Translation hypersurfaces of semi-Euclidean spaces with constant scalar curvature

1.   Introduction and preliminaries

Translation hypersurfaces are special Monge hypersurfaces. Many studies have been carried out with these hypersurfaces until today ^{[1,2,3,4,5,6,7,8,9,10,11]}.

In ^[1], Lima presented a complete description of all translation hypersurfaces with constant scalar curvature in the Euclidean space. In ^[2], they showed that every minimal translation and homothetical lightlike hypersurface is locally a hyperplane. In ^[3], the minimal translation hypersurfaces of $E^{4}$ were studied. Yang, Zhang and Fu obtained a characterization of a class of minimal translation graphs which are the generalization of minimal translation hypersurfaces in the Euclidean space ^[4]. In ^[5], the authors studied a characterization of minimal translation graphs in the semi-Euclidean space. Recently, homothetical and translation lightlike graphs, which are generalizations of homothetical and translation lightlike hypersurfaces were investigated in the semi-Euclidean space $\mathbb{R}_{q}^{n+2}$ ^[6]. Moreover Sağlam proved that all homothetical and all translation lightlike graphs are locally hyperplanes and according to this fact, both of these graphs are minimal. In ^[7], Seo gave a classification of the translation hypersurfaces with constant mean curvature or constant Gauss–Kronecker curvature in the Euclidean space and the Lorentz– Minkowski space. Moreover the author characterized the minimal translation hypersurfaces in the upper half-space model of the hyperbolic space. In 2019, Aydın and Ogrenmis studied translation hypersurfaces generated by translating planar curves and classified the translation hypersurfaces with constant Gauss-Kronecker curvature and constant mean curvature in the 4-dimensional isotropic space ^[8]. In ^[9], Ruiz-Hernandez investigated translation hypersurfaces in the (n+1)-dimensional Euclidean space whose Gauss-Kronecker curvature depends on its variables. In ^[10], Sousa, Lima and Vieira studied the geometry of generalized translation hypersurfaces immersed in Euclidean space equipped with a metric conformal to Euclidean metric and obtained results that characterize such hypersurfaces. In ^[11], Lima, Santos and Sousa gave a classification of the generalized translation graphs with constant mean curvature or constant Gauss–Kronecker curvature in the Euclidean space.

In the semi-Euclidean space $\mathbb{R}_{q}^{n+1},$ a translation hypersurface $M^{n}$ is a semi-Riemannian manifold with codimension 1 given by

where $f_{1}, \, f_{2}, \ldots, f_{n}$ are smooth functions. Each function $f_{i}$ depends on the real variable $x_{i}$ and is different from zero for $1\leq i\leq n.$ Or else it is a hyperplane.

In ^[1], Lima gave the parameterization of translation hypersurfaces with zero scalar curvature into $\mathbb{R}^{n+1}$ for $n\geq 3$. Moreover they showed that every translation hypersurface with constant scalar curvature must have zero scalar curvature in the Euclidean space $\mathbb{R}^{n+1}$ for $n\geq 3$ and proved the following theorem.

Theorem 1.1. Let $M^{n}$ be a translation hypersurface of $\mathbb{R }^{n+1}$ given by $\psi = (x_{1}, \ldots, x_{n}, F)$ for $n\geq 3.$ Then $M^{n}$ has zero scalar curvature iff it is congruent to the graph of the following functions:

1. $F(x_{1}, \ldots, x_{n}) = \sum_{i = 1}^{n-1}a_{i}x_{i}+f_{n}(x_{n})+b,$ on $\mathbb{R} ^{n-1}\times J,$ for some interval $J$ and $f_{n}:J\subset \mathbb{ R\rightarrow R}$ is a smooth function, which defines, after a suitable linear change of variables, a vertical cylinder.

2. A generalized periodic Enneper hypersurface given by

on $R^{n-3}\times I_{1}\times I_{2}\times I_{3},$ where $a, a_{1}, \ldots, a_{n-3}, b, b_{0}, c, d$ are real constants with $a, b, a+b\neq 0, \beta = 1+\sum_{i = 1}^{n-3}a_{i}^{2}$ and $I_{1}, I_{2}, I_{3}$ are the open intervals defined, respectively, by the conditions $\left \vert a\sqrt{\beta }x_{n-2}+a_{0}\right \vert < \pi /2,$ $\left \vert b\sqrt{\beta } x_{n-1}+b_{0}\right \vert < \pi /2$ and $\left \vert -\dfrac{ab}{a+b}\sqrt{ \beta }x_{n}+c\right \vert < \pi /2.$

In this paper, we obtain the parameterization of translation hypersurfaces with zero scalar curvature into $\mathbb{R}_{q}^{n+1}$. In addition we prove that translation hypersurfaces with constant scalar curvature must have zero scalar curvature in the semi-Euclidean space $\mathbb{R}_{q}^{n+1}$ for $n\geq 3$.

2.   Translation hypersurfaces of semi-Euclidean spaces with constant scalar curvature

Let $M^{n}$ be a semi-Riemannian manifold and $g_{ij}$ be the components of the metric tensor of $M^{n}$ and $g^{ij}$ be inverse of the functions $g_{ij}$ for $1\leq i, j\leq n.$ The Christoffel symbols or the affine connection of $M^{n}$ are given by

for $1\leq i, j, k\leq n.$ The Components of the Riemannian curvature tensor $R$ of a semi-Riemannian manifold $M^{n}$ are given by

for $1\leq i, j, k, l\leq n.$ The Components of the Ricci curvature tensor $Ric$ of a semi-Riemannian manifold $M^{n}$ are given by

for $1\leq i, j\leq n.$ The scalar curvature $S$ of a semi-Riemannian manifold $M^{n}$ are given by

Theorem 2.1. Let $M^{n}$ be a $n-$dimensional translation hypersurface of the semi-Euclidean space $\mathbb{R}_{q}^{n+1}$ with a natural orthonormal basis $\left \{ e_{1}, \ldots e_{n+1}\right \}$ determined by the following equations

Then the scalar curvature of $M^{n}$ given by

where $\varepsilon _{i} = \left \langle e_{i}, e_{i}\right \rangle = \pm 1$ for $1\leq i\leq n+1.$

Proof. It is easy to check that

and their inverse

with $Q = \varepsilon _{n+1}+\sum \limits_{k = 1}^{n}\varepsilon _{k}f_{k}^{^{\prime }2}$ and $i, j = 1, ..., n.$ By the direct calculation from the equations (2.1)–(2.4), we get (2.6).

Theorem 2.2. Let $M^{n}$ be a $n-$dimensional translation hypersurface of the semi-Euclidean space $\mathbb{R}_{q}^{n+1}$ for $n\geq 3$ determined by the following equations

Then $M^{n}$ has zero scalar curvature iff it is locally a hyperplane or it is parameterized by one of the following functions.

1.

on $\mathbb{R}^{n-1}\times I,$ for some open interval $I$, where $a_{i}, b\in \mathbb{R}$, $1\leq i\leq n-1$ and $f_{n}:I\subset \mathbb{ R\rightarrow R}$ is a smooth function. With a appropiate translation, it is a vertical hypercylinder.

2.

on $\mathbb{R}^{n-2}\times I_{1}\times I_{2},$ for some open intervals $I_{1}, I_{2}$, where $a_{i}, b\in \mathbb{R}$, $1\leq i\leq n-2$ with $\sum_{i = 1}^{n-2} \varepsilon _{i}a_{i}^{2} = -\varepsilon _{n+1}$ and $f_{n-1}:I_{1}\subset \mathbb{R\rightarrow R},$ $f_{n}:I_{2}\subset \mathbb{ R\rightarrow R}$ are smooth functions.

3. Let $a, a_{0}, a_{1}, \ldots, a_{n-3}, b, b_{0}, c_{0}, d$ be real constants with $a\neq 0, b\neq 0, a+b\neq 0, b-a\neq 0$, $\beta = \varepsilon _{n+1}+\sum_{i = 1}^{n-3}\varepsilon _{i}a_{i}^{2} > 0$ and $I_{1}, I_{2}, I_{3}, I_{4}$, $I_{5}$ be some open intervals defined, respectively, by the conditions $\left \vert a\sqrt{\beta }x_{n-2}+a_{0}\right \vert < \pi /2,$ $\left \vert b\sqrt{\beta }x_{n-1}+b_{0}\right \vert < \pi /2$, $\left \vert \dfrac{ab}{a+b}\sqrt{\beta }x_{n}+c_{0}\right \vert < \pi /2, \left \vert \dfrac{ab}{b-a}\sqrt{\beta }x_{n}+c_{0}\right \vert < \pi /2$ and $\left \vert -\dfrac{ab}{a+b}\sqrt{\beta }x_{n}+c_{0}\right \vert < \pi /2$.

a. If $\varepsilon _{n-1}\varepsilon _{n} = 1$ and $\varepsilon _{n-2}\varepsilon _{n} = 1,$ then

on $\mathbb{R}^{n-3}\times I_{1}\times I_{2}\times I_{3}$.

b. If $\varepsilon _{n-1}\varepsilon _{n} = -1$ and $\varepsilon _{n-2}\varepsilon _{n} = 1,$ then

on $\mathbb{R}^{n-3}\times I_{1}\times I_{2}\times I_{4}$.

c. If $\varepsilon _{n-1}\varepsilon _{n} = 1$ and $\varepsilon _{n-2}\varepsilon _{n} = -1,$ then

on $\mathbb{R}^{n-3}\times I_{1}\times I_{2}\times I_{4}.$

d. If $\varepsilon _{n-1}\varepsilon _{n} = -1$ and $\varepsilon _{n-2}\varepsilon _{n} = -1,$ then

on $\mathbb{R}^{n-3}\times I_{1}\times I_{2}\times I_{5}.$

If $\beta = 0,$ then $M^{n}$ is locally a hyperplane.

Proof. From Theorem 1.1, $M^{n}$ has zero scalar curvature iff

We will examine the proof according to the following cases.

Case 1. Let $\varepsilon _{n+1}+\sum \limits_{\substack{1\leq k\leq n \\ k\neq i, j}}\varepsilon _{k}f_{k}^{^{\prime }2} = 0$ for all $1\leq i < j\leq n,$ then the functions $f_{k}^{^{\prime }}$ are constant for all $1\leq k\leq n$. Consequently $M^{n}$ is locally a hyperplane.

Case 2. Let $f_{i}^{^{\prime \prime }}(x_{i}) = 0$ for all $i = 1, \ldots n-1,$ then $M^{n}$ is parameterized by the equation (2.9).

Case 3. Let $f_{i}^{^{\prime \prime }}(x_{i}) = 0$ for all $i = 1, \ldots n-2,$ then $f_{i}^{^{\prime }}(x_{i}) = a_{i},$ $a_{i}\in \mathbb{R}$. Also we can rewrite (2.15) by the following equation

According to this equation, we have the following cases:

ⅰ. If $f_{n-1}^{^{\prime \prime }} = 0,$ corresponding to Case 1.

ⅱ. If $f_{n}^{^{\prime \prime }} = 0,$ corresponding to Case 1.

ⅲ. If $\varepsilon _{n+1}+\sum \limits_{k = 1}^{n-2}\varepsilon _{k}a_{k}^{2} = 0,$ then $M^{n}$ is parameterized by the equation (2.10).

Case 4. Let $f_{i}^{^{\prime \prime }}(x_{i}) = 0$ for all $i = 1, \ldots n-3,$ then $f_{i}^{^{\prime }}(x_{i}) = a_{i},$ $a_{i}\in \mathbb{R}$. Also we can rewrite (2.15) by the following equation

where $\beta = \varepsilon _{n+1}+\sum \limits_{k = 1}^{n-3}\varepsilon _{k}a_{k}^{2}.$ If we multiply both sides of the above equation by $\varepsilon _{n-2}\varepsilon _{n-1}\varepsilon _{n}$, then we obtain

According to the assumption, the functions $f_{n-2}^{^{\prime \prime }},$ $f_{n-1}^{^{\prime \prime }}$ and $f_{n}^{^{\prime \prime }}$ are different from zero. Also we get $\beta +f_{k}^{^{\prime }2}\neq 0$ for $k = n-2, n-1, n.$ Hence we rewrite (2.16)

Differentiating the equation with respect to $x_{n-2}$ and $x_{n-1},$ we find

If $\left(\dfrac{f_{n-2}^{^{\prime \prime }}}{\beta +f_{n-2}^{^{\prime }2}} \right) ^{\prime } = 0,$ then there is a constant $a\neq 0$ such that

Substituting this equation into (2.17), we obtain

Differentiating the equation with respect to $x_{n-1}$ and $x_{n},$ we find

If $\left(\dfrac{f_{n-1}^{^{\prime \prime }}}{\beta +f_{n-1}^{^{\prime }2}} \right) ^{\prime } = 0,$ then there is a constant $b\neq 0$ such that

Substituting this equation into (2.19), we obtain

Since $ab\neq 0,$ from (2.21), then $\varepsilon _{n-1}a+\varepsilon _{n-2}b\neq 0.$ If we rearrange the equation, then we get

If we integrate the equations (2.18), (2.20) and (2.22), then we obtain

where $a_{0}, b_{0}$ and $c_{0}$ are constants. From these equations, we get

where $a_{1}, b_{1}$ and $c_{1}$ are constants. Therefore $M^{n}$ is parameterized by the equation

where $d = a_{1}+b_{1}+c_{1}$ is a constant. According to the values of $\varepsilon _{n-2}$, $\varepsilon _{n-1}$ and $\varepsilon _{n}$, if we rearrange the equation (2.23), then we get the following parameterizations.

ⅰ. If $\varepsilon _{n-1}\varepsilon _{n} = 1$ and $\varepsilon _{n-2}\varepsilon _{n} = 1,$ then the translation hypersurface $M^{n}$ is given by (2.11).

ⅱ. If $\varepsilon _{n-1}\varepsilon _{n} = -1$ and $\varepsilon _{n-2}\varepsilon _{n} = 1,$ then the translation hypersurface $M^{n}$ is given by (2.12).

ⅲ. If $\varepsilon _{n-1}\varepsilon _{n} = 1$ and $\varepsilon _{n-2}\varepsilon _{n} = -1,$ then the translation hypersurface $M^{n}$ is given by (2.13).

ⅳ. If $\varepsilon _{n-1}\varepsilon _{n} = -1$ and $\varepsilon _{n-2}\varepsilon _{n} = -1,$ then the translation hypersurface $M^{n}$ is given by (2.14).

Case 5. Let $f_{i}^{^{\prime \prime }}(x_{i}) = 0$ for $1\leq i\leq k\leq n-4,$ and $f_{j}^{^{\prime \prime }}(x_{j})\neq 0$ for any $j > k.$ We prove that this is not possible. Also we can rewrite (2.15) for any fixed $l\geq k+1$ by the following equation

Differentiating the equation (2.24) with respect to $x_{l}$, we obtain

According to the equation (2.25), we define

$A_{l}$ and $B_{l}$ are not dependent on $x_{l}.$ From (2.25) and (2.26), we have

Also there are two cases.

ⅰ. Let $A_{l} = 0$ for $l\geq k+1.$ From (2.26), we get

Differentiating the equation (2.28) with respect to $x_{p}$ for $p\geq k+1$ and $p\neq l$, we find

According to this equation, one must have

Otherwise the functions $f_{m}^{^{\prime }}$ are constant and we conclude that $f_{m}^{^{\prime \prime }} = 0$ for $1\leq m\leq n,$ $m\neq l, p.$ This is a contradiction with the assumption in Case 5. Since $A_{l} = 0,$ according to (2.25), we get

Since $\varepsilon _{l}\neq 0$ and $f_{l}^{^{\prime \prime }}\neq 0,$ we have

Differentiating the equation (2.32) with respect to $x_{p}$ for $p\geq k+1$ and $p\neq l$, we obtain

Differentiating the equation (2.33) with respect to $x_{q}$ for $q\geq k+1$ and $q\neq l, p$, we find $f_{p}^{^{\prime \prime \prime }}f_{q}^{^{\prime \prime \prime }} = 0.$ Therefore, at most one of the indexes $p\geq k+1$ and $p\neq l$ is nonzero, denoted by $p.$ Also we can get $f_{p}^{^{\prime \prime \prime }}\neq 0$ and $f_{q}^{^{\prime \prime \prime }} = 0$ for all $q\geq k+1$ and $q\neq l, p.$ From $f_{p}^{^{\prime \prime \prime }}\neq 0$ and the equation (2.33), we have

Substituting this equation into (2.29), since $\varepsilon _{n+1}+\sum \limits_{\substack{1\leq m\leq n \\ m\neq l, p}}\varepsilon _{m}f_{m}^{^{\prime }2}\neq 0,$ we get $f_{p}^{^{\prime \prime \prime }} = 0.$ This is a contradiction with $f_{p}^{^{\prime \prime \prime }}\neq 0.$ Also we get $f_{p}^{^{\prime \prime \prime }} = 0$ for all $p\geq k+1$ and $p\neq l.$ From (2.29), we conclude that

for all $p\geq k+1$ and $p\neq l.$ The above linear system has unique solution such that $f_{j}^{^{\prime \prime }} = 0$ for all $k+1\leq j\leq n$ and $j\neq l.$ This is a contradiction with the assumption in Case 5. Consequently, if $A_{l} = 0,$ then Case 5 is not possible.

ⅱ. Let $A_{l}\neq 0$ for $l\geq k+1.$ Since $A_{l}\neq 0$, from (2.27), we get

where $\alpha _{l} = \dfrac{B_{l}}{A_{l}}$ is a constant for $l\geq k+1.$ Substituting this equation into (2.25), we find

Since $f_{l}^{^{\prime \prime }}(x_{l})\neq 0$ for $l\geq k+1,$ we obtain

Differentiating the equation (2.37) with respect to $x_{s}$ for $s\geq k+1$ and $s\neq l$, we obtain

From (2.36), $f_{s}^{^{\prime \prime \prime }}+2\alpha _{s}f_{s}^{^{\prime }}f_{s}^{^{\prime \prime }} = 0$ for $s\geq k+1.$ Also we can rewrite the above equation

Since $f_{s}^{^{\prime \prime }}(x_{s})\neq 0$ for $s\geq k+1,$ we get

Differentiating the equation (2.38) with respect to $x_{t}$ for $t\geq k+1$ and $t\neq l$ and $t\neq s,$ we obtain

From (2.36), $f_{t}^{^{\prime \prime \prime }}+2\alpha _{t}f_{t}^{^{\prime }}f_{t}^{^{\prime \prime }} = 0$ for $t\geq k+1.$ Since $f_{t}^{^{\prime \prime }}(x_{t})\neq 0$ for $t\geq k+1,$ we obtain the above equation

with $t\neq l$, $t\neq s$ and $l\neq s.$ From ^[1], in a similar way to the proof of Theorem 1.2, this equality imply that at most one of the constants $\alpha _{l}$ is nonzero for $l\geq k+1.$ We assume that $\alpha _{l} = 0$ for $k+1\leq l\leq n-1.$ From (2.36), $f_{l}^{^{\prime \prime \prime }} = 0,$ then $f_{l}^{^{\prime \prime }}$ is constant for $k+1\leq l\leq n-1.$ From (2.37), we obtain

for $l\neq n.$ Therefore $f_{n}^{^{\prime \prime }}$ is constant and so $\alpha _{n} = 0.$ Thus, from (2.37), we get

According to the equality, at most one of the functions $f_{l}^{^{\prime \prime }}$ is nonzero for $k+1\leq l\leq n.$ This is a contradiction with the assumption in Case 5. Consequently, if $A_{l}\neq 0,$ then Case 5 is not possible.

Theorem 2.3. Let $M^{n}$ be a $n-$dimensional translation hypersurface of the semi-Euclidean space $\mathbb{R}_{q}^{n+1}$ for $n\geq 3$ determined by the following equations

Assume further that $M^{n}$ has constant scalar curvature. Then its constant scalar curvature must be zero.

Proof. We assume that a translation hypersurface $M^{n}$ has nonzero constant scalar curvature $S$. From (2.6) the scalar curvature of $M^{n}$ is given by

where $Q = \varepsilon _{n+1}+\sum \limits_{i = 1}^{n}\varepsilon _{i}f_{i}^{^{\prime }2}.$ Differentiating the equation (2.40) with respect to $x_{l},$ we obtain

If we rearrange this equation, then we get

Differentiating the equation (2.41) with respect to $x_{s}$ and $s\neq l,$ we find

From this equation, we get

Differentiating the equation (2.42) with respect to $x_{t},$ $t\neq l$ and $t\neq s,$ we have

We assume that $f_{l}^{^{\prime \prime }}f_{s}^{^{\prime \prime }}f_{t}^{^{\prime \prime }}\neq 0$ and $f_{l}^{^{\prime \prime \prime }} = 0.$ According to (2.43), we get $f_{s}^{^{\prime \prime \prime }} = 0$ or $f_{t}^{^{\prime \prime \prime }} = 0.$ From (2.42), we have $4f_{l}^{^{\prime }}f_{l}^{^{\prime \prime }}f_{s}^{^{\prime }}f_{s}^{^{\prime \prime }}S = 0.$ This contradicts $f_{l}^{^{\prime \prime }}f_{s}^{^{\prime \prime }}f_{t}^{^{\prime \prime }}\neq 0$ and $S\neq 0.$ Also $f_{l}^{^{\prime \prime \prime }}\neq 0$ and likewise $f_{s}^{^{\prime \prime \prime }}\neq 0$ and $f_{t}^{^{\prime \prime \prime }}\neq 0.$ From $f_{l}^{^{\prime \prime }}f_{s}^{^{\prime \prime }}f_{t}^{^{\prime \prime }}\neq 0$ and (2.43), we find

From (2.44), we get $f_{l}^{^{\prime \prime \prime }} = \alpha _{l}f_{l}^{^{\prime }}f_{l}^{^{\prime \prime }},$ with a nonzero constant $\alpha _{l}$. Substituting this equation into (2.41), we find

Differentiating the equation (2.45) with respect to $x_{l},$ we have $f_{l}^{^{\prime }}f_{l}^{^{\prime \prime }}S = 0.$ This contradicts $f_{l}^{^{\prime \prime }}f_{s}^{^{\prime \prime }}f_{t}^{^{\prime \prime }}\neq 0$ and $S\neq 0.$ Hence, it must be $f_{l}^{^{\prime \prime }}f_{s}^{^{\prime \prime }}f_{t}^{^{\prime \prime }} = 0.$ Also, at most two of the functions $f_{l}^{^{\prime \prime }}$ are nonzero for $1\leq l\leq n.$ Without loss of generality, we assume that $f_{n-1}^{^{\prime \prime }}\neq 0,$ $f_{n}^{^{\prime \prime }}\neq 0$ and $f_{l}^{^{\prime \prime }} = 0$ for $1\leq l\leq n-2,$ then $f_{l}^{^{\prime }} = a_{l}$ for $1\leq l\leq n-2$ and we arrange (2.6)

where $\alpha = 2\varepsilon _{n-1}\varepsilon _{n}\left(\varepsilon _{n+1}+\sum \limits_{k = 1}^{n-2}\varepsilon _{k}a_{k}^{2}\right)$ is a nonzero constant. Differentiating the equation (2.46) with respect to $x_{n-1},$ we have

Differentiating the equation with respect to $x_{n},$ we get

Also, there is a nonzero constant $\beta$ such that $f_{n-1}^{^{\prime \prime \prime }} = \beta f_{n-1}^{^{\prime }}f_{n-1}^{^{\prime \prime }}\neq 0$ and from (2.47)

Differentiating the equation (2.49) with respect to $x_{n-1},$ we get

This is a contradiction with $f_{n-1}^{^{\prime \prime }}\neq 0.$ Thus the constant scalar curvature must be zero.

3.   Conclusions

Translation hypersurfaces are special Monge hypersurfaces defined by the following equations

In this paper, we obtain the parameterization of translation hypersurfaces with zero scalar curvature into $\mathbb{R}_{q}^{n+1}$. Moreover we prove that translation hypersurfaces with constant scalar curvature must have zero scalar curvature in the semi-Euclidean space $\mathbb{R}_{q}^{n+1}$ for $n\geq 3$.