It is well known that study on hypersurfaces and submanifolds in euclid space is one of fundamental tasks of differential geometry. For oriented and compact isometric immersion hypersurfaces in euclid space, references [1-3] ever established a classical integral formula, that is the following Theorem 1.1.

In this paper, we study an oriented and compact $ n $-dimension isometric immersion submanifold $ M^{n} $ in the $ (n+p) $-dimension euclid space $ R^{n+p} $. Let $ \xi $ be the unit mean curvature vector field of $ M^{n} $. At first, we define the higher order mean curvature $ H_{r} (r = 0, 1, 2, \cdots, n) $ along the direction $ \xi $. And then, by applying the method of moving frame and exterior differential, we attain a new integral formula, that is the following Theorem 1.2. When codimension $ p = 1 $, Theorem 1.2 becomes Theorem 1.1.

Theorem 1.1(see [1-3]) Let $ \varphi : M\rightarrow R^{n+1} $ be an oriented and compact isometric immersion hypersurface without boundary. Then the following integral formulas hold

$ \begin{equation*} \int_{M}(H_{k}+H_{k+1}\langle \varphi , N \rangle)dM = 0, \; \; k = 0, 1, 2, \cdots, n-1, \end{equation*} $

here $ N $ is the unit normal vector field to $ M $, $ H_{k} $ is the $ k $-th higher order mean curvature of $ M $ and $ \langle , \rangle $ is the euclid inner product in $ R^{n+1} $, $ dM $ is the $ n $-dimension Riemann volume form for $ M $.

Theorem 1.2 Let $ \varphi : M^{n} \rightarrow R^{n+p} $ be an oriented and compact $ n $-dimension isometric immersion submanifold without boundary. Then the following integral formulas hold

$ \begin{equation*} \int_{M}(H_{k}+H_{k+1}\langle \varphi , \xi \rangle)dM = 0, \; \; k = 0, 1, 2, \cdots, n-1, \end{equation*} $

here $ \xi $ is the unit mean curvature vector field to $ M^{n} $, $ H_{k} $ is the $ k $-th higher order mean curvature along the direction $ \xi $ and $ \langle, \rangle $ is the euclid inner product in $ R^{n+p} $, $ dM $ is the $ n $-dimension Riemann volume form for $ M^{n} $.

Let $ R^{n+p} $ be the $ (n+p) $-dimension euclid space and $ (M^{n}, g) $ be a smooth $ n $-dimension Riemann manifold. Denote by $ \varphi : M^{n}\rightarrow R^{n+p} $ a smooth immersion mapping between smooth manifolds. If the equation $ g = \varphi^{*}(\langle , \rangle) $ holds everywhere on $ M^{n} $, then $ M^{n} $ or $ \varphi (M^{n}) $ is called an isometric immersion submanifold in $ R^{n+p} $. Here $ \langle, \rangle $ is the euclid inner product of $ R^{n+p} $ and $ \varphi ^{*} $ is the pull-back mapping for the immersion mapping $ \varphi $.

In this paper, we prescribe the index range as

$ \begin{equation*} 1\leq i, j, k, l \leq n, \; \; n+1\leq \alpha, \; \; \; \beta \leq n+p, \; \; 1\leq A, \; \; \; B \leq n+p. \end{equation*} $

Denote by $ \{e_{A}\} $ a local unit orthogonal frame field for $ R^{n+p} $ such that when being confined onto $ M^{n} $, $ \{ e_{i }\} $ is a local unit tangent frame field to $ M^{n} $ and $ \{e_{\alpha}\} $ is a local unit normal frame field to $ M^{n} $.

Denote by $ \{\omega ^{A}\} $ the dual frame field for $ \{e_{A}\} $, then the second fundamental form $ II $ for $ M^{n} $ can be expressed in component form as

$ \begin{equation*} II = \sum\limits_{\alpha, i, j} h_{ii}^{\alpha} \; \omega ^{i}\otimes \omega ^{j} \otimes e_{\alpha}. \end{equation*} $

Define the mean curvature vector field $ \sigma $ to $ M^{n} $ as

$ \begin{equation*} \sigma = \frac{1}{n}\sum\limits_{\alpha, i} h_{ii}^{\alpha}e_{\alpha}. \end{equation*} $

It is well-known that the definition of $ \sigma $ is independent on the choice of the local unit orthogonal frame field $ \{e_{A}\} $.

We consider the unit mean curvature vector field $ \xi = \sigma / |\sigma|. $ Let $ \{ \lambda _{i}\} $ be the principal curvature functions along the direction $ \xi $, then the $ r $-th higher order mean curvature $ H_{r} $ $ ( r = 1, 2, ..., n) $ is defined as

$ \begin{equation*} H_{r} = (^{n}_{r})^{-1}\cdot (\sum\limits _{1 \leq i_{1}<...<i_{r} \leq n} \lambda_{i_{1}} \cdot \lambda_{i_{2}} \cdot\cdot\cdot \lambda_{i_{r}}), \end{equation*} $

here $ (^{n}_{r}) $ is the ordinary combination number. At the same time we define $ H_{0}\equiv 1 $.

Reference [3] ever attained a fundamental integral formula that is the integral of the Codazzi tensor field on an isometric immersion hypersurface in $ R^{n+1} $. Similar to reference [3], for $ n $-dimension isometric immersion submanifold $ M^{n} $ of $ R^{n+p} $, we attain the following Lemma 2.2. Here we firstly recall some relevant fundamental concepts and properties. Assume that $ S $ is a tensor field of type $ (k, k) $ on a Riemann manifold $ (M^{n}, g) $. If $ S $ is anti symmetric both to its each pair of covariant indices and to its each pair of contravariant indices, then we write

$ \begin{equation*} S\in \Gamma ( \text{End} \Lambda ^{k} (TM)). \end{equation*} $

For $ S\in \Gamma (\text{End}\; \Lambda ^{k} (TM)), \; T\in \Gamma (\text{End}\; \Lambda ^{l} (TM)) $, we also consider the tensor field of type $ (k+l, k+l) $,

$ \begin{equation*} S\ast T \in \Gamma (\text{End}\; \Lambda ^{k+l}(TM)), \end{equation*} $

and the definition of $ S\ast T $ is that the exterior product of covariant components of $ S $ and the covariant components of $ T $, and respectively the exterior product of contravariant components of $ S $ and the contravariant components of $ T $. And by reference [3], this product $ \ast $ is associative and commutative.

Definition 2.1 (see [3], Codazzi tensor field) Let $ (M^{n}, g) $ be a $ n $-dimension Riemann manifold and $ S\in \Gamma(\text{End}\; \Lambda ^{k}(TM)) $. If for all $ C^{\infty} $ vector field $ X_{1}, X_{2}, \cdots, X_{k+1} \in \Gamma (TM) $ we have

$ \begin{equation*} \sum\limits _{j} (-1)^{j+1} (\nabla _{X_{j}}S (X_{1}\wedge X_{2} \wedge ... \wedge X_{j-1} \wedge X_{j+1}\wedge ... \wedge X_{k+1})) = 0, \end{equation*} $

then $ S $ is called a Codazzi tensor field on $ M^{n} $, here $ \nabla $ is the Levi-Civita connection of $ (M^{n}, g) $.

According to reference [3], we know that if $ S $ and $ T $ are Codazzi tensor field respectively of type $ (k, k) $ and type $ (l, l) $ on $ (M^{n}, g) $, then $ S\ast T $ must be a Codazzi field tensor field of type $ (k+l, k+l) $ on $ M^{n} $. From reference [3], we also define a Codazzi tensor field $ A $ of type $ (1, 1) $ on $ M^{n} $. Let $ (M^{n}, g) $ be a $ n $-dimension Riemann manifold and $ \psi: M^{n}\rightarrow R^{n+p} $ be an isometric immersion mapping. Let $ Y $ be the position vector field of $ \psi (M^{n}) $ in $ R^{n+p} $, then the Codazzi tensor field $ A $ of type $ (1, 1) $ is determined by

$ \begin{equation*} \langle A(X), Z \rangle = \langle A(Z), X \rangle = \langle II(X, Z), Y\rangle = \langle II(X, Z), \psi \rangle , \; \; \forall X, Z \in \Gamma (TM), \end{equation*} $

here $ \langle , \rangle $ is the euclid inner product of $ R^{n+p} $ and $ II $ is the second fundamental form for $ (M^{n}, g) $.

Now we are ready to prove the following Lemma 2.2.

Lemma 2.2 let $ (M^{n}, g) $ be a $ n $-dimension Riemann manifold and $ \psi: M^{n}\rightarrow R^{n+p} $ be an oriented and isometric immersion mapping. Let $ \psi (M^{n}) $ be compact and be without boundary. Assume that $ S $ is a Codazzi tensor field of type $ (k, k) $ on $ M^{n} $, then the following integral formulas hold

$ \begin{equation*} \int _{M} \{ (n-k)\cdot \text{trace} (S)- \text{trace} (S\ast A)\}dV = 0, \; \; k = 0, 1, 2, \cdots, n-1. \end{equation*} $

Here $ dV $ is the $ n $-dimension Riemann volume form of $ M^{n} $ and $ trace $ is the trace operator.

Proof Denote by $ dV $ the $ n $-dimension Riemann volume form of $ M^{n} $, then the following equation

$ \begin{equation*} \alpha (X_{1}, X_{2}, \cdots, X_{n-1}) = dV (Y^{\text{tan}}, X_{1}, X_{2}, \cdots, X_{n-1}), \; \; \forall X_{1}, X_{2}, \cdots, X_{n-1}\in \Gamma (TM) \end{equation*} $

determines a $ (n-1) $ form and it is written as $ \alpha = Y^{\text{tan}} \rfloor dV $, here $ Y^{\text{tan}} $ is the tangent component to $ M^{n} $ of the position vector $ Y $ for $ \psi(M^{n}) $ in $ R^{n+p} $.

From reference [3] and direct computation, we have

$ \begin{equation*} \nabla _{X} \alpha = \{ X - A(X)\} \rfloor dV, \; \; \forall X \in \Gamma (TM), \end{equation*} $

here $ \nabla $ is the Levi-Civita connection of $ (M^{n}, g) $. At first we assume that $ S $ is a Codazzi tensor field of type $ (n-1, n-1) $.

Let $ \{e_{i}\} $ is an unit orthogonal frame for $ M^{n} $ and write

$ \begin{equation*} E_{j} = e_{1}\wedge \cdot\cdot\cdot \wedge e_{j-1} \wedge e_{j+1} \wedge \cdot \cdot \cdot \wedge e_{n}, \; \; E = e_{1}\wedge e_{2} \wedge \cdot\cdot\cdot \wedge e_{n}. \end{equation*} $

We can see $ \omega = \alpha \circ S $ as a $ (n-1) $ form which takes value in $ \Gamma ( \text{End}\; \Lambda ^{k}(TM) $. By the computation in reference [3], we have

$ \begin{align*} d \omega (e_{1}, e_{2}, \cdots, e_{n}) & = \sum (-1)^{j+1} (\nabla _{e_{j}}\omega)(E_{j}) \\ & = \sum (-1)^{j+1} (\nabla _{e_{j}}\alpha)\circ S(E_{j}) +\alpha \; \circ \; \sum (-1)^{j+1} ( \nabla _{e_{j}} S)( E_{j}). \end{align*} $

Because $ S $ is a Codazzi tensor field, the second term of the above equation vanishes and so we have

$ \begin{align*} d \omega (e_{1}, e_{2}, \cdots, e_{n}) & = \sum (-1)^{j+1} \{ e_{j}- A(e_{j}) \}\; \rfloor\; dV \circ S(E_{j}) \\ & = \sum (-1)^{j+1} \langle e_{j} \wedge S(e_{j}), E \rangle - \sum (-1)^{j+1}\langle A(e_{j})\wedge S(E_{j}), E\rangle \\ & = \sum \langle e _{j} \wedge S(E_{j}), e_{j} \wedge E_{j}\rangle - \sum \langle A(e_{j}) \wedge S(E_{j}), e_{j}\wedge E_{j} \rangle \\ & = \text{trace} \; S - \text{trace}\; S \ast A . \end{align*} $

Because $ M^{n} $ is compact and is without boundary, $ \int_{M}\omega dV = 0 $. And so Lemma 2.1 holds in the case that $ S $ is a Codazzi tensor field of type $ (n-1, n-1) $. Now we assume that $ S $ is a Codazzi tensor field of type $ (k, k) $. Denote by $ I $ the identity element of $ \Gamma (\text{End}\; \Lambda ^{n-k-1}(TM)) $.

Because $ I $ is parallel, $ I\ast S $ is a Codizza tensor field of type $ (n-1, n-1) $. So from the above conclusion we have

$ \begin{equation*} \int_{M}\{\; \text{trace}\; (S \ast I)- \text{trace}\; (S \ast A \ast I)\; \}dV = 0, \; \; k = 0, 1, \cdots, n-1. \end{equation*} $

Finally we notice that

$ \begin{equation*} \text{trace}\; (S \ast I) = (n-k)\; \text{trace}\; (S)\; , \; \; \text{trace}\; (S \ast A \ast I) = \text{trace}\; (S \ast A), \end{equation*} $

we already finish the proof of Lemma 2.2.

Theorem 1.2 Let $ \varphi : M^{n} \rightarrow R^{n+p} $ be an oriented and compact $ n $-dimension isometric immersion submanifold without boundary. Then the following integral formulas hold.

$ \begin{equation*} \int_{M}(H_{k}+H_{k+1}\langle \varphi , \xi \rangle)dM = 0, \; \; k = 0, 1, 2, \cdots, n-1, \end{equation*} $

here $ \xi $ is the unit mean curvature vector field to $ M^{n} $, $ H_{k} $ is the $ k $-th higher order mean curvature along the direction $ \xi $ and $ \langle, \rangle $ is the euclid inner product in $ R^{n+p} $, $ dM $ is the $ n $-dimension Riemann volume form for $ M^{n} $.

Proof Let $ T_{\xi} $ be the shape operator of $ M^{n} $ along the direction of the unit mean curvature vector field $ \xi $, that is to say, $ T_{\xi} $ is a tensor field of type $ (1, 1) $ on $ M^{n} $ defined by

$ \begin{equation*} T_{\xi}(X) = - ( \overline{\nabla}_{X}\xi )^{\top}, \; \; \forall \; X\in \Gamma (TM), \end{equation*} $

here $ \overline{\nabla} $ is the Levi-Civita connection of $ R^{n+p} $.

Because the Levi-Civita connection $ \overline{\nabla} $ is flat, by the Codazzi equation for submanifold (see [4]), we know that $ T_{\xi} $ is a Codazzi tensor field of type $ (1, 1) $ on $ M^{n} $.

Denote by $ \lambda_{1}, \lambda_{2}, \cdots, \lambda_{n} $ the characteristic values of $ T_{\xi} $ and by $ \sigma _{r} $ the $ r $-th fundamental homogeneous symmetry polynomial, that is

$ \begin{equation*} \sigma _{r} = \sum\limits _{j_{1}<...<j_{r}} \lambda _{j_{1}} \cdot \cdot \cdot \lambda_{ j_{r}}, \; \; r = 1, 2, \cdots, n\; . \end{equation*} $

Let $ S = T^{k}_{\xi} = T_{\xi}\ast T_{\xi}\ast ...\ast T_{\xi} $ be the $ k $-times $ \ast\; $ product. According to reference [3], we know that $ S $ is a Codazzi tensor field of type $ (k, k) $. And by direct computation, we have

$ \begin{equation} \text{trace}\; (S) = k!\; \sigma_{k}\; . \end{equation} $

(3.1)

Denote by

$ \begin{equation*} h = h(x) = \langle \varphi (x), \xi (x) \rangle , \; \; x\in M \end{equation*} $

the support function of $ M^{n} $ along the direction $ \xi $. Then it is easy to see $ A = \; -hT_{\xi}. $ By direct computation, we have

$ \begin{equation} \text{trace}\; (S\ast A) = \; -h\; (k+1)!\; \sigma_{k+1}\; . \end{equation} $

(3.2)

Now we recall once again the definition of the higher order mean curvature $ H_{r} $ along the unit mean curvature vector field $ \xi $

$ \begin{equation*} H_{k} = (^{n}_{k})^{-1}\sigma _{k}, \; \; k = 1, 2, \cdots, n. \end{equation*} $

We notice the above (3.1), (3.2) and then we apply Lemma 2.2, we already finish the proof for Theorem 1.2.