{D^{\perp }}$ from the g, and then $g=g^D\oplus g^{D^{\perp }}$. Let ${\pi }^D:TM\to D$, ${\pi }^{D^{\perp }}:TM\to D^{\perp }$ be the projections. For $X^D,Y^D\in \Gamma (D)$, we define ${\nabla }_{X^D}^D{Y}^D={\pi }^D({\nabla }_{X^D}{Y}^D)$ and ${[X^D,Y^D]}^D={\pi }^D([X^D,Y^D])$ and ${[X^D,Y^D]}^{D^{\perp }}={\pi }^{D^{\perp }}([X^D,Y^D])$. When $X^D,Y^D\in \Gamma (D)$ and $f\in C^{\infty }(M)$

$${\nabla }_{f{X}^D}^D{Y}^D=f{\nabla }_{X^D}^D{Y}^D,{\nabla }_{X^D}^D(f{Y}^D)=X^D(f)Y^D+f{\nabla }_{X^D}^D{Y}^D,$$

(2)

$$T(X^D,Y^D)={\nabla }_{X^D}^D{Y}^D-{\nabla }_{Y^D}^D{X}^D-[X^D,Y^D]=-{[X^D,Y^D]}^{D^{\perp }},$$

(3)

and

$${\nabla }_{X^D}{Y}^D={\nabla }_{X^D}^D{Y}^D+B(X^D,Y^D),B(X^D,Y^D)={\pi }^{D^{\perp }}{\nabla }_{X^D}{Y}^D.$$

(4)

We note that $B(X^D,Y^D)\ne B(Y^D,X^D)$.

We define the connection (1) with coefficients $a=-b=\frac{1}{2}$ on M as a new quarter-symmetric non-metric connection [16], i.e.,

$${\nabla }_X^{Q}Y={\nabla }_{X}Y+\frac{1}{2}\omega (Y)AX-\frac{1}{2}\omega (X)AY,$$

(5)

whose torsion tensor given with

$$T^Q(X,Y)=\omega (Y)AX-\omega (X)AY,$$

(6)

and which satisfies

$$({\nabla }_X^{Q}g)(Y,Z)=-\frac{1}{2}(\omega (Y)F(X,Z)+\omega (Z)F(X,Y)).$$

(7)

For $X^D$,$Y^D$$\in \Gamma (D)$,

$$\begin{array}{c}\hfill {\nabla }_{X^D}^Q{Y}^D={ }^Q{\nabla }_{X^D}^D{Y}^D+B^Q(X^D,Y^D),\\ \hfill { }^Q{\nabla }_{X^D}^D{Y}^D={\pi }^D{\nabla }_{X^D}^Q{Y}^D,\\ \hfill B^Q(X^D,Y^D)={\pi }^{D^{\perp }}{\nabla }_{X^D}^Q{Y}^D.\end{array}$$

(8)

We call the $B(X^D,Y^D)$ as the second fundamental form with respect to the quarter-symmetric non-metric connection. By (4), (5) and (8), we have

$${ }^Q{\nabla }_{X^D}^D{Y}^D={\nabla }_{X^D}^D{Y}^D+\frac{1}{2}\omega (Y^D)A{X}^D-\frac{1}{2}\omega (X^D)A{Y}^D,B^Q(X^D,Y^D)=B(X^D,Y^D).$$

(9)

By (3) and (9), we have

$${ }^Q{T}^D(X^D,Y^D)=-{[X^D,Y^D]}^{D^{\perp }}+\omega (Y^D)A{X}^D-\omega (X^D)A{Y}^D.$$

(10)

Similarly to the case $D=TM$, we have the following.

Theorem 1. 

There exists a unique linear connection${ }^Q{\nabla }^D:\Gamma (D)\times \Gamma (D)\to \Gamma (D)$ on D, which satisfies${ }^Q{T}^D(X^D,Y^D)=-{[X^D,Y^D]}^{D^{\perp }}+\omega (Y^D)A{X}^D-\omega (X^D)A{Y}^D$.

Let $E_1,...,E_n$ be the orthonormal basis on D. We define the mean curvature vector associated with ${\nabla }^Q$ on D by $H^Q=\frac{1}{n}{\sum }_{i=1}^n{B}^Q(E_i,E_i)\in \Gamma (D^{\perp })$. By (9), then $H^Q=H$, where $H=\frac{1}{n}{\sum }_{i=1}^{n}B(E_i,E_i)\in \Gamma (D^{\perp })$. If $H^Q=0$, we say that D is minimal with respect to the quarter-symmetric non-metric connection ${\nabla }^Q$. If ${\nabla }_{\dot{\gamma }}^Q\dot{\gamma }=0$, we say the curve $\gamma$ is ${\nabla }^Q$-geodesic. If every ${\nabla }^Q$-geodesic with initial condition in D is contained in D, we say that D is totally geodesic with respect to the quarter-symmetric non-metric connection ${\nabla }^Q$.

Let $h(X,Y)=\frac{1}{2}[B(X,Y)+B(Y,X)]$ and $h^Q(X,Y)=\frac{1}{2}[B^Q(X,Y)+B^Q(Y,X)]$. If $h=H{g}^D$ (respectively, $h^Q=H^Q{g}^D$), we say that D is umbilical with respect to ∇ (respectively, ${\nabla }^Q$). We have the following.

Proposition 1. 

D is minimal (respectively, umbilical) with respect to ∇ if and only if D is minimal (respectively, umbilical) with respect to ${\nabla }^Q$.

Let $\xi \in \Gamma (D^{\perp })$ and $X^D\in \Gamma (D)$, then by (5), we have

$${\nabla }_{X^D}^Q\xi ={\nabla }_{X^D}\xi +\frac{1}{2}\omega (\xi )A{X}^D-\frac{1}{2}\omega (X^D)A\xi .$$

(11)

Let $A_{\xi }:\Gamma (D)\to \Gamma (D)$ be the shape operator with respect to ∇, defined by

$$g^D(A_{\xi }{X}^D,Y^D)=g^{D^{\perp }}(B(X^D,Y^D),\xi ).$$

(12)

Let $L_{X^D}^{\perp }\xi ={\pi }^{D^{\perp }}{\nabla }_{X^D}\xi$, then ${\nabla }_{X^D}\xi ={\pi }^D{\nabla }_{X^D}\xi +L_{X^D}^{\perp }\xi$, so

$${\pi }^D{\nabla }_{X^D}\xi =-A_{\xi }{X}^D,{\nabla }_{X^D}\xi =-A_{\xi }{X}^D+L_{X^D}^{\perp }\xi ,$$

(13)

which we call the Weingarten formula with respect to ∇ and $L_{X^D}^{\perp }\xi :\Gamma (D)\times \Gamma (D^{\perp })\to \Gamma (D^{\perp })$ is a metric connection on $D^{\perp }$ along $\Gamma (D)$.

Let $A_{\xi }^Q=(A_{\xi }-\frac{1}{2}\omega (\xi )A)I$ and${ }^Q{L}_{X^D}^{\perp }\xi =(L_{X^D}^{\perp }\xi -\frac{1}{2}\omega (X^D)A)I$; then, by (11) and (13), we obtain the Weingarten formula with respect to ${\nabla }^Q$.

$${\nabla }_{X^D}^Q\xi =-A_{\xi }^Q{X}^D{+}^Q{L}_{X^D}^{\perp }\xi ,$$

(14)

Given $X,Y,Z\in \Gamma (TM)$, the curvature tensor R with respect to ∇ is defined by

$$R^Q(X,Y)Z={\nabla }_X^Q{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z.$$

(15)

Given $X^D,Y^D,Z^D\in \Gamma (D)$, the curvature tensor $R^D$ on D with respect to ${\nabla }^D$ is defined by

$$\begin{array}{cc}\hfill { }^Q{R}^D(X^D,Y^D)Z^D=& { }^Q{\nabla }_{X^D}^D{ }^Q{\nabla }_{Y^D}^D{Z}^D{-}^Q{\nabla }_{Y^D}^D{ }^Q{\nabla }_X^D{Z}^D\hfill \\ & {-}^Q{\nabla }_{{[X^D,Y^D]}^D}^D{Z}^D-{\pi }^D{[[X^D,Y^D]}^{D^{\perp }},Z^D].\hfill \end{array}$$

(16)

In (16),${ }^Q{R}^D$ is a tensor field by adding the extra term $-{\pi }^D[{[X^D,Y^D]}^{D^{\perp }},Z^D]$. Given $X^D,Y^D,Z^D,W^D\in \Gamma (D)$, the Riemannian curvature tensors $R^Q$,${ }^Q{R}^D$ are defined by

$$\begin{array}{cc}\hfill R^Q(X^D,Y^D,Z^D,W^D)& =g(R^Q(X^D,Y^D)Z^D,W^D),\hfill \\ \hfill { }^Q{R}^D(X^D,Y^D,Z^D,W^D)& =g{(}^Q{R}^D(X^D,Y^D)Z^D,W^D).\hfill \end{array}$$

(17)

Theorem 2. 

Given $X^D,Y^D,Z^D,W^D\in \Gamma (D)$, we have the Gauss equation for D with respect to ${\nabla }^Q$ in the following.

$$\begin{array}{cc}\hfill R^Q(X^D,Y^D,Z^D,W^D)=& { }^Q{R}^D(X^D,Y^D,Z^D,W^D)\hfill \\ & -g(B(X^D,W^D),B(Y^D,Z^D))+g(B(Y^D,W^D),B(X^D,Z^D))\hfill \\ \hfill & +g(Y^D,Z^D)\omega (B(X^D,W^D))-g(X^D,Z^D)\omega (B(Y^D,W^D))\hfill \\ \hfill & +\frac{1}{2}g(A{X}^D,W^D)\omega (B(Y^D,Z^D))-\frac{1}{2}g(A{Y}^D,W^D)\omega (B(X^D,Z^D))\hfill \\ \hfill & +g(B(Z^D,W^D),[X^D,Y^D])+\frac{1}{2}\omega ({[X^D,Y^D]}^{D^{\perp }})g(A{Z}^D,W^D).\hfill \end{array}$$

(18)

Proof. 

From Equations (8) and (14), we have, for $X^D,Y^D,Z^D\in \Gamma (D)$,

$$\begin{array}{cc}\hfill {\nabla }_{X^D}^Q{\nabla }_{Y^D}^Q{Z}^D=& {\nabla }_{X^D}^Q{(}^Q{\nabla }_{Y^D}^D{Z}^D)+{\nabla }_{X^D}^Q(B^Q(Y^D,Z^D))\hfill \\ \hfill =& { }^Q{\nabla }_{X^D}^D{ }^Q{\nabla }_{Y^D}^D{Z}^D+B^Q(X^D{,}^Q{\nabla }_{Y^D}^D{Z}^D)\hfill \\ \hfill & -A_{B^Q(Y^D,Z^D)}{X}^D+\frac{1}{2}\omega (B^Q(Y^D,Z^D))A{X}^D{+}^Q{L}_{X^D}^{\perp }(B^Q(Y^D,Z^D)),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\nabla }_{Y^D}^Q{\nabla }_{X^D}^Q{Z}^D& ={ }^Q{\nabla }_{Y^D}^D{ }^Q{\nabla }_{X^D}^D{Z}^D+B^Q(Y^D{,}^Q{\nabla }_{X^D}^D{Z}^D)\hfill \\ & -A_{B^Q(X^D,Z^D)}{Y}^D+\frac{1}{2}\omega (B^Q(X^D,Z^D))A{Y}^D{+}^Q{L}_{Y^D}^{\perp }(B^Q(X^D,Z^D)).\hfill \end{array}$$

(20)

By (5) and ∇ with zero torsion, we have, for $X,Y\in \Gamma (TM)$,

$${\nabla }_{X_1}^Q{X}_2={\nabla }_{X_2}^Q{X}_1+[X_1,X_2]+\omega (X_2)A{X}_1-\omega (X_1)A{X}_2.$$

(21)

So, by (14) and (21), we obtain

$$\begin{array}{cc}\hfill {{\nabla }^Q}_{{[X^D,Y^D]}^{D^{\perp }}}{Z}^D=& {{\nabla }^Q}_{Z^D}({[X^D,Y^D]}^{D^{\perp }})+[{[X^D,Y^D]}^{D^{\perp }},Z^D]\hfill \\ & +\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}-\omega ({[X^D,Y^D]}^{D^{\perp }})A{Z}^D\hfill \\ \hfill =& -A_{{[X^D,Y^D]}^{D^{\perp }}}{Z}^D+\frac{1}{2}\omega ({[X^D,Y^D]}^{D^{\perp }})A{Z}^D{+}^Q{L}_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})\hfill \\ \hfill & +[{[X^D,Y^D]}^{D^{\perp }},Z^D]+\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}-\omega ({[X^D,Y^D]}^{D^{\perp }})A{Z}^D\hfill \\ \hfill =& -A_{{[X^D,Y^D]}^{D^{\perp }}}{Z}^D{+}^Q{L}_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})+\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}\hfill \\ \hfill & -\frac{1}{2}\omega ({[X^D,Y^D]}^{D^{\perp }})A{Z}^D+[{[X^D,Y^D]}^{D^{\perp }},Z^D].\hfill \end{array}$$

(22)

By ${{\nabla }^Q}_{[X^D,Y^D]}{Z}^D={{\nabla }^Q}_{{[X^D,Y^D]}^D}{Z}^D+{{\nabla }^Q}_{{[X^D,Y^D]}^{D^{\perp }}}{Z}^D$ and (12) and (8), we have

$$\begin{array}{cc}\hfill {\nabla }_{[X^D,Y^D]}^Q{Z}^D=& { }^Q{\nabla }_{{[X^D,Y^D]}^D}^D{Z}^D+B^Q({[X^D,Y^D]}^D,Z^D)-A_{{[X^D,Y^D]}^{D^{\perp }}}{Z}^D\hfill \\ & +L_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})+\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}\hfill \\ & -\frac{1}{2}\omega ({[X^D,Y^D]}^{D^{\perp }},A{Z}^D)+[{[X^D,Y^D]}^{D^{\perp }},Z^D].\hfill \end{array}$$

(23)

By (15), (16), (19), (20), and (23), we have

$$\begin{array}{cc}\hfill R^Q(X^D,Y^D)Z^D=& { }^Q{R}^D(X^D,Y^D)Z^D-{\pi }^{D^{\perp }}[{[X^D,Y^D]}^{D^{\perp }},Z^D]+B^Q(X^D{,}^Q{\nabla }_{Y^D}^D{Z}^D)\hfill \\ \hfill & -B^Q(Y^D{,}^Q{\nabla }_{X^D}^D{Z}^D)-B^Q({[X^D,Y^D]}^D,Z^D)-A_{B^Q(Y^D,Z^D)}{X}^D\hfill \\ & +A_{B^Q(X^D,Z^D)}{Y}^D{+}^Q{L}_{X^D}^{\perp }(B^Q(Y^D,Z^D)){-}^Q{L}_{Y^D}^{\perp }(B^Q(X^D,Z^D))\hfill \\ & +\frac{1}{2}\omega (B^Q(Y^D,Z^D))A{X}^D-\frac{1}{2}\omega (B^Q(X^D,Z^D))A{Y}^D\hfill \\ \hfill & +A_{{[X^D,Y^D]}^{D^{\perp }}}{Z}^D-L_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})\hfill \\ & -\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}+\frac{1}{2}\omega ({[X^D,Y^D]}^{D^{\perp }})A{Z}^D.\hfill \end{array}$$

(24)

By (12), (17), and (24) and the second equality in (9), we obtain (18). □

Corollary 1. 

If U = 0, $\omega =0$ and ${\nabla }^Q=\nabla$, we obtain

$$\begin{array}{cc}\hfill R(X^D,Y^D,Z^D,W^D)& =R^D(X^D,Y^D,Z^D,W^D)-g(B(X^D,W^D),B(Y^D,Z^D))\hfill \\ & +g(B(Y^D,W^D),B(X^D,Z^D))+g(B(Z^D,W^D),[X^D,Y^D]).\hfill \end{array}$$

(25)

Theorem 3. 

Given $X^D,Y^D,Z^D\in \Gamma (D)$, we have the Codazzi equation with respect to ${\nabla }^Q$ in the following.

$$\begin{array}{cc}\hfill {(R^Q(X^D,Y^D)Z^D)}^{D^{\perp }}=& {(}^Q{L}_{X^D}^{\perp }{B}^Q)(Y^D,Z^D)-{(}^Q{L}_{Y^D}^{\perp }{B}^Q)(X^D,Z^D)\hfill \\ & -\omega (X^D)B^Q((A{Y}^D),Z^D)+\omega (Y^D)B^Q((A{X}^D),Z^D)\hfill \\ & -{\pi }^{D^{\perp }}[{[X^D,Y^D]}^{D^{\perp }},Z^D]-{{ }^{Q}L}_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})\hfill \\ & -\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }},\hfill \end{array}$$

(26)

where ${(}^Q{L}_{X^D}^{\perp }{B}^Q)(Y^D,Z^D)={{ }^{Q}L}_{X^D}^{\perp }(B^Q(Y^D,Z^D))-B^Q{(}^Q{\nabla }_{X^D}^D{Y}^D,Z^D)-B^Q(Y^D,{{ }^Q\nabla }_{X^D}^D{Z}^D)$.

Proof. 

From (25), we have

$$\begin{array}{cc}\hfill {(R^Q(X^D,Y^D)Z^D)}^{D^{\perp }}=& -{\pi }^{D^{\perp }}\left[{[X^D,Y^D]}^{D^{\perp }},Z^D\right]+B^Q(X^D{,}^Q{\nabla }_{Y^D}^D{Z}^D)\hfill \\ \hfill & -B^Q(Y^D{,}^Q{\nabla }_{X^D}^D{Z}^D)-B^Q({[X^D,Y^D]}^D,Z^D)+L_{X^D}^{\perp }(B^Q(Y^D,Z^D))\hfill \\ \hfill & -L_{Y^D}^{\perp }(B^Q(X^D,Z^D))-L_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})-\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}.\hfill \end{array}$$

(27)

By (21), we have, for $X^D,Y^D\in \Gamma (D)$,

$${[X^D,Y^D]}^D{=}^Q{\nabla }_{X^D}^D{Y}^D{-}^Q{\nabla }_{Y^D}^D{X}^D-\omega (Y^D)A{X}^D+\omega (X^D)A{Y}^D.$$

(28)

By (28) and the definition of $(L_X^{\perp }{B}^Q)(Y^D,Z^D)$ and (27), we obtain

$$\begin{array}{cc}\hfill {(R^Q(X^D,Y^D)Z^D)}^{D^{\perp }}=& -{\pi }^{D^{\perp }}[{[X^D,Y^D]}^{D^{\perp }},Z^D]+B^Q(X^D{,}^Q{\nabla }_{Y^D}^D{Z}^D)-B^Q(Y^D{,}^Q{\nabla }_{X^D}^D{Z}^D)\hfill \\ \hfill & -B^Q{(}^Q{\nabla }_{X^D}^D{Y}^D{-}^Q{\nabla }_{Y^D}^D{X}^D-\omega (Y^D)A{X}^D+\omega (X^D)A{Y}^D,Z^D)\hfill \\ & {+}^Q{L}_{X^D}^{\perp }(B^Q(Y^D,Z^D)){-}^Q{L}_{Y^D}^{\perp }(B^Q(X^D,Z^D)){-}^Q{L}_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }})\hfill \\ & -\omega (Z^D)A{[X^D,Y^D]}^{D^{\perp }}.\hfill \end{array}$$

(29)

So, (26) holds. □

Corollary 2. 

If $U=0$, then we have

$$\begin{array}{cc}\hfill {(R(X^D,Y^D)Z^D)}^{D^{\perp }}& =(L_{X^D}^{\perp }B)(Y^D,Z^D)-(L_{Y^D}^{\perp }B)(X^D,Z^D)\hfill \\ & -{\pi }^{D^{\perp }}[{[X^D,Y^D]}^{D^{\perp }},Z^D]-L_{Z^D}^{\perp }({[X^D,Y^D]}^{D^{\perp }}).\hfill \end{array}$$

(30)

Theorem 4. 

Given $X^D,Y^D\in \Gamma (D),\xi \in \Gamma (D^{\perp })$, we have the Ricci equation for D with respect to ${\nabla }^Q$ in the following.

$${(R^Q(X^D,Y^D)\xi )}^{D^{\perp }}=-B^Q(X^D,A_{\xi }^Q{Y}^D)+B^Q(Y^D,A_{\xi }^Q{X}^D){+}^Q{R}^{L^{\perp }}(X^D,Y^D)\xi ,$$

(31)

where

$${{ }^{Q}R}^{L^{\perp }}(X^D,Y^D)\xi {=}^Q{L}_{X^D}^{\perp }{ }^Q{L}_{Y^D}^{\perp }\xi {-}^Q{L}_{Y^D}^{\perp }{ }^Q{L}_{X^D}^{\perp }\xi {-}^Q{L}_{{[X^D,Y^D]}^D}^{\perp }\xi -{\pi }^{D^{\perp }}{\nabla }_{{[X^D,Y^D]}^{\perp }}^Q\xi .$$

(32)

Proof. 

From (8) and (14), we have

$$\begin{array}{cc}\hfill {\nabla }_{X^D}^Q{\nabla }_{Y^D}^Q\xi & ={\nabla }_{X^D}^Q(-A_{\xi }^Q{Y}^D)+{\nabla }_{X^D}^Q{(}^Q{L}_{Y^D}^{\perp }\xi )\hfill \\ \hfill & ={-}^Q{\nabla }_{X^D}^D(A_{\xi }^Q{Y}^D)-B^Q(X^D,A_{\xi }^Q{Y}^D)-A_{{ }^Q{L}_{Y^D}^{\perp }\xi }^Q{X}^D{+}^Q{L}_{X^D}^{\perp }{ }^Q{L}_{Y^D}^{\perp }\xi ,\hfill \end{array}$$

(33)

$${\nabla }_{Y^D}^Q{\nabla }_{X^D}^Q\xi ={-}^Q{\nabla }_{Y^D}^D(A_{\xi }^Q{X}^D)-B^Q(Y^D,A_{\xi }^Q{X}^D)-A_{{ }^Q{L}_{X^D}^{\perp }\xi }^Q{Y}^D{+}^Q{L}_{Y^D}^{\perp }{ }^Q{L}_{X^D}^{\perp }\xi ,$$

(34)

$$\begin{array}{cc}\hfill {\nabla }_{[X^D,Y^D]}^Q\xi =& {\nabla }_{{[X^D,Y^D]}^D}^Q\xi +{\nabla }_{{[X^D,Y^D]}^{D^{\perp }}}^Q\xi \hfill \\ \hfill =& -A_{\xi }^Q({[X^D,Y^D]}^D){+}^Q{L}_{{[X^D,Y^D]}^D}^{\perp }\xi +{\pi }^D{\nabla }_{{[X^D,Y^D]}^{D^{\perp }}}^Q\xi \hfill \\ \hfill & +{\pi }^{D^{\perp }}{\nabla }_{{[X^D,Y^D]}^{D^{\perp }}}^Q\xi .\hfill \end{array}$$

(35)

From (32)–(35), we obtain (31). □

Corollary 3. 

If $U=0$, then we have

$${\left(R(X^D,Y^D)\xi \right)}^{D^{\perp }}=-B(X^D,A_{\xi }{Y}^D)+B(Y^D,A_{\xi }{X}^D)+R^{L^{\perp }}(X^D,Y^D)\xi ,$$

(36)

where

$$R^{L^{\perp }}(X^D,Y^D)\xi =L_{X^D}^{\perp }{L}_{Y^D}^{\perp }\xi -L_{Y^D}^{\perp }{L}_{X^D}^{\perp }\xi -L_{{[X^D,Y^D]}^D}^{\perp }\xi -{\pi }^{D^{\perp }}{\nabla }_{{[X^D,Y^D]}^{\perp }}\xi .$$

(37)

4. Chen’s Inequalities for Non-Integrable Distributions with QSNMC

In the following, we prove Chen’s inequalities with respect to D and ${\nabla }^Q$. For $X,Y\in \Gamma (TM)$, we let

$$\alpha (X,Y)=({\nabla }_X\omega )(Y)-\frac{1}{4}\omega (AX)\omega (Y)+\frac{1}{4}\omega (X)\omega (AY),$$

where $({\nabla }_X\omega )(Y)=X(\omega (Y))-\omega ({\nabla }_{X}Y)$. We have

$$\begin{array}{cc}\hfill R^Q(X,Y,Z,W)=& R(X,Y,Z,W)+\alpha (X,Z)g(AY,W)-\alpha (Y,Z)g(AX,W)\hfill \\ & +\frac{1}{2}\omega (Z)g(({\nabla }_{X}A)Y,W)-\frac{1}{2}\omega (Z)g(({\nabla }_{Y}A)X,W)\hfill \\ & +\frac{1}{2}\omega (X)g(({\nabla }_{Y}A)Z,W)-\frac{1}{2}\omega (Y)g(({\nabla }_{X}A)Z,W)\hfill \\ & +({\nabla }_{Y}W)(X)g(AZ,W)-({\nabla }_{X}W)(Y)g(AZ,W))\hfill \\ & +\frac{1}{4}\omega (Y)\omega (Z)g(AAX,W)-\frac{1}{4}\omega (X)\omega (Z)g(AAY,W).\hfill \end{array}$$

(38)

In M, we can choose a local orthonormal frame $E_1,...,E_n,E_{n+1},...,E_m$, such that $E_1,...,E_n$ are orthonormal frames of D. And, we let $\lambda ={\sum }_{j=1}^n\alpha (E_j,E_j)$ and $h_{ij}^r{=}^Q{h}_{ij}^r=g(B(E_i,E_j),E_r)$ for $1\le i,j\le n$ and $n+1\le r\le m$.

The squared length of B is $\begin{array}{c}\hfill {\parallel B\parallel }^2={\parallel B^Q\parallel }^2=\sum _{i,j=1}^{n}g(B(E_i,E_j),B(E_i,E_j))\end{array}$. Let M be an m-dimensional real space form of constant sectional curvature c endowed with a quarter-symmetric non-metric connection ${\nabla }^Q$. The curvature tensor R with respect to the Levi-Civita connection on M is expressed by

$$R(X,Y,Z,W)=c\{g(X,W)g(Y,Z)-g(X,Z)g(Y,W)\}.$$

(39)

By (38) and (39), we obtain

$$\begin{array}{cc}\hfill R^Q(X,Y,Z,W)=& c\{g(X,W)g(Y,Z)-g(X,Z)g(Y,W)\}\hfill \\ & +\alpha (X,Z)g(AY,W)-\alpha (Y,Z)g(AX,W)\hfill \\ & +\frac{1}{2}\omega (Z)g(({\nabla }_{X}A)Y,W)-\frac{1}{2}\omega (Z)g(({\nabla }_{Y}A)X,W)\hfill \\ & +\frac{1}{2}\omega (X)g(({\nabla }_{Y}A)Z,W)-\frac{1}{2}\omega (Y)g(({\nabla }_{X}A)Z,W)\hfill \\ & +({\nabla }_{Y}W)(X)g(AZ,W)-({\nabla }_{X}W)(Y)g(AZ,W))\hfill \\ & +\frac{1}{4}\omega (Y)\omega (Z)g(AAX,W)-\frac{1}{4}\omega (X)\omega (Z)g(AAY,W).\hfill \end{array}$$

(40)

Let $\Pi \subset D$, be a two-plane section. The sectional curvature of ${ }^Q{K}^D(\Pi )$ with respect to D and${ }^Q{\nabla }^D$ is defined by

$${ }^Q{K}^D(\Pi )=\frac{1}{2}{[}^Q{R}^D(E_1,E_2,E_2,E_1){-}^Q{R}^D(E_1,E_2,E_1,E_2)],$$

(41)

where $E_1,E_2$ are orthonormal basis of $\Pi$ and${ }^Q{K}^D(\Pi )$ is independent of the choice of $E_1,E_2$. For any orthonormal basis $E_1,...,E_n$ of D, denote by${ }^Q{\tau }^D$ the scalar curvature of D with respect to${ }^Q{\nabla }^D$, defined by

$${ }^Q{\tau }^D=\frac{1}{2}\sum _{1\le i,j\le n}{ }^Q{R}^D(E_i,E_j,E_j,E_i).$$

(42)

By Theorem 2, we have

$$\begin{array}{cc}\hfill R^Q(X^D,Y^D,Z^D,W^D)=& { }^Q{R}^D(X^D,Y^D,Z^D,W^D)\hfill \\ & -g(B(X^D,W^D),B(Y^D,Z^D))+g(B(Y^D,W^D),B(X^D,Z^D))\hfill \\ \hfill & +g(Y^D,Z^D)\omega (B(X^D,W^D))-g(X^D,Z^D)\omega (B(Y^D,W^D))\hfill \\ \hfill & +\frac{1}{2}g(A{X}^D,W^D)\omega (B(Y^D,Z^D))-\frac{1}{2}g(A{Y}^D,W^D)\omega (B(X^D,Z^D))\hfill \\ \hfill & +g(B(Z^D,W^D),[X^D,Y^D])+\frac{1}{2}\omega ({[X^D,Y^D]}^{D^{\perp }})g(A{Z}^D,W^D).\hfill \end{array}$$

(43)

Let $E_1,E_2$ be the orthonormal basis of $\Pi \subset D$, and we have the following definition.

$$\begin{array}{cc}\hfill & \sum _{i=1}^{n}F(E_i,E_i)=\mu ,\hfill \\ \hfill & tr(B{|}_{\Pi })=B(E_1,E_1)+B(E_2,E_2),\hfill \\ \hfill & A^D=\frac{1}{2}\sum _{1\le i,j\le n}g(B(E_j,E_i),[E_j,E_i]),\hfill \\ \hfill {\Omega }^{\Pi }=& -\frac{1}{2}\alpha (E_1,E_2)F(E_2,E_1)+\frac{1}{2}\alpha (E_2,E_2)F(E_1,E_1)+\frac{1}{2}\alpha (E_1,E_1)F(E_2,E_2)\hfill \\ & -\frac{1}{2}\alpha (E_2,E_1)F(E_1,E_2)-\frac{1}{2}g(B(E_1,E_2)-B(E_2,E_1),[E_1,E_2]),\hfill \\ \hfill C_1=& \frac{1}{4}[\sum _{3\le j}g(\omega (E_1)({\nabla }_{E_j}A)E_j-\omega (E_j)({\nabla }_{E_j}A)E_1,E_1)\hfill \\ & +\sum _{3\le i}g(\omega (E_i)({\nabla }_{E_1}A)E_1-\omega (E_1)({\nabla }_{E_1}A)E_i,E_i)\hfill \\ & +\sum _{2\le i\ne j\le n}g(\omega (E_i)({\nabla }_{E_j}A)E_j-\omega (E_j)({\nabla }_{E_j}A)E_i,E_i)],\hfill \\ \hfill C_2=& \frac{1}{2}[\sum _{3\le j}F(E_j,E_1)[({\nabla }_{E_j}\omega )(E_1)-({\nabla }_{E_1}\omega )(E_j)]\hfill \\ & +\sum _{3\le i}F(E_1,E_i)[({\nabla }_{E_1}\omega )(E_i)-({\nabla }_{E_i}\omega )(E_1)]\hfill \\ \hfill & +\sum _{2\le i\ne j\le n}F(E_j,E_i)[({\nabla }_{E_j}\omega )(E_i)-({\nabla }_{E_i}\omega )(E_j)]],\hfill \\ \hfill C_3=& \frac{1}{8}[\sum _{3\le j}\omega (E_j)F(A{E}_1\omega (E_j)-A{E}_j\omega (E_1),E_1)\hfill \\ & +\sum _{3\le i}\omega (E_1)F(A{E}_i\omega (E_1)-A{E}_1\omega (E_i),E_i)\hfill \\ \hfill & +\sum _{2\le i\ne j\le n}\omega (E_j)F(A{E}_i\omega (E_j)-A{E}_j\omega (E_i),E_i)],\hfill \\ \hfill C_4=& \frac{1}{4}[\sum _{3\le j}F(\omega (B(E_1,E_j))E_j-\omega (B(E_j,E_j))E_1,E_1)\hfill \\ & +\sum _{3\le i}F(\omega (B(E_i,E_1))E_1-\omega (B(E_1,E_1))E_i,E_i)\hfill \\ \hfill & +\sum _{2\le i\ne j\le n}F(\omega (B(E_i,E_j))E_j-\omega (B(E_j,E_j))E_i,E_i),\hfill \\ \hfill C_5=& -\frac{1}{4}[\sum _{3\le j}\omega ({[E_1,E_j]}^{D^{\perp }})F(E_j,E_1)+\sum _{2\le i\le n,1\le j\le n}\omega ({[E_i,E_j]}^{D^{\perp }})F(E_j,E_i)]\hfill \\ & -\frac{1}{4}\sum _{3\le j}\omega ({[E_1,E_2]}^{D^{\perp }})F(E_1,E_2).\hfill \end{array}$$

(44)

Then, $A^D$ and ${\Omega }^{\Pi }$ are independent of the choice of the orthonormal basis. Let $\parallel H^Q{\parallel }^2=g(H^Q,H^Q)$. For the distribution D of the real space form M endowed with ${\nabla }^Q$, we establish the following inequality called Chen’s first inequality.

Theorem 5. 

Let $TM=D⨁D^{\perp }$, $dimD=n\ge 3$, and M is a manifold with a constant sectional curvature c and a connection ${\nabla }^Q$, then

$$\begin{array}{cc}\hfill { }^Q{\tau }^D{-}^Q{K}^D(\Pi )\le & \frac{(n+1)(n-2)}{2}c+\frac{1}{2}\sum _{1\le i\ne j\le n}^n\alpha (E_i,E_j)F(E_j,E_i)-\frac{n-1}{2}\lambda \mu \hfill \\ \hfill & -\frac{n(n-1)}{2}\omega (H)+\frac{1}{4}g(tr(B{|}_{\Pi }),P)-A^D+{\Omega }^{\Pi }\hfill \\ \hfill & +C_1+C_2+C_3+C_4+C_5\hfill \\ & +\frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2+\frac{1}{2}{\left|\right|B\left|\right|}^2.\hfill \end{array}$$

(45)

Proof. 

Let $E_1,...,E_n$ and $E_{n+1},...,E_m$ be orthonormal basis of D and $D^{\perp }$, respectively. Let $E_1,E_2$ be the orthonormal basis of $\Pi \subset D$. By (40), (41) and (43), we obtain

$$\begin{array}{cc}\hfill { }^Q{K}^D(\Pi )=& c-{\Omega }^{\Pi }-\frac{1}{4}g(tr(B{|}_{\Pi }),P)+\sum _{r=n+1}^m[h_{11}^r{h}_{22}^r-h_{12}^r{h}_{21}^r]\hfill \\ \hfill & +\frac{1}{4}g(\omega (E_1)({\nabla }_{E_2}A)E_2-\omega (E_2)({\nabla }_{E_2}A)E_1,E_1\hfill \\ \hfill & +\frac{1}{4}g(\omega (E_2)({\nabla }_{E_1}A)E_1-\omega (E_1)({\nabla }_{E_1}A)E_2,E_2\hfill \\ \hfill & +\frac{1}{2}F(E_2,E_1)[({\nabla }_{E_2}\omega )(E_1)-({\nabla }_{E_1}\omega )(E_2)]\hfill \\ \hfill & +\frac{1}{2}F(E_2,E_1)[({\nabla }_{E_2}\omega )(E_1)-({\nabla }_{E_1}\omega )(E_2)]\hfill \\ \hfill & +\frac{1}{2}F(E_1,E_2)[({\nabla }_{E_1}\omega )(E_2)-({\nabla }_{E_2}\omega )(E_1)]\hfill \\ \hfill & +\frac{1}{8}\omega (E_2)F(A{E}_1\omega (E_2)-A{E}_2\omega (E_1),E_1)\hfill \\ \hfill & +\frac{1}{8}\omega (E_1)F(A{E}_2\omega (E_1)-A{E}_1\omega (E_2),E_2)\hfill \\ \hfill & +\frac{1}{4}F(\omega (B(E_1,E_2))E_2-\omega (B(E_2,E_2))E_1,E_1)\hfill \\ \hfill & +\frac{1}{4}F(\omega (B(E_2,E_1))E_1-\omega (B(E_1,E_1))E_2,E_2)\hfill \\ \hfill & +\frac{1}{4}\omega ({[E_1,E_2]}^{D^{\perp }})F[(E_1,E_2)-F(E_2,E_1)].\hfill \end{array}$$

(46)

Similarly, we obtain

$$\begin{array}{cc}\hfill { }^Q{\tau }^D=& \frac{1}{2}\sum _{1\le i\ne j\le n}{{ }^{Q}R}^D(E_i,E_j,E_j,E_i)\hfill \\ \hfill =& \frac{n(n-1)}{2}c+\frac{1}{2}\sum _{1\le i\ne j\le n}^n\alpha (E_i,E_j)F(E_j,E_i)-\frac{n-1}{2}\lambda \mu -A^D\hfill \\ \hfill & +\sum _{r=n+1}^m\sum _{1\le i<j\le n}[h_{ii}^r{h}_{jj}^r-h_{ij}^r{h}_{ji}^r]-\frac{n(n-1)}{2}\omega (H)\hfill \\ \hfill & +\frac{1}{4}\sum _{1\le i\ne j\le n}g(\omega (E_i)({\nabla }_{E_j}A)E_j-\omega (E_j)({\nabla }_{E_j}A)E_i,E_1\hfill \\ \hfill & +\frac{1}{2}\sum _{1\le i\ne j\le n}F(E_j,E_i)[({\nabla }_{E_j}\omega )(E_i)-({\nabla }_{E_i}\omega )(E_j)]\hfill \\ \hfill & +\frac{1}{8}\sum _{1\le i\ne j\le n}\omega (E_j)F(A{E}_i\omega (E_j)-A{E}_j\omega (E_i),E_i)\hfill \\ \hfill & +\frac{1}{4}\sum _{1\le i\ne j\le n}F(\omega (B(E_i,E_j))E_j-\omega (B(E_j,E_j))E_i,E_i)\hfill \\ \hfill & -\frac{1}{4}\sum _{1\le i\ne j\le n}\omega ({[E_i,E_j]}^{D^{\perp }})F(E_j,E_i).\hfill \end{array}$$

(47)

So,

$$\begin{array}{cc}\hfill { }^Q{\tau }^D{-}^Q{K}^D(\Pi )\le & \frac{(n+1)(n-2)}{2}c+\frac{1}{2}\sum _{1\le i\ne j\le n}^n\alpha (E_i,E_j)F(E_j,E_i)-\frac{n-1}{2}\lambda \mu \hfill \\ \hfill & -\frac{n(n-1)}{2}\omega (H)+\frac{1}{4}g(tr(B{|}_{\Pi }),P)-A^D+{\Omega }^{\Pi }\hfill \\ \hfill & +C_1+C_2+C_3+C_4+C_5\hfill \\ \hfill & +\sum _{r=n+1}^m[(h_{11}^r+h_{22}^r)\sum _{3\le j\le n}{h}_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r-\sum _{1\le i<j\le n}{h}_{ij}^r{h}_{ji}^r+h_{12}^r{h}_{21}^r].\hfill \end{array}$$

(48)

By Lemma 2.4 in [17], we obtain

$$\begin{array}{c}\hfill \sum _{r=n+1}^m[(h_{11}^r+h_{22}^r)\sum _{3\le j\le n}{h}_{jj}^r+\sum _{3\le i<j\le n}{h}_{ii}^r{h}_{jj}^r]\le \frac{n^2(n-2)}{2(n-1)}{\parallel H\parallel }^2.\end{array}$$

(49)

We note that

$$\begin{array}{cc}\hfill & \sum _{r=n+1}^m\left[-\sum _{1\le i<j\le n}{h}_{ij}^r{h}_{ji}^r+h_{12}^r{h}_{21}^r\right]\hfill \\ \hfill =& \sum _{r=n+1}^m\left[-\sum _{3\le j\le n}{h}_{1j}^r{h}_{j1}^r-\sum _{2\le i<j\le n}{h}_{ij}^r{h}_{ji}^r\right]\hfill \\ \hfill \le & \sum _{r=n+1}^m\left[\sum _{3\le j\le n}\frac{{(h_{1j}^r)}^2+{(h_{j1}^r)}^2}{2}+\sum _{2\le i<j\le n}\frac{{(h_{ij}^r)}^2+{(h_{ji}^r)}^2}{2}\right]\hfill \\ \hfill \le & \sum _{r=n+1}^m\left[\sum _{3\le j\le n}\frac{{(h_{1j}^r)}^2+{(h_{j1}^r)}^2}{2}+\sum _{2\le i<j\le n}\frac{{(h_{ij}^r)}^2+{(h_{ji}^r)}^2}{2}\right.\hfill \\ \hfill & \left.+\sum _{i=1}^n\frac{{(h_{ii}^r)}^2}{2}+\frac{{(h_{12}^r)}^2+{(h_{21}^r)}^2}{2}\right]\hfill \\ \hfill & =\frac{{\parallel B\parallel }^2}{2}.\hfill \end{array}$$

(50)

□

By (48)–(50), we obtain (45).

Corollary 4. 

The equality case of (45) holds if and only if D is totally geodesic with respect to ${\nabla }^Q$ and $h_{12}^r=h_{21}^r=0$.

Proof. 

The equality case of (50) holds if and only if $h_{ii}^r=0$, for $1\le i\le n$, $h_{12}^r=h_{21}^r=0$ and $h_{1j}^r=-h_{j1}^r$ for $3\le j\le n$ and $h_{kl}^r=-h_{lk}^r$ for $2\le k<l\le n$.

The equality case of (49) holds if and only if $h_{11}^r+h_{22}^r=h_{ii}^r$ for $3\le i\le n$. So, Corollary 4 is proven. □

For each unit vector field $X^D\in \Gamma (D)$, we choose the orthonormal basis $E_1,...,E_n$ of D, such that $E_1=X^D$. We define

$$\begin{array}{cc}\hfill & {{ }^{\mathrm{Q}}\mathrm{Ric}}^D(X^D)=\sum _{j=2}^n{ }^Q{R}^D(X^D,E_j,E_j,X^D),\hfill \\ \hfill & A^D(X^D)=\sum _{j=2}^{n}g(B(E_j,X^D),[E_j,X^D]),\hfill \\ \hfill & \parallel B^{X^D}{\parallel }^2=\sum _{i=2}^n[g(B(X^D,E_j),B(X^D,E_j))+g(B(E_j,X^D),B(E_j,X^D))].\hfill \end{array}$$

(51)

Theorem 6. 

Let $TM=D\oplus D^{\perp }$, dim $D=n\ge 2$, and M is a manifold with a constant sectional curvature c and a connection ${\nabla }^Q$, then

$$\begin{array}{cc}\hfill { }^Q{\mathrm{Ric}}^D(X^D)\le & (n-1)c-\lambda F(X^D,X^D)+\alpha (X^D,X^D)F(X^D,X^D)\hfill \\ & -n\omega (H)+\omega (B(X^D,X^D))-A^D(X^D)\hfill \\ \hfill & +\frac{1}{2}\sum _{j=2}^{n}g(\omega (X^D)({\nabla }_{E_j}A)E_j-\omega (E_j)({\nabla }_{E_j}A)X^D,X^D)\hfill \\ \hfill & +\sum _{j=2}^n({\nabla }_{E_j}\omega )(X)F(E_j,X^D)-({\nabla }_X\omega )(E_j)F(E_j,X^D)\hfill \\ \hfill & +\frac{1}{4}\sum _{j=2}^n\omega (E_j)F(\omega (E_j)A{X}^D-\omega (X)A{E}_j,X^D)\hfill \\ \hfill & +\frac{1}{2}\sum _{j=2}^{n}F(\omega (B(X^D,E_j))E_j-\omega (B(E_j,E_j))X^D,X^D).\hfill \\ \hfill & -\frac{1}{2}\sum _{j=2}^n\omega ({[X^D,E_j]}^{D^{\perp }}F(E_j,X^D))+\frac{n^2}{4}{\parallel H\parallel }^2+\frac{\parallel B^{X^D}{\parallel }^2}{2}.\hfill \end{array}$$

(52)

Proof. 

By (40), (41) and (43), we have

$$\begin{array}{cc}\hfill { }^Q{\mathrm{Ric}}^D(X^D)\le & (n-1)c-\lambda F(X^D,X^D)+\alpha (X^D,X^D)F(X^D,X^D)\hfill \\ & -n\omega (H)+\omega (B(X^D,X^D))-A^D(X^D)\hfill \\ \hfill & +\frac{1}{2}\sum _{j=2}^{n}g(\omega (X^D)({\nabla }_{E_j}A)E_j-\omega (E_j)({\nabla }_{E_j}A)X^D,X^D)\hfill \\ \hfill & +\sum _{j=2}^n({\nabla }_{E_j}\omega )(X^D)F(E_j,X^D)-({\nabla }_{X^D}\omega )(E_j)F(E_j,X^D)\hfill \\ \hfill & +\frac{1}{4}\sum _{j=2}^n\omega (E_j)F(\omega (E_j)A{X}^D-\omega (X^D)A{E}_j,X^D)\hfill \\ \hfill & +\frac{1}{2}\sum _{j=2}^{n}F(\omega (B(X^D,E_j))E_j-\omega (B(E_j,E_j))X^D,X^D).\hfill \\ \hfill & -\frac{1}{2}\sum _{j=2}^n\omega ({[X^D,E_j]}^{D^{\perp }}F(E_j,X^D))+\sum _{r=n+1}^m\sum _{j=2}^n[h_{11}^r{h}_{jj}^r-h_{1j}^r{h}_{j1}^r].\hfill \end{array}$$

(53)

By Lemma 2.5 in [17], we obtain

$$\begin{array}{cc}\hfill \sum _{r=n+1}^{n+p}\sum _{j=2}^n{h}_{11}^r{h}_{jj}^r& \le \frac{n^2}{4}{\parallel H\parallel }^2.\hfill \end{array}$$

(54)

We note that

$$-\sum _{r=n+1}^m\sum _{j=2}^n{h}_{1j}^r{h}_{j1}^r\le \sum _{r=n+1}^m\sum _{j=2}^n\frac{{(h_{1j}^r)}^2+{(h_{j1}^r)}^2}{2}=\frac{\parallel B^{X^D}{\parallel }^2}{2}.$$

(55)

By (53)–(55), we obtain (52). □

Corollary 5. 

The equality case of (51) holds if and only if $h_{1j}^r=-h_{j1}^r$ for $2\le j\le n$ and $h_{11}^r-h_{22}^r-...-h_{nn}^r=0$.

5. Examples

In this section, two examples, namely the Heisenberg group $(H_3,g_{H_3})$ and the three-dimensional unit sphere $S_3$ of non-integrable distributions in a Riemannian manifold with quarter-symmetric non-metric connections, follow.

Example 1. 

Let $(H_3,g_{H_3})$ be the Heisenberg group $H_3$ endowed with the Riemannian metric g, which has an orthonormal basis $X_1,X_2,X_3$, satisfy the commutation relations

$$[X_1,X_2]=X_3,[X_1,X_3]=0,[X_2,X_3]=0.$$

(56)

Let ∇ be the Levi-Civita connection on $H_3$. By (56) and the Koszul formula, we have

$$\begin{array}{cc}\hfill {\nabla }_{X_j}{X}_j& =0,1\le j\le 3,{\nabla }_{X_1}{X}_2=\frac{1}{2}{X}_3,{\nabla }_{X_2}{X}_1=-\frac{1}{2}{X}_3,\hfill \\ \hfill {\nabla }_{X_1}{X}_3& ={\nabla }_{X_3}{X}_1=-\frac{1}{2}{X}_2,{\nabla }_{X_2}{X}_3={\nabla }_{X_3}{X}_2=\frac{1}{2}{X}_1.\hfill \end{array}$$

(57)

Consider the distribution $D=span\{X_1^D,X_2^D\}$, which is not integrable by (56). The metric of D is induced by the metric on $H_3$. Let $U=X_1^D+X_2^D+X_3^{D^{\perp }}$. By (57), we have

$$\begin{array}{c}\hfill {\nabla }_{X_i^D}^D{X}_j^D=0,\forall i,j=1,2,B(X_1^D,X_1^D)=B(X_2^D,X_2^D)=0,\\ \hfill B(X_1^D,X_2^D)=\frac{1}{2}{X}_3^{D^{\perp }},B(X_2^D,X_1^D)=-\frac{1}{2}{X}_3^{D^{\perp }}.\end{array}$$

(58)

By (9), we obtain

$$\begin{array}{c}\hfill { }^Q{\nabla }_{X^D}^D{Y}^D={\nabla }_{X^D}^D{Y}^D+\frac{1}{2}g(X_1^D+X_2^D,Y^D)A{X}^D-\frac{1}{2}g(X_1^D+X_2^D,X^D)A{Y}^D,\\ \hfill B^Q(X^D,Y^D)=B(X^D,Y^D).\end{array}$$

(59)

And let $A{X}_1^D=X_1^D,A{X}_2^D=-X_2^D,A{X}_3^{D^{\perp }}=-X_3^{D^{\perp }}$. We have

$$\begin{array}{cc}\hfill & { }^Q{\nabla }_{X_1^D}^D{X}_1^D=0,{ }^Q{\nabla }_{X_1^D}^D{X}_2^D=-\frac{1}{2}{X}_1^D+\frac{1}{2}{X}_2^D,\hfill \\ \hfill & {{ }^Q\nabla }_{X_2^D}^D{X}_1^D=-\frac{1}{2}{X}_1^D-\frac{1}{2}{X}_2^D,{{ }^Q\nabla }_{X_2^D}^D{X}_2^D=0\hfill \\ \hfill & \begin{array}{c}\hfill B^Q(X_1^D,X_1^D)=0,B^Q(X_1^D,X_2^D)=\frac{1}{2}{X}_3^{D^{\perp }},\end{array}\hfill \\ \hfill & \begin{array}{c}\hfill B^Q(X_2^D,X_1^D)=-\frac{1}{2}{X}_3^{D^{\perp }},B^Q(X_2^D,X_2^D)=0,H^Q=0.\end{array}\hfill \end{array}$$

(60)

By (17), (41), (42) and (62), we have

$$\begin{array}{c}\hfill {{ }^{Q}R}^D(X_1^D,X_2^D)X_1^D=\frac{1}{4}{X}_1^D-\frac{1}{4}{X}_2^D,\\ \hfill {{ }^{Q}R}^D(X_1^D,X_2^D)X_2^D=-\frac{1}{4}{X}_1^D-\frac{1}{4}{X}_2^D,\\ \hfill {{ }^{Q}K}^D(D)=0,{{ }^Q\tau }^D=0.\end{array}$$

(61)

Example 2. 

Consider the three-dimensional unit sphere $S_3$ as a Riemannian manifold endowed with the metric induced from $R^4$. The tangent space of $S_3$ at each point has an orthonormal basis $e_1,e_2,e_3$, which satisfies

$$[e_1,e_2]=2e_3,[e_1,e_3]=-2e_2,[e_2,e_3]=2e_1.$$

(62)

The Levi-Civita connection ∇ of $S_3$ is given by

$$\begin{array}{c}\hfill {\nabla }_{e_1}{e}_2=e_3,{\nabla }_{e_2}{e}_1=-e_3,{\nabla }_{e_1}{e}_1={\nabla }_{e_2}{e}_2={\nabla }_{e_3}{e}_3=0,\\ \hfill {\nabla }_{e_1}{e}_3=-e_2,{\nabla }_{e_3}{e}_1=e_2,{\nabla }_{e_2}{e}_3=e_1,{\nabla }_{e_3}{e}_2=-e_1.\end{array}$$

(63)

Let $D_1=span\{e_1^{D_1},e_2^{D_1}\}$, by (63), then ${\nabla }_{e_i^{D_1}}^{D_1}{e}_j^{D_1}=0,\forall i,j=1,2$. Let $U=e_1^{D_1}+e_3^{D_1^{\perp }}$, $A{e}_1^{D_1}=e_1^{D_1},A{e}_2^{D_1}=-e_2^{D_1},A{e}_3^{D_1^{\perp }}=-e_3^{D_1^{\perp }}$, then

$$\begin{array}{c}{ }^Q{\nabla }_{e_1^D}^{D_1}{e}_1^{D_1}=0,{ }^Q{\nabla }_{e_1^D}^{D_1}{e}_2^{D_1}=\frac{1}{2}{e}_2^{D_1},{{ }^Q\nabla }_{e_2^{D_1}}^{D_1}{e}_1^{D_1}=-\frac{1}{2}{e}_2^{D_1},{{ }^Q\nabla }_{e_2^{D_1}}^{D_1}{e}_2^{D_1}=0\hfill \\ B^Q(e_1^{D_1},e_1^{D_1})=0,B^Q(e_1^{D_1},e_2^{D_1})=e_3^{D_1^{\perp }},B^Q(e_2^{D_1},e_1^{D_1})=-e_3^{D_1^{\perp }},B^Q(e_2^{D_1},e_2^{D_1})=0,H^Q=0.\hfill \\ {{ }^{Q}R}^{D_1}(e_1^{D_1},e_2^{D_1})e_1^{D_1}=-4e_2^{D_1},{{ }^{Q}R}^{D_1}(e_1^{D_1},e_2^{D_1})e_2^{D_1}=-4e_1^{D_1},{{ }^{Q}K}^{D_1}(D_1)=4,{{ }^Q\tau }^{D_1}=4.\hfill \end{array}$$

(64)

6. Conclusions

This paper proved the Gauss, Codazzi and Ricci equations for non-integrable distributions with respect to a quarter-symmetric non-metric connection. As applications, we deduced Chen’s inequalities for non-integrable distributions of real space forms with a quarter-symmetric metric connection and a quarter-symmetric non-metric connection.

For further work, it would be interesting to research the various affine connection of non-integrable distributions and deal with the applications of the results obtained in the paper. Also, many inequalities of a similar type will be possibly established for different kinds of submanifolds in various ambient manifolds [18].