On Submanifolds in a Riemannian Manifold with a Semi-Symmetric Non-Metric Connection

Abstract

In this paper, we study submanifolds in a Riemannian manifold with a semi-symmetric non-metric connection. We prove that the induced connection on a submanifold is also semi-symmetric non-metric connection. We consider the total geodesicness and minimality of a submanifold with respect to the semi-symmetric non-metric connection. We obtain the Gauss, Cadazzi, and Ricci equations for submanifolds with respect to the semi-symmetric non-metric connection.

1. Introduction

In 1924, Friedmann and Schouten [1] introduced the idea of semi-symmetric connection on a differentiable manifold. A linear connection $\widehat{\nabla }$ on a differentiable manifold $\tilde{M}$ is said to be a semi-symmetric connection if the torsion $\widehat{T}$ of the connection $\widehat{\nabla }$ satisfies

$$\widehat{T}(\tilde{X},\tilde{Y})=\pi \left(\tilde{Y}\right)\tilde{X}-\pi \left(\tilde{X}\right)\tilde{Y},$$

(1)

where $\pi$ is a 1-form.

In 1932, Hayden [2] introduced the notion of a semi-symmetric metric connection on a Riemannian manifold $(\tilde{M},g)$. A semi-symmetric connection $\widehat{\nabla }$ is said to be a semi-symmetric metric connection if

$$\widehat{\nabla }g=0.$$

(2)

Yano [3] studied some properties of a Riemannian manifold endowed with a semi-symmetric metric connection. Submanifolds of a Riemannian manifold with a semi-symmetric metric connection were studied by Nakao [4].

After a long gap, the study of a semi-symmetric connection $\widehat{\nabla }$ satisfying

$$\widehat{\nabla }g\ne 0$$

(3)

was initiated by Prvanovic [5] with the name pseudo-metric semi-symmetric connection, and was just followed by Smaranda and Andonie [6].

A semi-symmetric connection $\widehat{\nabla }$ is said to be a semi-symmetric non-metric connection if it satisfies the condition Equation (3).

In 1992, Agashe and Chafle [7] introduced a semi-symmetric non-metric connection on a Riemannian manifold $(\tilde{M},g)$ given by

$${\check{\tilde{\nabla }}}_{\tilde{X}}\tilde{Y}={\tilde{\nabla }}_{\tilde{X}}\tilde{Y}+\pi \left(\tilde{X}\right)\tilde{Y},$$

where $\tilde{\nabla }$ is the Levi-Civita connection of $(\tilde{M},g)$ and $\pi$ is a 1-form. Agashe and Chafle [8] studied submanifolds of a Riemannian manifold with this semi-symmetric non-metric connection. In 2000, Sengupta, De, and Binh [9] gave another type of semi-symmetric non-metric connection. Özgür [10] studied properties of submanifolds of a Riemannian manifold with this semi-symmetric non-metric connection. Recently, De, Han, and Zhao [11] introduced a new type of semi-symmetric non-metric connection which is given by

$${\check{\tilde{\nabla }}}_{\tilde{X}}\tilde{Y}={\tilde{\nabla }}_{\tilde{X}}\tilde{Y}+a\omega \left(\tilde{X}\right)\tilde{Y}+b\omega \left(\tilde{Y}\right)\tilde{X},$$

(4)

where a and b are two non-zero real numbers and $\omega$ is a 1-form. They proved the existence of this new type of linear connection and studied a Riemannian manifold admitting this type of semi-symmetric non-metric connection in [11].

Motivated by [8] and [10], we have studied submanifolds of a Riemannian manifold endowed with the semi-symmetric non-metric connection defined by Equation (4) in this paper. The paper has been organized as follows: In Section 2, we give some properties of the semi-symmetric non-metric connection; In Section 3, we consider a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection and show that the induced connection on the submanifold is also a semi-symmetric non-metric connection. We also consider the total geodesicness and minimality of a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection; In Section 4, we deduce the Gauss, Codazzi, and Ricci equations with respect to the semi-symmetric non-metric connection. Using this Gauss equation, we give the relation between the sectional curvatures with respect to the semi-symmetric non-metric connection of a Riemannian manifold and a submanifold, which is analogous to Synger’s inequality [12]. Finally, we consider these fundamental equations of a submanifold in a space form with constant curvature with the semi-symmetric non-metric connection.

2. Preliminaries

Let $\tilde{M}$ be an $(n+d)$-dimensional Riemannian manifold with a Riemannian metric g and $\tilde{\nabla }$ be the Levi-Civita connection of $(\tilde{M},g)$. De, Han, and Zhao [11] defined a special type of linear connection on $\tilde{M}$ by

$${\check{\tilde{\nabla }}}_{\tilde{X}}\tilde{Y}={\tilde{\nabla }}_{\tilde{X}}\tilde{Y}+a\omega \left(\tilde{X}\right)\tilde{Y}+b\omega \left(\tilde{Y}\right)\tilde{X},$$

(5)

where a and b are two non-zero real numbers and $\omega$ is a 1-form on $\tilde{M}$. Denote by $\tilde{U}={\omega }^{♯}$, i.e., the vector field $\tilde{U}$ is defined by $\omega \left(\tilde{X}\right)=g(\tilde{X},\tilde{U})$ for all $\tilde{X}\in \mathcal{X}\left(\tilde{M}\right)$, $\mathcal{X}\left(\tilde{M}\right)$ is the set of all differentiable vector fields on $\tilde{M}$.

By Equation (5), the torsion tensor $\check{\tilde{T}}$ with respect to the connection $\check{\tilde{\nabla }}$ is given by

$$\check{\tilde{T}}(\tilde{X},\tilde{Y})=(b-a)\omega \left(\tilde{Y}\right)\tilde{X}-(b-a)\omega \left(\tilde{X}\right)\tilde{Y}=\pi \left(\tilde{Y}\right)\tilde{X}-\pi \left(\tilde{X}\right)\tilde{Y},$$

where $\pi \left(\tilde{X}\right)=(b-a)\omega \left(\tilde{X}\right)$ is a 1-form.

Therefore, the connection $\check{\tilde{\nabla }}$ is a semi-symmetric connection. Additionally,

$$\left({\check{\tilde{\nabla }}}_{\tilde{X}}g\right)(\tilde{Y},\tilde{Z})=-2a\omega \left(\tilde{X}\right)g(\tilde{Y},\tilde{Z})-b\omega \left(\tilde{Y}\right)g(\tilde{X},\tilde{Z})-b\omega \left(\tilde{Z}\right)g(\tilde{X},\tilde{Y})\ne 0.$$

Hence, the semi-symmetric connection $\check{\tilde{\nabla }}$ defined by Equation (5) is a semi-symmetric non-metric connection.

Analogous to the definition of the curvature tensor $\tilde{R}$ of $\tilde{M}$ with respect to the Levi-Civita connection $\tilde{\nabla }$, we define the curvature tensor $\check{\tilde{R}}$ of $\tilde{M}$ with respect to the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$ given by

$$\check{\tilde{R}}(\tilde{X},\tilde{Y})\tilde{Z}={\check{\tilde{\nabla }}}_{\tilde{X}}{\check{\tilde{\nabla }}}_{\tilde{Y}}\tilde{Z}-{\check{\tilde{\nabla }}}_{\tilde{Y}}{\check{\tilde{\nabla }}}_{\tilde{X}}\tilde{Z}-{\check{\tilde{\nabla }}}_{[\tilde{X},\tilde{Y}]}\tilde{Z},$$

(6)

where $\tilde{X},\tilde{Y},\tilde{Z}\in \mathcal{X}\left(\tilde{M}\right)$.

Using Equations (5) and (6), we have

$$\begin{array}{cc}\hfill \check{\tilde{R}}(\tilde{X},\tilde{Y})\tilde{Z}& =\tilde{R}(\tilde{X},\tilde{Y})\tilde{Z}-a\left({\tilde{\nabla }}_{\tilde{Y}}\omega \right)\left(\tilde{X}\right)\tilde{Z}+a\left({\tilde{\nabla }}_{\tilde{X}}\omega \right)\left(\tilde{Y}\right)\tilde{Z}-b\left({\tilde{\nabla }}_{\tilde{Y}}\omega \right)\left(\tilde{Z}\right)\tilde{X}\hfill \\ & +b\left({\tilde{\nabla }}_{\tilde{X}}\omega \right)\left(\tilde{Z}\right)\tilde{Y}+b^2\omega \left(\tilde{Y}\right)\omega \left(\tilde{Z}\right)\tilde{X}-b^2\omega \left(\tilde{X}\right)\omega \left(\tilde{Z}\right)\tilde{Y}.\hfill \end{array}$$

(7)

The Riemannian Christoffel tensors of the connections $\tilde{\nabla }$ and $\check{\tilde{\nabla }}$ are defined by

$$\tilde{R}(\tilde{X},\tilde{Y},\tilde{Z},\tilde{W})=g(\tilde{R}(\tilde{X},\tilde{Y})\tilde{Z},\tilde{W})$$

and

$$\check{\tilde{R}}(\tilde{X},\tilde{Y},\tilde{Z},\tilde{W})=g(\check{\tilde{R}}(\tilde{X},\tilde{Y})\tilde{Z},\tilde{W}),$$

respectively.

3. Submanifolds of a Riemannian Manifold with the Semi-Symmetric Non-Metric Connection $\check{\tilde{\nabla }}$

Let M be an n-dimensional submanifold of an $(n+d)$-dimensional Riemannian manifold with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$. We decompose the vector field $\tilde{U}$ on M uniquely into their tangent and normal components $U^{\top }$, $U^{\perp }$.

The Gauss formula for the submanifold M with respect to the Levi-Civita connection $\tilde{\nabla }$ is given by

$${\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\forall X,Y\in \mathcal{X}\left(M\right),$$

(8)

where h is the second fundamental form of M in $\tilde{M}$.

For the second fundament form h, the covariant of h is defined by

$$\left({\overline{\nabla }}_{X}h\right)(Y,Z)={\nabla }_X^{\perp }h(Y,Z)-h({\nabla }_{X}Y,Z)-h(Y,{\nabla }_{X}Z),\forall X,Y,Z\in \mathcal{X}\left(M\right).$$

(9)

Then, $\overline{\nabla }h$ is a normal bundle valued tensor of type $(0,3)$ and is called the third fundamental form of M. $\overline{\nabla }$ is called the van der Waerden–Bortolotti connection of M; i.e., $\overline{\nabla }$ is the connection in $TM\oplus T^{\perp }M$ built with $\nabla$ and ${\nabla }^{\perp }$.

Let $\check{\nabla }$ be the induced connection from the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$. We define

$${\check{\tilde{\nabla }}}_{X}Y={\check{\nabla }}_{X}Y+\check{h}(X,Y),\forall X,Y\in \mathcal{X}\left(M\right),$$

(10)

where $\check{h}$ is a $(1,2)$-tensor field in $T^{\perp }M$, the normal part of M. The Equation (10) may be called the Gauss formula for M with respect to the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$.

Using Equations (5), (8), and (10), we have

$${\check{\nabla }}_{X}Y+\check{h}(X,Y)={\nabla }_{X}Y+h(X,Y)+a\omega \left(X\right)Y+b\omega \left(Y\right)X.$$

(11)

Comparing the tangential and normal parts of Equation (11), we obtain

$${\check{\nabla }}_{X}Y={\nabla }_{X}Y+a\omega \left(X\right)Y+b\omega \left(Y\right)X$$

(12)

and

$$\check{h}(X,Y)=h(X,Y).$$

(13)

From Equation (12), we have

$$\begin{array}{cc}\hfill \check{T}(X,Y)& ={\check{\nabla }}_{X}Y-{\check{\nabla }}_{Y}X-[X,Y]=(b-a)\omega \left(Y\right)X-(b-a)\omega \left(X\right)Y\hfill \end{array}$$

(14)

where $\check{T}$ is the torsion tensor of the connection $\check{\nabla }$ on M. Moreover, using Equation (12), we have

$$\begin{array}{cc}\hfill \left({\check{\nabla }}_{X}g\right)(Y,Z)& ={\check{\nabla }}_X\left(g(Y,Z)\right)-g({\check{\nabla }}_{X}Y,Z)-g(Y,{\check{\nabla }}_{X}Z)\hfill \\ & =-2a\omega \left(X\right)g(Y,Z)-b\omega \left(Y\right)g(X,Z)-b\omega \left(Z\right)g(X,Y)\hfill \\ & \ne 0.\hfill \end{array}$$

(15)

In view of Equations (12), (14), and (15), we can state the following theorem:

Theorem 1.

The induced connection $\check{\nabla }$ on a submanifold of a Riemannian manifold endowed with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$ is also a semi-symmetric non-metric connection.

If $\check{h}(X,Y)=0$ for all $X,Y\in \mathcal{X}\left(M\right)$, then M is called totally geodesic with respect to the semi-symmetric non-metric connection. Let $\{e_1,\cdots ,e_n\}$ be an orthonormal basis of the tangent space of M. We define the mean curvature vector $\check{H}$ of M with respect to the semi-symmetric non-metric connection by

$$\check{H}=\frac{1}{n}\sum _{i=1}^n\check{h}(e_i,e_i).$$

(16)

From Equation (13) we know that

$$\check{H}=H,$$

(17)

where H is the mean curvature vector of the submanifold M. If $\check{H}=0$, then M is called minimal with respect to the semi-symmetric non-metric connection.

From Equations (13) and (17), we have the following result:

Theorem 2.

Let M be an n-dimensional submanifold of an $(n+d)$-dimensional Riemannian manifold $\check{M}$ with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$. Then,

(1) 

M is totally geodesic with respect to the semi-symmetric non-metric connection if and only if M is totally geodesic with respect to the Levi-Civita connection.

(2) 

M is minimal with respect to the semi-symmetric non-metric connection if and only if M is minimal with respect to the Levi-Civita connection.

Let $\xi$ be a normal vector field on M. From Equation (5), we have

$${\check{\tilde{\nabla }}}_X\xi ={\tilde{\nabla }}_X\xi +a\omega \left(X\right)\xi +b\omega \left(\xi \right)X.$$

(18)

It is well known that the Weingarten formula for a submanifold of a Riemannian manifold is given by

$${\tilde{\nabla }}_X\xi =-A_{\xi }X+{\nabla }_X^{\perp }\xi ,$$

(19)

where $A_{\xi }$ is the shape operator of M in the direction of $\xi$.

Using Equation (19), we can write Equation (18) as

$${\check{\tilde{\nabla }}}_X\xi =-A_{\xi }X+b\omega \left(\xi \right)X+{\nabla }_X^{\perp }\xi +a\omega \left(X\right)\xi .$$

(20)

Now we define a $(1,1)$-tensor field on M by

$${\check{A}}_{\xi }=(A_{\xi }-b\omega \left(\xi \right))I.$$

(21)

Then, Equation (20) turns into

$${\check{\tilde{\nabla }}}_X\xi =-{\check{A}}_{\xi }X+{\nabla }_X^{\perp }\xi +a\omega \left(X\right)\xi .$$

(22)

Equation (22) is called the Weingarten formula for M with respect to the semi-symmetric non-metric connection.

Since $A_{\xi }$ is symmetric, it is easy to verify that

$$g({\check{A}}_{\xi }X,Y)=g(X,{\check{A}}_{\xi }Y)$$

and

$$g([{\check{A}}_{\xi },{\check{A}}_{\eta }]X,Y)=g([A_{\xi },A_{\eta }]X,Y),$$

(23)

where $[{\check{A}}_{\xi },{\check{A}}_{\eta }]={\check{A}}_{\xi }{\check{A}}_{\eta }-{\check{A}}_{\eta }{\check{A}}_{\xi }$, $[A_{\xi },A_{\eta }]=A_{\xi }{A}_{\eta }-A_{\eta }{A}_{\xi }$ and $\xi ,\eta$ are normal vector fields on M.

From Equations (21) and (23), we can also obtain the following theorems:

Theorem 3.

Principal directions of the unit normal vector ξ with respect to the Levi-Civita connection $\tilde{\nabla }$ and the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$, and the principle curvatures are equal if and only if ξ is orthogonal to $U^{\perp }$.

Theorem 4.

Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$. Then, the shape operators with respect to the semi-symmetric non-metric connection are simultaneously diagonalizable if and only if the shape operators with respect to the Levi-Civita connection are simultaneously diagonalizable.

4. Gauss, Codazzi, and Ricci Equations with Respect to the Semi-Symmetric Non-Metric Connection

We denote the curvature tensor of a submanifold M of a Riemannian manifold $\tilde{M}$ with respect to the induced semi-symmetric non-metric connection $\check{\nabla }$ and the induced Levi-Civita connection ∇ by

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

(24)

and

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

respectively, where $X,Y,Z\in \mathcal{X}\left(M\right)$.

Theorem 5.

Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection $\check{\nabla }$. Then, for all $X,Y,Z,W\in \mathcal{X}\left(M\right)$, we have

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y,Z,W)& =\check{R}(X,Y,Z,W)-g(\check{h}(Y,Z),\check{h}(X,W))+g(\check{h}(X,Z),\check{h}(Y,W))\hfill \\ & +b\omega \left(\check{h}(Y,Z)\right)g(X,W)-b\omega \left(\check{h}(X,Z)\right)g(Y,W).\hfill \end{array}$$

(25)

Here Equation (25) is called the Gauss equation for the submanifold M with respect to the semi-symmetric non-metric connection.

Proof. 

From Equations (10) and (20), we have

$$\begin{array}{cc}\hfill {\check{\tilde{\nabla }}}_X{\check{\tilde{\nabla }}}_{Y}Z& ={\check{\nabla }}_X{\check{\nabla }}_{Y}Z+\check{h}(X,{\check{\nabla }}_{Y}Z)-A_{\check{h}(Y,Z)}X\hfill \\ & +b\omega \left(\check{h}(Y,Z)\right)X+{\nabla }_X^{\perp }\check{h}(Y,Z)+a\omega \left(X\right)\check{h}(Y,Z),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\check{\tilde{\nabla }}}_Y{\check{\tilde{\nabla }}}_{X}Z& ={\check{\nabla }}_Y{\check{\nabla }}_{X}Z+\check{h}(Y,{\check{\nabla }}_{X}Z)-A_{\check{h}(X,Z)}Y\hfill \\ & +b\omega \left(\check{h}(X,Z)\right)Y+{\nabla }_Y^{\perp }\check{h}(X,Z)+a\omega \left(Y\right)\check{h}(X,Z),\hfill \end{array}$$

(27)

and

$${\check{\tilde{\nabla }}}_{[X,Y]}Z={\check{\nabla }}_{[X,Y]}Z+\check{h}([X,Y],Z).$$

(28)

Using Equations (24), (26)–(28), we obtain

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y)Z& =\check{R}(X,Y)Z+\check{h}(X,{\check{\nabla }}_{Y}Z)-\check{h}(Y,{\check{\nabla }}_{X}Z)-\check{h}([X,Y],Z)\hfill \\ & -A_{\check{h}(Y,Z)}X+A_{\check{h}(X,Z)}Y+b\omega \left(\check{h}(Y,Z)\right)X-b\omega \left(\check{h}(X,Z)\right)Y\hfill \\ & +{\nabla }_X^{\perp }\check{h}(Y,Z)-{\nabla }_Y^{\perp }\check{h}(X,Z)+a\omega \left(X\right)\check{h}(Y,Z)-a\omega \left(Y\right)\check{h}(X,Z).\hfill \end{array}$$

(29)

Since $g(A_{\xi }X,Y)=g(h(X,Y),\xi )$ and $h=\check{h}$, from Equation (29) we find

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y,Z,W)& =\check{R}(X,Y,Z,W)-g(A_{\check{h}(Y,Z)}X,W)+g(A_{\check{h}(X,Z)}Y,W)\hfill \\ & +b\omega \left(\check{h}(Y,Z)\right)g(X,W)-b\omega \left(\check{h}(X,Z)\right)g(Y,W)\hfill \\ & =\check{R}(X,Y,Z,W)-g(\check{h}(Y,Z),\check{h}(X,W))+g(\check{h}(X,Z),\check{h}(Y,W))\hfill \\ & +b\omega \left(\check{h}(Y,Z)\right)g(X,W)-b\omega \left(\check{h}(X,Z)\right)g(Y,W).\hfill \end{array}$$

☐

Recalling that if $\pi \in T_{p}M$ is a 2-dimensional subspace of $T_{p}M$ spanned by an orthonormal base $\{X,Y\}$, we define the sectional curvature $\check{K}\left(\pi \right)$ with respect to the semi-symmetric non-metric connection as $\check{R}(X,Y,Y,X)$. Let $\check{\tilde{K}}\left(\pi \right)$ denote the corresponding sectional curvature in $\tilde{M}$. As an application of the Gauss Equation (25), we can obtain the following Synger’s inequality with respect to the semi-symmetric non-metric connection.

Corollary 1.

Let M be a submanifold of a Riemannian manifold $\tilde{M}$ with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$ and γ be a geodesic in $\tilde{M}$ which lies in M, and T be a unit tangent vector field of γ. π is a subspace of the tangent space $T_{p}M$ spanned by $\{X,T\}$. Then,

(1) 

$\check{\tilde{K}}\left(\pi \right)\ge \check{K}\left(\pi \right)$ along γ.

(2) 

if X is a unit tangent vector field on M which is parallel along γ and orthogonal to T, then the equality of (1) holds if and only if X is parallel along γ in $\tilde{M}$.

Proof. 

(1) Let $\gamma$ be a geodesic in $\tilde{M}$ which lies in M and T be a unit tangent vector field of $\gamma$. Then, we have

$$h(T,T)=0.$$

(30)

Let $\pi$ be a subspace of the tangent space $T_{p}M$ spanned by an orthonormal base $\{X,T\}$. Applying the Gauss Equation (25) and $h=\check{h}$, we obtain

$$\begin{array}{cc}\hfill \check{\tilde{K}}\left(\pi \right)& =\check{\tilde{R}}(X,T,T,X)\hfill \\ & =\check{R}(X,T,T,X)-g(h(X,X),h(T,T))+g(h(X,T),(X,T))+b\omega \left(h(T,T)\right)\hfill \\ & =\check{K}\left(\pi \right)+g(h(X,T),(X,T))\hfill \\ & \ge \check{K}\left(\pi \right).\hfill \end{array}$$

(31)

(2) If X be parallel along $\gamma$, we have ${\nabla }_{T}X=0.$ Thus, we have

$${\tilde{\nabla }}_{T}X=h(T,X).$$

Then, the equality of Equation (31) holds if and only if $h(X,T)=0$; i.e., ${\tilde{\nabla }}_{T}X=0$. ☐

Theorem 6.

Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection $\check{\nabla }$. Then, for all $X,Y,Z\in \mathcal{X}\left(M\right)$, we have

$${\left(\check{\tilde{R}}(X,Y)Z\right)}^{\perp }={\check{\overline{\nabla }}}_X\check{h}(Y,Z)-{\check{\overline{\nabla }}}_Y\check{h}(X,Z)-(b-a)\omega \left(X\right)\check{h}(Y,Z)+(b-a)\omega \left(Y\right)\check{h}(X,Z),$$

(32)

where ${\check{\overline{\nabla }}}_X\check{h}(Y,Z)={\nabla }_X^{\perp }\check{h}(Y,Z)-\check{h}({\check{\nabla }}_{X}Y,Z)-\check{h}(Y,{\check{\nabla }}_{X}Z)$. Equation (32) is called the Codazzi equation with respect to the semi-symmetric non-metric connection.

Proof. 

From Equation (29), the normal component of $\check{\tilde{R}}(X,Y)Z$ is given by

$$\begin{array}{cc}\hfill {\left(\check{\tilde{R}}(X,Y)Z\right)}^{\perp }& =\check{h}(X,{\check{\nabla }}_{Y}Z)-\check{h}(Y,{\check{\nabla }}_{X}Z)-\check{h}([X,Y],Z)+{\nabla }_X^{\perp }\check{h}(Y,Z)\hfill \\ & -{\nabla }_Y^{\perp }\check{h}(X,Z)+a\omega \left(X\right)\check{h}(Y,Z)-a\omega \left(Y\right)\check{h}(X,Z)\hfill \\ & ={\nabla }_X^{\perp }\check{h}(Y,Z)-{\nabla }_Y^{\perp }\check{h}(X,Z)-\check{h}(Y,{\check{\nabla }}_{X}Z)+\check{h}(X,{\check{\nabla }}_{Y}Z)\hfill \\ & -\check{h}({\check{\nabla }}_{X}Y-{\check{\nabla }}_{Y}X+(b-a)\omega \left(X\right)Y-(b-a)\omega \left(Y\right)X,Z)\hfill \\ & +a\omega \left(X\right)\check{h}(Y,Z)-a\omega \left(Y\right)\check{h}(X,Z)\hfill \\ & ={\check{\overline{\nabla }}}_X\check{h}(Y,Z)-{\check{\overline{\nabla }}}_Y\check{h}(X,Z)\hfill \\ & -(b-2a)\omega \left(X\right)\check{h}(Y,Z)+(b-2a)\omega \left(Y\right)\check{h}(X,Z),\hfill \end{array}$$

where ${\check{\overline{\nabla }}}_X\check{h}(Y,Z)={\nabla }_X^{\perp }\check{h}(Y,Z)-\check{h}({\check{\nabla }}_{X}Y,Z)-\check{h}(Y,{\check{\nabla }}_{X}Z)$. ☐

Remark 1.

$\check{\overline{\nabla }}$ is the connection in $TM\oplus T^{\perp }M$ built with $\check{\nabla }$ and ${\nabla }^{\perp }$. It may be called the van der Waerden–Bortolotti connection with respect to the semi-symmetric non-metric connection.

Theorem 7.

Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection $\check{\nabla }$. Then, for all $X,Y\in \mathcal{X}\left(M\right)$ and normal vector fields $\xi ,\mu$ on M, we have

$$\begin{array}{c}\hfill \check{\tilde{R}}(X,Y,\xi ,\mu )=R^{\perp }(X,Y,\xi ,\mu )-g([A_{\xi },A_{\mu }]X,Y)+a[g(Y,{\nabla }_X{U}^{\top })-g(X,{\nabla }_Y{U}^{\top })]g(\xi ,\mu ).\end{array}$$

(33)

Equation (33) is called the Ricci equation for the submanifold M with respect to the semi-symmetric non-metric connection.

Proof. 

From Equations (10) and (22), we get

$$\begin{array}{cc}\hfill {\check{\tilde{\nabla }}}_X{\check{\tilde{\nabla }}}_Y\xi & =-{\check{\nabla }}_X\left({\check{A}}_{\xi }Y\right)-\check{h}(X,{\check{A}}_{\xi }Y)-{\check{A}}_{{\nabla }_Y^{\perp }\xi }X+{\nabla }_X^{\perp }{\nabla }_Y^{\perp }\xi \hfill \\ & +a\omega \left(X\right){\nabla }_Y^{\perp }\xi +ag({\tilde{\nabla }}_{X}Y,U^{\top })\xi +ag(Y,{\tilde{\nabla }}_X{U}^{\top })\xi \hfill \\ & +a\omega \left(Y\right){\tilde{\nabla }}_X\xi +a^2\omega \left(X\right)\omega \left(Y\right)\xi +ab\omega \left(Y\right)\omega \left(\xi \right)X,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\check{\tilde{\nabla }}}_Y{\check{\tilde{\nabla }}}_X\xi & =-{\check{\nabla }}_Y\left({\check{A}}_{\xi }X\right)-\check{h}(Y,{\check{A}}_{\xi }X)-{\check{A}}_{{\nabla }_X^{\perp }\xi }Y+{\nabla }_Y^{\perp }{\nabla }_X^{\perp }\xi \hfill \\ & +a\omega \left(Y\right){\nabla }_X^{\perp }\xi +ag({\tilde{\nabla }}_{Y}X,U^{\top })\xi +ag(X,{\tilde{\nabla }}_Y{U}^{\top })\xi \hfill \\ & +a\omega \left(X\right){\tilde{\nabla }}_Y\xi +a^2\omega \left(Y\right)\omega \left(X\right)\xi +ab\omega \left(X\right)\omega \left(\xi \right)Y\hfill \end{array}$$

(35)

and

$${\check{\tilde{\nabla }}}_{[X,Y]}\xi =-{\check{A}}_{\xi }[X,Y]+{\nabla }_{[X,Y]}^{\perp }\xi +ag([X,Y],U^{\top })\xi .$$

(36)

Using Equations (34)–(36), we have

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y,\xi ,\mu )& =g\left(\check{\tilde{R}}(X,Y)\xi \right),\mu )\hfill \\ & =R^{\perp }(X,Y,\xi ,\mu )-g(\check{h}(X,{\check{A}}_{\xi }Y),\mu )+g(\check{h}(Y,{\check{A}}_{\xi }X),\mu )\hfill \\ & +a[g(Y,{\tilde{\nabla }}_X{U}^{\top })-g(X,{\tilde{\nabla }}_Y{U}^{\top })]g(\xi ,\mu ).\hfill \end{array}$$

In view of Equations (10), (13), and (21), the above equation turns into

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y,\xi ,\mu )& =R^{\perp }(X,Y,\xi ,\mu )-g(h(X,A_{\xi }Y),\mu )+g(h(Y,A_{\xi }X),\mu )\hfill \\ & +a[g(Y,{\nabla }_X{U}^{\top })-g(X,{\nabla }_Y{U}^{\top })]g(\xi ,\mu )\hfill \\ & =R^{\perp }(X,Y,\xi ,\mu )-g((A_{\xi }{A}_{\mu }-A_{\mu }{A}_{\xi })X,Y)\hfill \\ & +a[g(Y,{\nabla }_X{U}^{\top })-g(X,{\nabla }_Y{U}^{\top })]g(\xi ,\mu )\hfill \\ & =R^{\perp }(X,Y,\xi ,\mu )-g([A_{\xi },A_{\mu }]X,Y)\hfill \\ & +a[g(Y,{\nabla }_X{U}^{\top })-g(X,{\nabla }_Y{U}^{\top })]g(\xi ,\mu )\hfill \end{array}$$

☐

It will be useful to examine the form of our fundamental equations with respect to the semi-symmetric non-metric connection when the ambient space $\tilde{M}$ has constant curvature. Now, assume that $\tilde{M}$ is an $(n+d)$-dimensional space form of constant curvature C with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$. Let M be a submanifold of $\tilde{M}$. Then, from Equation (7) we have

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y)Z& =C[g(Y,Z)X-g(X,Z)Y]-a\left({\tilde{\nabla }}_Y\omega \right)\left(X\right)Z+a\left({\tilde{\nabla }}_X\omega \right)\left(Y\right)Z\hfill \\ & -b\left({\tilde{\nabla }}_Y\omega \right)\left(Z\right)X+b\left({\tilde{\nabla }}_X\omega \right)\left(Z\right)Y+b^2\omega \left(Y\right)\omega \left(Z\right)X-b^2\omega \left(X\right)\omega \left(Z\right)Y,\hfill \end{array}$$

(37)

where $X,Y,Z\in \mathcal{X}\left(M\right)$.

Hence from Equation (25) we know that the Gauss equation becomes

$$\begin{array}{cc}\hfill \check{R}(X,Y)Z& =C[g(Y,Z)X-g(X,Z)Y]-a\left({\tilde{\nabla }}_Y\omega \right)\left(X\right)Z+a\left({\tilde{\nabla }}_X\omega \right)\left(Y\right)Z\hfill \\ & -b\left({\tilde{\nabla }}_Y\omega \right)\left(Z\right)X+b\left({\tilde{\nabla }}_X\omega \right)\left(Z\right)Y+b^2\omega \left(Y\right)\omega \left(Z\right)X-b^2\omega \left(X\right)\omega \left(Z\right)Y\hfill \\ & +g(\check{h}(Y,Z),\check{h}(X,W))-g(\check{h}(X,Z),\check{h}(Y,W))\hfill \\ & -b\omega \left(\check{h}(Y,Z)\right)g(X,W)+b\omega \left(\check{h}(X,Z)\right)g(Y,W).\hfill \end{array}$$

From Equation (37) we know

$${\left(\check{\tilde{R}}(X,Y)Z\right)}^{\perp }=0$$

So from Equation (32) we know that the Codazzi equation becomes

$${\check{\overline{\nabla }}}_X\check{h}(Y,Z)-{\check{\overline{\nabla }}}_Y\check{h}(X,Z)=(b-2a)\omega \left(X\right)\check{h}(Y,Z)-(b-2a)\omega \left(Y\right)\check{h}(X,Z).$$

Since $\tilde{M}$ is a space form of constant C, it follows that $\tilde{R}(X,Y,\xi ,\mu )=0$. On the other hand, from Equation (37) we have

$$\begin{array}{cc}\hfill \check{\tilde{R}}(X,Y,\xi ,\mu )& =a[\left({\check{\nabla }}_X\omega \right)Y-\left({\check{\nabla }}_Y\omega \right)X]g(\xi ,\mu )\hfill \\ & =a[X\left(g(U^{\top },Y)\right)-g({\nabla }_{X}Y,U^{\top })-Y\left(g(U^{\top },X)\right)-g({\nabla }_{Y}X,U^{\top })]g(\xi ,\mu )\hfill \\ & =a[g({\nabla }_X{U}^{\top },Y)-g({\nabla }_Y{U}^{\top },X)g(\xi ,\mu ).\hfill \end{array}$$

(38)

Then, using Equations (33) and (38), we obtain that the Ricci equation becomes

$$R^{\perp }(X,Y,\xi ,\mu )=g([A_{\xi },A_{\mu }]X,Y)$$

(39)

Using Equations (23) and (39), we can state the following result:

Corollary 2.

Let M be a submanifold of a space form of constant curvature with the semi-symmetric non-metric connection $\check{\tilde{\nabla }}$. Then, the normal connection ${\nabla }^{\perp }$ is flat if and only if all second fundamental tensors with respect to the Levi-Civita connection and the semi-symmetric non-metric connection are simultaneously diagonalizable.

Example. Let ${\mathbb{T}}^2:S^1\left(1\right)\times S^1\left(1\right)\in {\mathbb{R}}^4$ be a torus embedded in ${\mathbb{R}}^4$ defined by

$${\mathbb{T}}^2=\{(cosu,sinu,cosv,sinv):u,v\in \mathbb{R}\}.$$

For $p=(cosu,sinu,cosv,sinv)$, $T_P\left({\mathbb{T}}^2\right)$ is spanned by

$$e_1=(-sinu,cosu,0,0),$$

$$e_2=(0,0-sinv,cosv)$$

and $T_P^{\perp }\left({\mathbb{T}}^2\right)$ is spanned by

$$e_3=(cosu,sinu,0,0),$$

$$e_4=(0,0,cosv,sinv).$$

Differentiating these, we get

$$\begin{array}{c}{\tilde{\nabla }}_{e_1}{e}_1=-e_3,{\tilde{\nabla }}_{e_1}{e}_2=0,{\tilde{\nabla }}_{e_1}{e}_3=e_1,{\tilde{\nabla }}_{e_1}{e}_4=0,\hfill \\ {\tilde{\nabla }}_{e_2}{e}_1=0,{\tilde{\nabla }}_{e_2}{e}_2=-e_4,{\tilde{\nabla }}_{e_2}{e}_3=0,{\tilde{\nabla }}_{e_2}{e}_4=e_2.\hfill \end{array}$$

(40)

Let $\omega$ be a 1-form on ${\mathbb{R}}^4$. A semi-symmetric non-metric connection $\check{\tilde{\nabla }}$ on ${\mathbb{R}}^4$ is given by

$${\check{\tilde{\nabla }}}_{\tilde{X}}\tilde{Y}={\tilde{\nabla }}_{\tilde{X}}\tilde{Y}+a\pi \left(\tilde{X}\right)\tilde{Y}+b\pi \left(\tilde{Y}\right)\tilde{X}.$$

(41)

From Equations (40) and (41), we have

$$\begin{array}{c}{\check{\tilde{\nabla }}}_{e_1}{e}_1=-e_3+(a+b)\omega \left(e_1\right)e_1,{\check{\tilde{\nabla }}}_{e_1}{e}_2=a\omega \left(e_1\right)e_2+b\omega \left(e_2\right)e_1,\hfill \\ {\check{\tilde{\nabla }}}_{e_2}{e}_1=a\omega \left(e_2\right)e_1+b\omega \left(e_1\right)e_2,{\check{\tilde{\nabla }}}_{e_2}{e}_2=-e_4+(a+b)\omega \left(e_2\right)e_2.\hfill \end{array}$$

(42)

Using Equation (42), we obtain

$$\begin{array}{c}{\check{\nabla }}_{e_1}{e}_1=(a+b)\omega \left(e_1\right)e_1,{\check{\nabla }}_{e_1}{e}_2=a\omega \left(e_1\right)e_2+b\omega \left(e_2\right)e_1,\hfill \\ {\check{\nabla }}_{e_2}{e}_1=a\omega \left(e_2\right)e_1+b\omega \left(e_1\right)e_2,{\check{\nabla }}_{e_2}{e}_2=(a+b)\omega \left(e_2\right)e_2\hfill \end{array}$$

(43)

and

$$\check{h}(e_1,e_1)=-e_3,\check{h}(e_1,e_2)=\check{h}(e_2,e_1)=0,\check{h}(e_2,e_2)=-e_4.$$

(44)

From Equation (43), we have

$$\begin{array}{cc}\hfill \check{T}(e_1,e_2)& ={\check{\nabla }}_{e_1}{e}_2-{\check{\nabla }}_{e_2}{e}_1-[e_1,e_2]\hfill \\ =(a-b)\omega \left(e_2\right)e_1-(a-b)\omega \left(e_1\right)e_2.\hfill \end{array}$$

(45)

and

$$\begin{array}{c}\left({\check{\nabla }}_{e_1}g\right)(e_1,e_1)=-2(a+b)\omega \left(e_1\right),\left({\check{\nabla }}_{e_1}g\right)(e_1,e_2)=-b\omega \left(e_2\right),\hfill \\ \left({\check{\nabla }}_{e_1}g\right)(e_2,e_2)=-2a\omega \left(e_1\right),\left({\check{\nabla }}_{e_2}g\right)(e_1,e_1)=-2a\omega \left(e_2\right),\hfill \\ \left({\check{\nabla }}_{e_2}g\right)(e_1,e_2)=-b\omega \left(e_2\right),\left({\check{\nabla }}_{e_2}g\right)(e_2,e_2)=-2(a+b)\omega \left(e_2\right).\hfill \end{array}$$

(46)

Equations (45) and (46) show that the induced connection $\check{\nabla }$ is also a semi-symmetric non-metric connection.

Using Equation (44), we know that the mean curvature vector of ${\mathbb{T}}^2$ with respect to the semi-symmetric non-metric connection is

$$\check{H}=\frac{1}{2}[\check{h}(e_1,e_1)+\check{h}(e_2,e_2)]=-\frac{1}{2}(e_3+e_4).$$