## 2. Preliminaries

The theory of Chen invariants and Chen inequalities was initiated by B.-Y. Chen [

1,

2].

Let

$(M,g)$ be an

n-dimensional (

$n\ge 2$) Riemannian manifold and ∇ its Levi–Civita connection. One denotes by

R the Riemannian curvature tensor field on

M. For any

$p\in M$ and

$\pi \subset {T}_{p}M$ a plane section, the

sectional curvature $K\left(\pi \right)$ of

$\pi $ is defined by

$K\left(\pi \right)=R({e}_{1},{e}_{2},{e}_{1},{e}_{2})$, where we use the convention

$R({e}_{1},{e}_{2},{e}_{1},{e}_{2})=g(R({e}_{1},{e}_{2}){e}_{2},{e}_{1}),$ with

$\{{e}_{1},{e}_{2}\}$ an orthonormal basis of

$\pi $. Let

$\{{e}_{1},\dots ,{e}_{n}\}$ be an orthonormal basis of

${T}_{p}M$. The

scalar curvature $\tau $ at

p is given by

where

$K({e}_{i}\wedge {e}_{j})$ is the sectional curvature of the plane section spanned by

${e}_{i}$ and

${e}_{j}$ (other authors consider

$\tau \left(p\right)={\sum}_{1\le i\ne j\le n}K({e}_{i}\wedge {e}_{j})$).

The

Chen first invariant ${\delta}_{M}$ is defined by

The Chen invariant

$\delta (2,2)$, given by

was studied in [

3].

We shall consider the Chen invariant

$\underset{k\mathrm{terms}}{\underbrace{\delta (2,\dots ,2)}}$, denoted by

${\delta}^{k}(2,\dots ,2)$, which is given by

where

${\pi}_{1},\dots ,{\pi}_{k}$ are mutually orthogonal plane sections at

p.

Obviously, ${\delta}^{1}\left(2\right)={\delta}_{M}$.

In the next section, we shall prove an algebraic inequality and study its equality case. As an application we shall give a simple proof of the Chen inequality for the invariant ${\delta}^{k}(2,\dots ,2)$.

## 4. A Chen Inequality

As an application of Proposition 1, we give a simple proof of the Chen inequality for the Chen invariant ${\delta}^{k}(2,\dots ,2)$ of submanifolds in Riemannian space forms.

Let $\tilde{M}\left(c\right)$ be an m-dimensional Riemannian space form of constant sectional curvature c. The standard examples are the Euclidean space ${\mathbb{E}}^{m}$, the sphere ${S}^{m}$ and the hyperbolic space ${H}^{m}$.

Let

M be an

n-dimensional submanifold of

$\tilde{M}\left(c\right)$ and denote by

h the second fundamental form of

M in

$\tilde{M}\left(c\right)$. Recall that the mean curvature vector

$H\left(p\right)$ at

$p\in M$ is given by

where

$\{{e}_{1},\dots ,{e}_{n}\}$ is an orthonormal basis of

${T}_{p}M$.

The submanifold M is said to be minimal if $H\left(p\right)=0,\forall p\in M$.

The Gauss equation is (see [

4])

for any vector fields

$X,Y,Z,W$ tangent to

M.

**Theorem** **1.** Let $\tilde{M}\left(c\right)$ be an m-dimensional Riemannian space form of constant sectional curvature c and M an n-dimensional submanifold of $\tilde{M}\left(c\right)$. Then one has the following Chen inequality: Moreover, the equality holds at a point $p\in M$ if and only if there exist suitable orthonormal bases $\{{e}_{1},\dots ,{e}_{n}\}\subset {T}_{p}M$ and $\{{e}_{n+1},\dots ,{e}_{m}\}\subset {T}_{p}^{\perp}M$ such that the shape operators take the formswhere ${A}_{j}^{r}$ are symmetric $2\times 2$ matrices with trace ${A}_{j}^{r}=0$, $\forall j=1,\dots ,k$. **Proof.** Let $p\in M$, ${\pi}_{1},\dots ,{\pi}_{k}\subset {T}_{p}M$ be mutually orthogonal plane sections and $\{{e}_{1},{e}_{2}\}\subset {\pi}_{1},\dots ,\{{e}_{2k-1},{e}_{2k}\}\subset {\pi}_{k}$ be orthonormal bases. We construct $\{{e}_{1},\dots ,{e}_{2k},{e}_{2k+1},\dots ,{e}_{n}\}\subset {T}_{p}M$ and $\{{e}_{n+1},\dots ,{e}_{m}\}\subset {T}_{p}^{\perp}M$ orthonormal bases, respectively.

Denote by ${h}_{ij}^{r}=g(h({e}_{i},{e}_{j}),{e}_{r})$, $i,j=1,\dots ,n$, $r\in \{n+1,\dots ,m\}$, the components of the second fundamental form.

By the Gauss equation, we have

The Gauss equation also implies

By using the algebraic inequality from the previous section, we obtain

which implies the desired inequality.

If the equality case holds at a point

$p\in M$, then we have equalities in all the inequalities in the proof, i.e.,

for any

$r\in \{n+1,\dots ,m\}.$We choose ${e}_{n+1}$ parallel to $H\left(p\right)$. Then the shape operators take the above forms. □

**Corollary** **1.** Let $\tilde{M}\left(c\right)$ be an m-dimensional Riemannian space form of constant sectional curvature c and M an n-dimensional submanifold of $\tilde{M}\left(c\right)$. If there exists a point $p\in M$ such that ${\delta}^{k}(2,\dots ,2)\left(p\right)>\left(\right)open="["\; close="]">\frac{n(n-1)}{2}-k$, then M is not minimal.

For

$k=1$, one derives Chen’s first inequality (see [

1]).

**Corollary** **2.** Let $\tilde{M}\left(c\right)$ be an m-dimensional Riemannian space form of constant sectional curvature c and M an n-dimensional submanifold of $\tilde{M}\left(c\right)$. Then one has Equality holds if and only if, with respect to suitable frame fields $\{{e}_{1},\dots ,{e}_{n},$${e}_{n+1},\dots ,{e}_{m}\}$, the shape operators take the following forms: **Example** **1.** The generalized Clifford torus.

Let $T={S}^{k}\left(\frac{1}{\sqrt{2}}\right)\times {S}^{k}\left(\frac{1}{\sqrt{2}}\right)\subset {S}^{2k+1}\subset {\mathbb{E}}^{2k+2}$.

It is known that T is a minimal hypersurface of ${S}^{2k+1}$, but a non-minimal submanifold of ${\mathbb{E}}^{2k+2}$.

Obviously ${\delta}^{k}(2,\dots ,2)=\tau =2k(k-1)$.

Then $T\subset {S}^{2k+1}$ does not satisfy the equality case of Theorem 1.

If we consider $T\subset {\mathbb{E}}^{2k+2}$, then it satisfies the equality case of Theorem 1.

**Remark** **1.** By using the inequality from Proposition 1, we can obtain Chen inequalities for the invariant ${\delta}^{k}(2,\dots ,2)$ on submanifolds in other ambient spaces, for instance, complex space forms, Sasakian space forms, Hessian manifolds of constant Hessian curvature, etc.