By using the operator

The study of submanifolds of finite type began in 1970, with Chen's attempts to find the best possible estimate of the total mean curvature of a compact submanifold of the Euclidean space and to find a notion of “degree” for Euclidean submanifolds [

In algebraic geometry varieties are the main objects to study. Since an algebraic variety is defined by using algebraic equations, one can define the degree of an algebraic variety by its algebraic structure, and it is well known that the concept of degree plays a fundamental role in algebraic geometry [

On the other hand, in differential geometry, the main objects to study are Riemannian (sub) manifolds. According to Nash’s immersion Theorem, every Riemannian manifold can be realized as a submanifold of the Euclidean space via an isometric immersion [

So inspired by algebraic geometry, in 1970, Chen defined the notions of order and type for submanifolds of the Euclidean space by the use of the Laplace operator. After that, Chen was able to obtain some sharp estimates of the total mean curvature for compact submanifolds of the Euclidean space in terms of their orders. Moreover, he could introduce submanifolds and maps of finite type [

On one hand finite-type submanifolds provides a natural way to exploit the spectral theory to study the geometry of submanifolds and smooth maps, in particular the Gauss map. On the other hand the techniques of the submanifold theory can be used in the study of spectral geometry via the study of finite-type submanifolds.

As is well known, the Laplace operator of a hypersurface

In contrast to the operator

In this paper, by using the operator

In this section, we recall some prerequisites about Newton transformations

Consider an orientable isometrically immersed hypersurface

The classical Newton transformations

Each

Associated to each Newton transformation

Since

As mentioned in the introduction, there is no notion of degree for submanifolds of the Euclidean space in general. However, Chen could use the induced Riemannian structure on a submanifold to introduce a pair of well-defined numbers

Consider an isometrically immersed closed orientable hypersurface

It is well known that the eigenvalues of

For each function

Consider the set

For an isometrically immersed closed hypersurface

By using the above notation, we have the following spectral decomposition of

The immersion

The following Lemma states that for an isometrically immersed closed orientable hypersurface

Let

Consider the decomposition

This shows that

On the set of all

For an isometric immersion

Since

Before we give our main result and to facilitate the reader, we quote Theorem

Let

There is no compact Euclidean hypersurface of

If

The formula (

Since

In this section, we will relate the notion of the

A classical theorem of Hadamard [

Let

The second fundamental form is definite at every point of

The Gauss-Kronecker curvature never vanishes on

Moreover, any of the above conditions implies that

Here we observe that the convexity of a hypersurface in

Let

The Ricci curvature of

Let

In [

Let

Now we establish the corresponding result for the operator

Let

We will follow the techniques introduced by Chen (Theorem

By applying Theorem

Let

By using the concept of order Chen in [

Let

Now we establish the corresponding result for the operator

Let

We will follow the techniques introduced by Chen (Theorem

Combining (

For every

Combining (

By applying Theorem

Let

An immediate consequence of Corollaries

Let