Since the celebrated theory of Nash [1] of isometric immersion of a Riemannian manifold into a suitable Euclidean space gave very important and effective motivation to view each Riemannian manifold as a submanifold in a Euclidean space, the problem of discovering simple sharp relationships between intrinsic and extrinsic invariants of a Riemannian submanifold becomes one of the most fundamental problems in submanifold theory. The main extrinsic invariant of a submanifold is the squared mean curvature and the main intrinsic invariants of a manifold include the Ricci curvature and the scalar curvature. There were also many other important modern intrinsic invariants of (sub)manifolds introduced by Chen such as $k$-Ricci curvature (see [2-4]).

In 1999, Chen [5] proved a basic inequality involving the Ricci curvature and the squared mean curvature of submanifolds in a real space form $R^{m}(C)$. This inequality is now called Chen-Ricci inequality [6]. In [5], Chen also defined the $k$-Ricci curvature of a $k$-plane section of $T_{x}M^n$, $x\in M$, where $M^n$ is a submanifold of the real space form $R^{n+p}(C)$. And he proved a basic inequality involving the $k$-Ricci curvature and the squared mean curvature of the submanifold $M^n$. These inequalities described relationships between the intrinsic invariants and the extrinsic invariants of a Riemannian submanifold and drew attentions of many people. Similar inequalities are studied for different submanifolds in various ambient manifolds (see [7-10]).

On the other hand, Hayden [11] introduced a notion of a semi-symmetric connection on a Riemannian manifold. Yano [12] studied Riemannaian manifolds endowed with a semi-symmetric connection. Nakao [13] studied submanifolds of Riemannian manifolds with a semi-symmetric metric connection. Recently, Mihai and Özgür [14, 15] studied Chen inequalities for submanifolds of real space forms admitting a semi-symmetric metric connection and Chen inequalities for submanifolds of complex space forms and Sasakian space forms with a semi-symmetric metric connection, respectively. Motivated by studies of the above authors, in this paper we establish Chen-Ricci inequalities for submanifolds in generalized complex forms with a semi-symmetric metric connection.

Let $ N^{n+p}$be an $(n+p)$-dimensional Riemannian manifold with Riemannian metric $g$ and a linear connection $\overline{\nabla}$ on $N^ {n+p}$. If the torsion tensor $\overline{T}$ of $\overline{\nabla} $, defined by

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

for any vector fields $\overline{X}$ and $\overline{Y}$ on $N^{n+p}$, satisfies

$\overline{T}(\overline {X}, \overline {Y})= \phi (\overline{Y})\overline{X}-\phi(\overline{X})\overline{Y}$

for a 1-form $\phi$, then the connection $\overline{\nabla}$ is called a semi-symmetric connection. Furthermore, if $\overline{\nabla}$ satisfies $\overline{\nabla}g=0$, then $\overline{\nabla}$ is called a semi-symmetric metric connection. Let $\overline{\nabla}'$ denote the Levi-Civita connection with respect to the Riemannian metric $g$. In [12] Yano gave a semi-symmetric metric connection $\overline{\nabla}$ which can be written as

$\begin{equation} \overline{\nabla}_{\overline{X}}\overline{Y}=\overline{\nabla}'_{\overline{X}}\overline{Y}+\phi(\overline{Y})\overline{X} -g(\overline{X}, \overline{Y})U \end{equation}$

(2.1)

for any vector field $\overline{X}$, $\overline{Y}$ on $ N^{n+p}$, where U is a vector field given by $g(U, \overline{X})=\phi(\overline{X})$.

Let $M^n$ be an $n$-dimensional submanifold of $N^{n+p}$ with a semi-symmetric metric connection $\overline{\nabla}$ and the Levi-Civita connection $\overline{\nabla}'$. On the submanifold $M^n$ we consider the induced semi-symmetric metric connection denoted by $\nabla$ and the induced Levi-Civita connection denoted by $\nabla'$. The Gauss formulas with respect to $\nabla$ and $\nabla'$, respectively, can be written as

${\bar \nabla _X}Y = {\nabla _X}Y + h(X, Y), \quad {\bar \nabla '_X}Y = {\nabla '_X}Y + h'(X, Y)$

(2.2)

for any vector fields $X, Y$ on $M^n$, where $h'$ is the second fundamental form of $M^n$ in $N^ {n+p}$ and $h$ is a (0, 2)-tensor on $M^n$. According to formula (7) in [13], $h$ is also symmetric.

Let $\overline{R}$ be the curvature tensor of $N^ {n+p}$ with respect to $\overline{\nabla}$ and $\overline{R}'$ be the curvature tensor of \(N^ {n+p}\) with respect to $\overline{\nabla}'$. We also denote by $R$ and $R' $ the curvature tensor of $\nabla$ and $\nabla'$, respectively, on \(M^n\). From [13], we know the curvature tensor $\overline{R}$ with respect to the semi-symmetric metric $\overline{\nabla}$ on $N^{n+p}$ can be written as

$\begin{equation} \begin{split} \overline{R}( X, Y, Z, W)=&\overline{R}' (X, Y, Z, W)-\alpha( Y, Z)g( X, W)+\alpha( X, Z)g( Y, W)\\ &-\alpha( X, W)g( Y, Z)+\alpha( Y, W)g( X, Z)\\ \end{split} \end{equation}$

(2.3)

for any vector fields $ X, Y, Z, W $on $M^{n}$, where $\alpha$ is a (0, 2)-tensor field defined by

$\alpha( X, Y)=(\overline{\nabla}'_{ X}{\phi}) Y-\phi( X)\phi( Y)+ \frac{1}{2}\phi(U)g( X, Y).$

Denote by $\lambda$ the trace of $\alpha$. The Gauss equation for the submanifold $M^n$ in $N^{n+p}$ is

$\begin{equation} \overline{R}'(X, Y, Z, W)=R'(X, Y, Z, W)+g(h'(X, Z), h'(Y, W))-g(h'(X, W), h'(Y, Z))\end{equation}$

(2.4)

for any vector fields $X, Y, Z, W $on $M^{n}$. In [13], the Gauss equation with respect to the semi-symmetric metric connection is

$\begin{equation} \overline{R}(X, Y, Z, W)=R(X, Y, Z, W)+g(h(X, Z), h(Y, W))-g(h(X, W), h(Y, Z)).\end{equation}$

(2.5)

In $N^ {n+p}$ we can choose a local orthonormal frame $\{e_1, \cdots, e_n, e_{n+1}, \cdots, e_{n+p}\}$ such that restricting to $M^n$, $e_1, \cdots, e_n$ are tangent to $M^n$. Setting $h_{ij}^r=g(h(e_i, e_j), e_r)$, then the squared length of $h$ is

$||h|{|^2} = \sum\limits_{i,j = 1}^n g (h({e_i},{e_j}),h({e_i},{e_j})) = \sum\limits_{r = n + 1}^{n + p} {\sum\limits_{i,j = 1}^n {(h_{ij}^r} } {)^2}.$

The mean curvature vector of $M^n$ associated to $\overline{\nabla}$ is $H=\frac{1}{n}\sum\limits_{i=1}^{n}h(e_i, e_i)$ and the mean curvature vector of $M^n$ associated to $\overline{\nabla}'$ is $H'$=$\frac{1}{n}\sum\limits_{i=1}^{n}h'(e_i, e_i)$.

Let $\pi \subset T_{x}M^n$ be a 2-plane section for any $x \in M^n$ and $K(\pi)$ be the sectional curvature of $\pi$ associated to the induced semi-symmetric metric connection ${\nabla}$. The scalar curvature $\tau$ at $x$ with respect to ${\nabla}$ is defined by

$\tau (x) = \sum\limits_{1 \le i{\rm{ }}＜j \le n} {K({e_i} \wedge {e_j}).} $

(2.6)

The following lemmas will be used in the paper.

Lemma 2.1(see [13]) If U is a tangent vector field on $M^n$, we have $H=H', \hspace{2mm} h=h'$.

Lemma 2.2(see [13]) Let $M^n$ be an $n$-dimensional submanifold of an $(n+p)$-dimensional Riemannian manifold $N^{n+p}$ with the semi-symmetric metric connection $\overline{\nabla}$. Then

(i) $M^n$ is totally geodesic with respect to the Levi-Civita connection and with respect to the semi-symmetric metric connection if and only if $U$ is tangent to $M^n$.

(ii) $M^n$ is totally umbilical with respect to the Levi-Civita connection if and only if $M^n$ is totally umbilical with respect to the semi-symmetric metric connection.

Lemma 2.3(see [10]) Let $f(x_1, x_2, \cdots, x_n)$ be a function on $R^n$ defined by

$f({x_1},{x_2}, \cdots ,{x_n}) = {x_1}\sum\limits_{i = 2}^n {{x_i}} .$

If $x_1+x_2+\cdots+x_n=2\varepsilon$, then we have

$f(x_1, x_2, \cdots, x_n)\leq \varepsilon^2$

with the equality holding if and only if $x_1=x_2+x_n+\cdots+x_n=\varepsilon $.

A 2$m$-dimensional almost Hermitian manifold $(N, J, g)$ is said to be a generalized complex space form (see [16,17]) if there exists two functions $F_1$ and $F_2$ on $N$ such that

$\begin{array}{l} \bar R'(\bar X,\bar Y,\bar Z,\bar W) = \\ {F_1}[g(\bar Y,\bar Z)g(\bar X,\bar W) - g(\bar X,\bar Z)g(\bar Y,\bar W)]\\ + {F_2}[g(\bar X,J\bar Z)g(J\bar Y,\bar W)\\ - g(\bar Y,J\bar Z)g(J\bar X,\bar W) + 2g(\bar X,J\bar Y)g(J\bar Z,\bar W)] \end{array}$

(2.7)

for any vector fields $\overline X, \overline Y, \overline Z, \overline W$ on $N$, where $\overline{R}'$ is the curvature tensor of $N$ with respect to the Levi-Civita connection $\overline{\nabla}'$. In such a case, we will write $N(F_1, F_2)$. If $N(F_1, F_2)$ is a generalized complex space form with a semi-symmetric metric connection $\overline{\nabla}$, using (2.3) and (2.7), the curvature tensor $\overline{R}$ with respect to the semi-symmetric metric connection $\overline{\nabla}$ of $N(F_1, F_2)$ can be written as

$\begin{array}{l} \bar R(X,Y,Z,W) = \\ {F_1}[g(Y,Z)g(X,W) - g(X,Z)g(Y,W)]\\ + {F_2}[g(X,JZ)g(JY,W) - g(Y,JZ)g(JX,W)\\ + 2g(X,JY)g(JZ,W)]\\ - \alpha (Y,Z)g(X,W) + \alpha (X,Z)g(Y,W)\\ - \alpha (X,W)g(Y,Z) + \alpha (Y,W)g(X,Z) \end{array}$

(2.8)

for $X, Y, Z, W$ on $M$, where $M$ is a submanifold of $N$.

Let $M$ be an $n$-dimensional submanifold of a 2$m$-dimensional generalized complex space form $N(F_1, F_2)$. We set $JX=PX+FX$ for any vector field $X$ tangent to $M$, where $PX$ and $FX$ are tangential and normal components of $JX$, respectively.

In this section, we establish a sharp relation between the Ricci curvature along the direction of an unit tangent vector $X$ and the mean curvature $||H||$ with respect to the semi-symmetric metric connect $\overline{\nabla}$.

Theorem 3.1 Let $M^n, \hspace{2mm}n\geq 2$, be an $n$-dimensional submanifold of a $2m$-dimensional generalized complex space form $N(F_1, F_2)$ endowed with the semi-symmetric metric connection $\overline{\nabla}$. For each unit vector $X \in T_{x}M$, we have

(1)

$\begin{equation}{\rm Ric}(X)\leq (n-1)F_1+3F_2||PX||^2-(n-2)\alpha(X, X)-\lambda+\frac{n^2}{4}||H||^2. \end{equation}$

(3.1)

(2) If $H(x)=0$, then a unit tangent vector $X$ at $x$ satisfies the equality case of (3.1) if and only if $X\in N(x)=\{X\in T_{x}M:h(X, Y)=0, \hspace{2mm}\forall Y\in T_{x}M\}$.

(3) The equality of inequality (3.1) holds identically for all unit tangent vectors at $x$ if and only if in the case of $n\neq 2, \hspace{2mm}h_{ij}^{r}=0, \hspace{2mm}i, j=1, 2\cdots, n;\hspace{2mm} r=n+1, \cdots, 2m$, or in the case of $n=2, \hspace{2mm}h_{11}^{r}=h_{22}^{r}, \hspace{2mm}h_{12}^{r}=h_{21}^{r}=0, \hspace{2mm}r=3, \cdots, 2m$.

Proof (1) Let $X\in T_{x}M$ be an unit tangent vector at $x$. We choose an orthonormal basis $e_1, \cdots, \hspace{2mm}e_n, e_{n+1}\cdots, e_{2m}$ such that $e_1, \cdots, e_n$ are tangent to $M$ at $x$ and $e_{1}=X$.

When we set $X= W=e_i, \hspace{2mm} Y= Z=e_j, \hspace{2mm}i, j=1, \cdots, n, \hspace{2mm} i\neq j$ in (2.5) and (2.8), we have

$R_{ijji}=F_1+3F_{2}g^2(Je_i, e_j)-\alpha(e_i, e_i)-\alpha(e_j, e_j)+\sum\limits_{r = n + 1}^{2m} {[h_{ii}^rh_{jj}^r-{{(h_{ij}^r)}^2}].} $

(3.2)

Using (3.2), we get

$\begin{equation} \begin{split} {\rm Ric}(X)=&\sum_{j=2}^{n}R_{1jj1}=(n-1)F_1+\sum_{j=2}^{n}3F_{2}g^2({JX, e_j)}\\&-(n-1)\alpha(X, X)-\sum_{j=2}^{n}\alpha(e_j, e_j) +\sum_{r=n+1}^{2m}\sum_{i=2}^{n}[h_{11}^{r}h_{ii}^{r}-(h_{1i}^{r})^2]\\ \leq& (n-1)F_1+3F_2||PX||^2-(n-2)\alpha(X, X)-\lambda+\sum_{r=n+1}^{2m}\sum_{i=2}^{n}h_{11}^{r}h_{ii}^{r}.\\ \end{split} \end{equation}$

(3.3)

We consider the maximum of the function

${f_r}(h_{11}^r, \cdots, h_{nn}^r) = \sum\limits_{i = 2}^n {h_{11}^r} h_{ii}^r$

under the condition $h_{11}^{r}+h_{22}^{r}+\cdots +h_{nn}^{r}=k^r, $ where $k^r$ is a real constant.

From Lemma 2.3 we know the solution $(h_{11}^{r}, \cdots, h_{nn}^{r})$ of this problem must satisfy

$h_{11}^r = \sum\limits_{i = 2}^r {h_{ii}^r} = \frac{{{k^r}}}{2}.$

(3.4)

So it follows that

${f_r} \le \frac{{{{({k^r})}^2}}}{4} = \frac{1}{4}{(\sum\limits_{i = 1}^n {h_{ii}^r} )^2}.$

(3.5)

From (3.3) and (3.5) we have

$\begin{array}{l} {\rm{Ric}}(X) \le (n - 1){F_1} + 3{F_2}||PX|{|^2} - (n - 2)\alpha (X, X) - \lambda + \sum\limits_{r = n + 1}^{2m} {\frac{1}{4}} {(\sum\limits_{i = 1}^n {h_{ii}^r} )^2}\\ = (n - 1){F_1} + 3{F_2}||PX|{|^2} - (n - 2)\alpha (X, X) - \lambda + \frac{{{n^2}}}{4}||H|{|^2}. \end{array}$

(2) For the unit vector $X$ at $x$, if the equality case of inequality (3.1) holds, using (3.3), (3.4) and (3.5) we have

$h_{1i}^r = 0, i \ne 1, \forall r;$

(3.6)

$h_{11}^r + h_{22}^r + \cdots + h_{nn}^r - 2h_{11}^r = 0, \forall r.$

(3.7)

From $H(x)=0$, we have $h_{11}^{r}=0, $ then $h_{1j}^{r}=0, \hspace{2mm}\forall j, r.$ So we get X$\in N(x)=\{X\in T_{x}M:h(X, Y)=0, \hspace{2mm}\forall Y\in T_{x}M\}$.

The converse is obvious.

(3) For all unit vector $X$ at $x$, the equality case of inequality (3.1) holds. Let $X=e_i, \hspace{2mm}i=1, 2\cdots n$, as in (2), we have

$\begin{array}{l} h_{ij}^r = 0, i \ne j, \forall r;\\ h_{11}^r + h_{22}^r + \cdots + h_{nn}^r - 2h_{ii}^r = 0, \forall i = 1, \cdots, n;r = n + 1, \cdots, 2m. \end{array}$

We can distinguish two cases:

(a) in the case of $n\neq 2$, we have $h_{ij}^{r}=0, \hspace{2mm} i, j=1, 2, \cdots, n, \hspace{2mm} r=n+1, \cdots, 2m $.

(b) in the case of $n=2, $ we have $h_{11}^{r}=h_{22}^{r}, \hspace{2mm}h_{12}^{r}=h_{21}^{r}=0, \hspace{2mm} r=3, \cdots, 2m.$

The converse is trivial.

Corollary 3.2 If the equality case of inequality (3.1) holds for all unit tangent vector X of $M^n$, then we have

(1) the equality case of inequality (3.1) holds for all unit tangent vector X of $M^n$ if and only if $M^n$ is a totally umbilical submanifold;

(2) if $U$ is a tangent field on $M^n$ and $n\geq 3$, $M^n$ is a totally geodesic submanifold.

Proof (1) For $n=2$, from Theorem 3.1 we know the equality case of inequality (3.1) holds for all unit tangent vector $X$ of $M^2$ if and only if $M^2$ is a totally umbilical submanifold with respect to the semi-symmetric metric connection. Then from Lemma 2.2, $M^2$ is a totally umbilical submanifold with respect to the Levi-Civita connection.

For $n\geq 3$, from Theorem 3.1 we know the equality case of inequality (3.1) holds for all unit tangent vector $X$ of $M^n$ if and only $h_{ij}^{r}=0, \hspace{2mm} \forall i, j, r$. According to formula (7) from [13], we have $h_{ij}^{'r}= h_{ij}^r+k^{r}g_{ij}, $ where $k^r$ are real-valued functions on $M$. Thus we have $h_{ij}^{'r}=k^{r}g_{ij}$. So $M^n$ is a totally umbilical submanifold.

(2) If $U$ is a tangent vector field on $M^n$, from Lemma 2.1 we have $h'=h$. For $n\geq 3$, from Theorem 3.1 the equality case of inequality (3.1) holds for all unit tangent vector $X$ of $M^n$ if and only if $h_{ij}^{r}=0, \hspace{2mm} \forall i, j, r$. Thus we have $h_{ij}^{'r}=0, \hspace{2mm} \forall i, j, r$. So $M^n$ is a totally geodesic submanifold.

In this section, we establish a sharp relation between the $k$-Ricci curvature and the mean curvature $||H||$ with respect to the semi-symmetric metric connect $\overline{\nabla}$.

Let $L$ be a $k$-plane section of $T_{x}M^n, \hspace{2mm}x\in M^n$, and $X$ be a unit vector in $L$. We choose an orthonormal frame $e_1, \cdots, e_k$ of $L$ such that $e_1=X$. In [5] the $k$-Ricci curvature of $L$ at $X$ is defined by

${\rm Ric}_L(X)=K_{12}+K_{13}+\cdots+K_{1k}.$

(4.1)

The scalar curvature of a $k$-plane section $L$ is given by

$\tau (L) = \sum\limits_{1 \le i ＜ j \le k} {{K_{ij}}.} $

(4.2)

For an integer $k, \hspace{2mm}2\leq k\leq n, $ the Riemannian invariant $\Theta_k$ on $M^n$ at $x\in M^n$ defined by

$\Theta_{k}(x)=\frac{1}{k-1}\inf\{{\rm Ric}_L(X): L, X\}, $

(4.3)

where $L$ runs over all $k$-plane sections in $T_xM$ and $X$ runs over all unit vectors in $L$. From (2.6), (4.1) and (4.2) for any $k$-plane section $L_{i_1\cdots i_k}$ spanned by $\{e_{i_1}, \cdots, e_{i_k\}}$, it follows that

$\tau ({L_{{i_1} \cdots {i_k}}}) = \frac{1}{2}\sum\limits_{i \in \{ {i_1}, \cdots {i_k}\} } {{\rm{Ri}}{{\rm{c}}_{{i_1}, \cdots {i_k}}}({e_i})} $

(4.4)

and

$\tau (x) = \frac{1}{{C_{n - 2}^{k - 2}}}\sum\limits_{1 \le {i_1} \cdots {i_k} \le n} {\tau ({L_{{i_1} \cdots {i_k}}})} .$

(4.5)

From (4.3), (4.4) and (4.5), we have

$\tau(x)\geq \frac{n(n-1)}{2}\Theta_{k}(x).$

(4.6)

Theorem 4.1 Let $M^n, \hspace{2mm}n\geq 3, $ be an $n$-dimensional submanifold of a $2m$-dimensional generalized complex space form $N(F_1, F_2)$ endowed with a semi-symmetric connection $\overline{\nabla}$. Then we have

$||H||^2\geq \frac{2\tau}{n(n-1)}+\frac{2}{n}\lambda -F_1-\frac{3F_2}{n(n-1)}\|P\|^2.$

Proof For $x\in M^n$, let $\{e_1, \cdots, e_n\}$ and $\{e_{n+1}, \cdots, e_{2m }\}$ be an orthonormal basis of $T_{x}^ M$ and $T_{x}^\bot M$, respectively, where $e_{n+1}$ is parallel to the mean curvature vector $H$.

From (3.2), we have

$R_{ijji}=F_1+3F_2g^2(Je_i, e_j)-\alpha(e_i, e_i)-\alpha(e_j, e_j)+\sum\limits_{r = n + 1}^{2m} {[h_{ii}^rh_{jj}^r-{{(h_{ij}^r)}^2}].} $

(4.7)

Setting $||P||^2=\sum\limits_{i, j=1}^{n}{g}^2(Je_i, e_j)$. From (2.6), it follows that

$2\tau(x)=n(n-1)F_1+3F_2||P||^2-2(n-1)\lambda +n^2||H||^2-||h||^2.$

(4.8)

Then equation (4.8) can be also written as

$n^2||H||^2=2\tau+||h||^2+2(n-1)\lambda-n(n-1)F_1-3F_2||P||^2.$

(4.9)

We choose an orthonormal basis $\{e_1, \cdots, e_n, e_{n+1}, \cdots, e_{2m }\}$ such that $e_1, \cdots, e_n$ diagonalize the shape operator $Ae_{n+1}$, i.e.,

$A{e_{n + 1}} = \left( {\begin{array}{*{20}{c}} {{a_1}}&0& \cdots &0\\ 0&{{a_2}}& \cdots &0\\ \vdots&\vdots&\vdots&\vdots \\ 0&0& \cdots &{{a_n}} \end{array}} \right)$

and $Ae_r=(h_{ij}^r), \hspace{2mm}i, j=1\cdots\cdots n;\hspace{2mm}r=n+2, \cdots, 2m, \hspace{2mm}\mbox{trace} Ae_r=0$. So (4.9) turns into

${n^2}||H|{|^2} = 2\tau + \sum\limits_{i = 1}^n {a_i^2} + \sum\limits_{r = n + 2}^{2m} {\sum\limits_{i, j = 1}^n {{{(h_{ij}^r)}^2}} } + 2(n - 1)\lambda - n(n - 1){F_1} - 3{F_2}||P|{|^2}.$

(4.10)

On the other hand, we get

${(n||H||)^2} = {(\sum\limits_{i = 1}^n {{a_i}} )^2} \le n\sum\limits_{i = 1}^n {a_i^2}, $

which implies

$\sum\limits_{i = 1}^n {a_i^2} \ge n||H|{|^2}.$

(4.11)

From (4.10) and (4.11), it follows that

$n^2||H||^2\geq 2\tau +n||H||^2+2(n-1)\lambda -n(n-1)F_1-3F_2||P||^2, $

which means

$||H||^2\geq \frac{2\tau}{n(n-1)}+\frac{2}{n}\lambda -F_{1}- \frac{3F_2}{n(n-1)}||P||^2.$

(4.12)

Using Theorem 4.1 and (4.6) we can obtain the following theorem.

Theorem 4.2 Let $M^n, \hspace{2mm}n\geq 3$, be an $n$-dimensional submanifold of a $2m$-dimensional generalized complex space form $N(F_1, F_2)$ endowed with a semi-symmetric connection $\overline{\nabla}$. Then for any integer $k, \hspace{2mm}2\leq k \leq n$, and for any point $x\in M$, we have

$||H||^2(x) \geq \Theta_{k}(x)+ \frac{2}{n}\lambda -F_1-\frac{3F_2}{n(n-1)}||P||^2.$

Proof Let $\{e_1, \cdots, e_n\}$ be an orthonormal basis of $T_xM^n$ at $x\in M^n$. The $k$-plane section spanned by $e_{i_1}, \cdots, e_{i_k}$ is denoted by $L_{i_1\cdots i_k}$.

Then from (4.6) and (4.12), we have

$||H||^2(x)\geq \Theta_k{(x)}+\frac{2}{n}\lambda -F_1-\frac{3F_2}{n(n-1)}||P||^2.$

Remark 4.3 For $F_1=F_2=C$ ($C$ is constant) in Theorem 3.1, we obtain a Chen-Ricci inequality for submanifolds of complex space forms with a semi-symmetric metric connection.

For $F_1=F_2=C$ ($C$ is constant) in Theorem 4.1 and Theorem 4.2, the results can be found in [15].