It is well known that translation hypersurfaces are very important in Differential Geometry, providing an interesting class of constant mean curvature hypersurfaces and minimal hypersurfaces in a number of spaces endowed with good symmetries and even in certain applications in Microeconomics. There are many results about them, for instance, Chen et al. (2003), Dillen et al. (1991), Inoguchi et al. (2012), Lima et al. (2014), Liu (1999), López (2011), López and Moruz (2015), López and Munteanu (2012), Seo (2013) and Chen (2011), for an interesting application in Microeconomics.
Scherk (1835) obtained the following classical theorem: Let be a translation surface in , if is minimal then it must be a plane or the Scherk surface defined by
where is a nonzero constant. In a different aspect, Liu (1999) considered the translation surfaces with constant mean curvature in -dimensional Euclidean space and Lorentz-Minkowski space and Inoguchi et al. (2012) characterized the minimal translation surfaces in the Heisenberg group , and López and Munteanu, the minimal translation surfaces in .
The concept of translation surfaces was also generalized to hypersurfaces of by Dillen et al. (1991), who obtained a classification of minimal translation hypersurfaces of the -dimensional Euclidean space. A classification of the translation hypersurfaces with constant mean curvature in -dimensional Euclidean space was made by Chen et al. (2003).
The absence of an affine structure in hyperbolic space does not permit to give an intrinsic concept of translation surface as in the Euclidean setting. Considering the half-space model of hyperbolic space, López (2011), introduced the concept of translation surface and presented a classification of the minimal translation surfaces. Seo (2013) has generalized the results obtained by Lopez to the case of translation hypersurfaces of the -dimensional hyperbolic space.
Definition 1. We say that a hypersurface of the Euclidean space is a translation hypersurface if it is the graph of a function given by
where are cartesian coordinates and each is a smooth function of one real variable for .
Now, let be an oriented hypersurface and denote the principal curvatures of . For each , we can consider similar problems to the above ones, related with the -th elementary symmetric polynomials, , given by
In particular, is the mean curvature, the scalar curvature and the Gauss-Kronecker curvature, up to normalization factors. A very useful relationship involving the various is given in the [Proposition 1, Caminha (2006)]. This result will play a central role along this paper.
Recently, some authors have studied the geometry of translational hypersurfaces under a condition in the curvature, where . Namely, Leite (1991) gave a new example of a translation hypersurface of with zero scalar curvature. Lima et al. 2014 presented a complete description of all translation hypersurfaces with zero scalar curvature in the Euclidean space and Seo 2013 proved that if is a translation hypersurface with constant Gauss-Kronecker curvature in , then is congruent to a cylinder, and hence .
In this paper, we obtain a complete classification of translation hypersurfaces of with . We prove the following
Theorem 1. Let be a translation hypersurface in . Then, for , has zero curvature if, and only if, it is congruent to the graph of the following functions
on , for certain intervals , and arbitrary smooth functions . Which defines, after a suitable linear change of variables, a vertical cylinder, or
on , with and are real constants where and nonzero, , are open intervals defined by the conditions while is defined by .
Theorem 2. Any translation hypersurface in with constant, for , must have .
Finally, we observe that, when one considers the upper half-space model of the -dimensional hyperbolic space , that is,
endowed with the hyperbolic metric then, unlike in the Euclidean setting, the coordinates are interchangeable, but the same does not happen with the coordinate and, due to this observation, López 2011 and Seo 2013 considered two classes of translation hypersurfaces in :
A hypersurface is called a translation hypersurface of type I (respectively, type II) if it is given by an immersion satisfying
where each is a smooth function of a single variable. Respectively, in case of type II,
Theorem 3(Theorem 3.2, Seo 2013). There is no minimal translation hypersurface of type I in .
and with respect to type II surfaces he proved
Theorem 4(Theorem 3.3, Seo 2013). Let be a minimal translation surface of type II given by the parametrization . Then the functions and are as follows:
where , , and are constants.
We emphasize that the result proved by Seo, Theorem 3.2 of Seo 2013, implies that our result (Theorem 2) is not valid in the hyperbolic space context.
PRELIMINARIES AND BASIC RESULTS
Let be a connected Riemannian manifold. In the remainder of this paper, we will be concerned with isometric immersions, , from a connected, -dimensional orientable Riemannian manifold, , into . We fix an orientation of , by choosing a globally defined unit normal vector field, , on . Denote by , the corresponding shape operator. At each , restricts to a self-adjoint linear map . For each , let be the smooth function such that denotes the -th elementary symmetric function on the eigenvalues of , which can be defined by the identity
where by definition. If and is a basis of , given by eigenvectors of , with corresponding eigenvalues , one immediately sees that
where is the -th elementary symmetric polynomial on . Consequently,
In the next result we present an expression for the curvature of a translation hypersurface in the Euclidean space. This expression will play an essential role in this paper.
Proposition 1. Let be a smooth function, defined as , where each is a smooth function of one real variable. Let be the graphic of , given in coordinates by
The curvature of is given by
where the dot represents derivative with respect to the corresponding variable, that is, for each , one has and
Proof. Let be as stated in the Proposition, denote by the Euclidean gradient of and the standard Euclidean inner product. Then, we have
and the coordinate vector fields associated to the parametrization given in (2) have the following form
Hence, the elements of the metric of are given by
implying that the matrix of the metric has the following form
where is the identity matrix of order . Note that the -th column of , which will be denoted by , has the expression given by the column vector
An easy calculation shows that the unitary normal vector field of satisfies
where . Thus, the second fundamental form of satisfies
implying that the matrix of is diagonal
with -th column given by the column vector
If denotes the matrix of the Weingarten mapping, then . In (1), changing by gives
Thus, we conclude that the expression for curvature can be found by the following calculation
Due to the multilinearity of function , on its column vectors, it follows immediately that
leading to the conclusion
Now, applying the expressions (4) and (5) in (6) we reach to the expression
Calculating the determinant on the right in the equality above, we get
Consequently, the expression for in (7) assumes the following form
Finally, using that we obtain the desired expression
In order to prove Theorem 1 we need the following lemma.
Lemma 1. Let be smooth functions of one real variable satisfying the differential equation
where is a positive real constant and the big hat means an omitted term. If , for each then
where , are real constants with .
Since the derivatives it follows that . Thus dividing (8) by this product we get the equivalent equation:
which implies, after taking derivative with respect to for each , that , thus for some non null constant . Thus, setting
which can be easily solved to give:
Now, since it implies that , from (9) it follows that
where . ∎
With this lemma at hand we can go to the proof of Theorem 1.
Proof of the Theorem 1 From Proposition 1, we have that has zero curvature if, and only if,
In order to ease the analysis, we divide the proof in four cases.
Case 1: Suppose , . In this case, we have no restrictions on the functions . Thus
where and for , the functions are arbitrary smooth functions of one real variable. Note that the parametrization obtained comprise hyperplanes.
Case 2: Suppose , , then, there are constants such that , for . From (10) we have
from which we conclude that for some and thus, this case is contained in the Case .
Case 3: Now suppose , and , for every . Observe that if we had for some the analysis would reduce to the Cases 1 and 2. In this case, there are constants such that for any . From (10) we have
where and the hat means an omitted term. Then, from Lemma 1 we have that
where and are real constants, and , and are nonzero.
Case 4: Finally, suppose , where and , and for any . We will show that this case cannot occur. In fact, note that for any fixed
Derivative with respect to the variable , in the above equality, gives
That is, if we set
then, it follows that do not depend on the variable and we can write
We have two possible situations to take into account: Case I. , and Case II. there is an such that .
Case I. : Under this assumption, there are constants such that equation (12) becomes . Furthermore, it can be shown that for
Since it follows that , and using that we obtain
Now, for , substitute in (14) to obtain the identity
for any . Hence we conclude that,
These equalities, from [Proposition 1, Caminha (2006)], imply that at most of the constants are nonzero. If with , in the expression obtained for , making and taking derivatives with respect to the variables we get
for all . As for all , we obtain that
for all . Consequently,
Since , at most of the functions are nonzero, for and , leading to a contradiction. So, for all , which implies that is constant for all . Now, again from equation (11) we get
From which, we conclude that
Therefore, at most of the functions are nonzero, leading to a contradiction. Thus, it follows that Case 4 cannot occur, if for every .
Case :In this case, we have implying
Derivative of with respect to variable , for and , gives
Now, for with distinct indices, taking the derivatives of with respect to gives
Consequently, for at most indices, say , we can have , , and for every , with . Thus , with , together with equation implies that the sum
Now, if we have by equation (16) that
From which, we conclude that
Thus, for at most indices we must have , for every , and . This contradicts the hypothesis assumed in Case 4. Hence, cannot occur. Since the case , cannot occur as well, it follows that Case 4 is not possible. This completes the proof of the theorem. ∎
[Proof of the Theorem 2] Let be a translation hypersurface with constant curvature. First, note that
We have as a consequence of the proof of Theorem 1, see (13), the identity
where . With this we conclude, by Proposition 1, that
Now, we have two cases to consider: odd and even.
Case odd: Suppose that there are such that . Then,
On the other hand, using (17) we obtain
Since is odd, we conclude that and , for any and, therefore, .
Now, if for at most indices we have for example then
for some constant . Thus,
If , then . Otherwise,
As implies that does not depend on the variables , it follows that leading to a contradiction.
Case even: In this case, there is a natural such that . Then and consequently
Therefore, by (18) we get
Suppose that there are such that . In this case,
We conclude that for each there is a constant such that . Now, it is easy to verify (see (11)) that
Differentiating this identity with respect to the variable , gives
Finally, suppose that for any -tuple of indices, say it holds that . Then,
Implying that at least derivatives vanish, i.e., there are at most functions such that for example . Thus, by Proposition 1
for some constant . We conclude that analogously to the way it was presented for the case odd. ∎