# Hypersurfaces with pointwise 1-type Gauss map in Lorentz--Minkowski space/ Huperpinnad punktiti 1-tuupi Gaussi kujutusega Lorentzi-Minkowski ruumis.

1. INTRODUCTION

The notion of finite type submanifolds in Euclidean space or pseudo-Euclidean space was introduced by B. Y. Chen in the late 1970s (cf. [5,6]). Since then the theory of submanifolds of finite type has been studied by many geometers and many interesting results have been obtained (see  for a report on this subject).

In  the notion of finite type was extended to differentiable maps, in particular, to Gauss map of submanifolds. The notion of finite type Gauss map is especially a useful tool in the study of submanifolds (cf. [1-4,9,14,15,19]).

If a submanifold M of a pseudo-Euclidean space [E.sub.s.sup.m] has 1-type Gauss map G, then G satisfies [DELTA]G = [lambda] (G+C) for some [lambda] [member of] R and some constant vector C. However, the Laplacian of the Gauss map of several surfaces and hypersurfaces, such as catenoids and right cones in [E.sup.3] , generalized catenoids and right n-cones in [E.sup.n+1] , and helicoids of the 1st, 2nd, and 3rd kind, conjugate Enneper's surfaces of the second kind, and B-scrolls in [E.sub.1.sup.3]  take the form

[DELTA]G = f(G+C) (1.1)

for some non-constant function f on M and some constant vectorC. A submanifold is said to have pointwise 1-type Gauss map if its Gauss map satisfies (1.1) for some smooth function f on M and some constant vector C. A pointwise 1-type Gauss map is called proper if the function f is non-constant. A submanifold with pointwise 1-type Gauss map is said to be of the first kind if the vectorC in (1.1) is the zero vector. Otherwise, a submanifold with pointwise 1-type Gauss map is said to be of the second kind.

In , Kim and Yoon gave the complete classification of ruled surfaces in a 3-dimensional Minkowski space with pointwise 1-type Gauss map; in  they characterized ruled surfaces of an m-dimensional Minkowski space [E.sub.1.sup.m] in terms of the notion of pointwise 1-type Gauss map, and moreover, they studied rotation surfaces of the pseudo-Euclidean space [E.sub.2.sup.4] with pointwise 1-type Gauss map in . Recently, in , U-H. Ki, D.-S. Kim, Y. H. Kim, and Y.-M. Roh gave a complete classification of rational surfaces of revolution in Minkowski 3-space with pointwise 1-type Gauss map.

In this paper our aim is to study hypersurfaces of a Lorentz-Minkowski space [L.sup.n+1] with pointwise 1-type Gauss map. We first obtain a characterization of hypersurfaces [M.sub.q] of index q of [L.sup.n+1] with pointwise 1-type Gauss map, that is, we show that an oriented hypersurface [M.sub.q] of a Lorentz-Minkowski space [L.sup.n+1] has pointwise 1-type Gauss map of the first kind if and only if [M.sub.q] has constant mean curvature. As a consequence of this, all oriented isoparametric hypersurfaces of [L.sup.n+1] have 1-type Gauss map. Then we classify rational rotation hypersurfaces of [L.sup.n+1] with pointwise 1-type Gauss map which extend the results given in  on rational surfaces of revolution in [L.sup.3] to the hypersurfaces of [L.sup.n+1]. We also give examples of a rational rotation hypersurface with pointwise 1-type Gauss map of the first and second kind.

2. PRELIMINARIES

Let [L.sup.n+1] denote the (n+1)-dimensional Lorentz-Minkowski space, that is, the real vector space [R.sup.n+1] endowed with the Lorentzian metric <,> = [(d[x.sub.1]).sup.2] + ... + [(d[x.sub.n]).sup.2] - [(d[x.sub.n+1]).sup.2], where ([x.sub.1], ... ,[x.sub.n+1]) are the canonical coordinates in [R.sup.n+1]. A vector x of [L.sup.n+1] is said to be space-like if <x,x> > 0 or x = 0, time-like if <x,x> < 0, or light-like (or null) if <x,x> = 0 and x [not equal to] 0.

An immersed hypersurface [M.sub.q] of [L.sup.n+1] with index q (q = 0,1) is called space-like (Riemannian) or timelike (Lorentzian) if the induced metric which, as usual, is also denoted by <,> on [M.sub.q] has the index 0 or 1, respectively. The de Sitter n-space [S.sub.n.sup.n] ([x.sub.o], c) centred at [x.sub.o] [member of] [L.sup.n+1], c > 0; is a Lorentzian hypersurface of [L.sup.n+1] defined by

[S.sub.1.sup.n]([x.sub.o], c) = {x [member of] [L.sup.n+1] | < x - [x.sub.o], x - [x.sub.o]> = [c.sup.2]}

and the hyperbolic space [H.sup.n]([x.sub.o],c) centred at [x.sub.o] [member of] [L.sup.n+1], c > 0, is a space-like hypersurface of [L.sup.n+1] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [x.sub.n+1] - [x.sub.n+1.sup.0] is the (n+1)-th component of x - [x.sub.0].

Let [PI] be a 2-dimensional subspace of [L.sup.n+1] passing through the origin. We will say that [PI] is non-degenerate if the metric <,> restricted to [PI] is a non-degenerate quadratic form. A curve in [L.sup.n+1] is called space-like, time-like, or light-like if the tangent vector at any point is space-like, time-like, or light-like, respectively.

Here we will define non-degenerate rotation hypersurfaces in [L.sup.n+1] with a time-like, space-like, or light-like axis. For an open interval I [subset] R, let [gamma] : I [right arrow] [PI] be a regular smooth curve in a non-degenerate 2-plane [PI] of [L.sup.n+1] and let l be a line in [PI] that does not meet the curve [gamma]. A rotation hypersurface [M.sub.q] with index q in [L.sup.n+1] with a rotation axis l is defined as the orbit of a curve [gamma] under the orthogonal transformations of [L.sup.n+1] with a positive determinant that leaves the rotation axis l fixed (for details see ). The curve [gamma] is called a profile curve of the rotation hypersurface. As we consider non-degenerate rotation hypersurfaces, it is sufficient to consider the case that the profile curve is space-like or time-like. The explicit parametrizations for non-degenerate rotation hypersurfaces [M.sub.q] in [L.sup.n+1] were given in  according to the axis l being time-like, space-like, or light-like.

Let {[[eta].sub.1], ... ,[[eta].sub.n+1]} be the standard orthonormal basis of [L.sup.n+1], that is, <[[eta].sub.i], [[eta].sub.j]> = [[delta].sub.ij], < [[eta].sub.n+1], [[eta].sub.n+1] = -1, <[[eta].sub.i], [[eta].sub.n+1]> = 0, i, j = 1,2, ... ,n. Let [THETA] ([u.sub.1], ... [u.sub.n-2]) denote an orthogonal parametrization of the unit sphere [S.sup.n-2](1) in the Euclidean space [E.sup.n-1] generated by {[[eta].sub.1], ... ,[[eta].sub.n-1]}:

[THETA]([u.sub.1], ... ,[u.sub.n-2]) = cos[u.sub.1][[eta].sub.1]+sin [u.sub.1] cos [u.sub.2] [[eta].sub.2]

+ ... + sin [u.sub.1] ... sin [u.sub.n-3] cos [u.sub.n-2] [[eta].sub.n-2] + sin [u.sub.1] ... sin [u.sub.n-3] sin [u.sub.n-2] [[eta].sub.n-1], (2.1)

where 0 < [u.sub.i] < [pi] (i = 1, ... ,n-3); 0 < [u.sub.n] < 2 [pi].

Remark 2.1. When n = 2, the term [THETA] ([u.sub.1], ... ;[u.sub.n-2]) in the following definitions of rotation hypersurfaces is replaced by [[eta].sub.1]:

Case 1. l is time-like. In this case the plane [PI] that contains the line l and a profile curve [gamma] is Lorentzian. Without loss of generality, we may suppose that l is the [x.sub.n+1]-axis and [PI] is the [x.sub.n][x.sub.n+1]-plane which is Lorentzian.

Let [gamma](t) = [phi](t) [[eta].sub.n] + [psi](t)[[eta].sub.n+1] be a parametrization of [gamma] in the plane [PI] with [x.sub.n] = [phi](t) > 0, t [member of] I [subset]R. The curve is space-like if [epsilon] = sgn([[phi].sup.'2] - [[psi].sup.'2]) = 1 and time-like if [epsilon] = sgn([[phi].sup.'2] - [[psi].sup.'2]) = -1. So a parametrization of a rotation hypersurface [M.sub.q],T of [L.sup.n+1] with a time-like axis is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

where 0 < [u.sub.n-1] < [pi]. The second index in [M.sub.q,T] stands for the time-like axis. The hypersurface [M.sub.q,T] is also called a spherical rotation hypersurface of [L.sup.n+1] as parallels of [M.sub.q,T] are spheres [S.sup.n-1](0, [phi](t)).

Case 2. l is space-like. In this case the plane [PI] which contains a profile curve is Lorentzian or Riemannian. So there are rotation hypersurfaces of the first and second kind labelled by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] in [L.sup.n+1] with a space-like axis.

Subcase 2.1. The plane [PI] is Lorentzian. Without losing generality we may suppose that l is the [x.sub.n]-axis, that is, the vector [[eta].sub.n] = (0,0, ... ,0,1,0) is the direction of the rotation axis, and [PI] is the [x.sub.n][x.sub.n+1]-plane. Let [gamma](t) = [psi](t) [[eta].sub.n] + [phi](t) [[eta].sub.n+1] be a parametrization of [gamma] in the plane [PI] with [x.sub.n+1] = [phi] (t) > 0, t [member of] I [subset] R. Thus a parametrization of a rotation hypersurface of the first kind [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] with a space-like axis is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

0 < [u.sub.n-1] < [infinity], which is also called a hyperbolic rotation hypersurface of [L.sup.n+1] as parallels of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are hyperbolic spaces [H.sup.n-1](0,-[phi](t)).

Subcase 2.2. The plane [PI] is Riemannian. We may suppose that l is the [x.sub.n]-axis and [PI] is the [x.sub.n-1][x.sub.n]-plane without loss of generality. Let [gamma](t) = [phi](t) [[eta].sub.n-1] + [psi](t) [[eta].sub.n] be a parametrization of [gamma] in the plane [PI] with [x.sub.n-1] = [phi](t) > 0, t [member of] I [subset] R. In this case the curve [gamma] is space-like. Similarly, a parametrization of a rotation hypersurface of the second kind [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] with a space-like axis is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

-[infinity] < [u.sub.n-1] < [infinity], which is called a pseudo-spherical rotation hypersurface of [L.sup.n+1] as parallels of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are pseudo-spheres [S.sub.1.sup.n-1] (0, [phi](t)) when n > 2. (If n = 2, then [S.sub.1.sup.1] [equivalent to] [H.sup.1].) Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] has index 1, that is, q = 1.

Case 3. l is light-like. Let {[[??].sub.i], ... , [[??].sub.n+1]} be a pseudo-Lorentzian basis of [L.sup.n+1], that is, < [[??].sub.i], [[??].sub.j] > = [[delta].sub.ij], i,j = 1, ... ,n-1, < [[??].sub.i], [[??].sub.n] > = < [[??].sub.i], [[??].sub.n+1] > = 0; i = 1,2, ... n - 1, < [[??].sub.n], [[??].sub.n+1] > = 1, < [[??].sub.n], [[??].sub.n] > = 0, < [[??].sub.n+1], [[??].sub.n+1] > = 0. We can choose [[??].sub.1] = (1,0, ..., 0), ... [[??].sub.n-1] = (0, ..., 1,0,0)m [[??].sub.n] = 1/[square root of (2)](0, ..., 0,1,-1), [[??].sub.n+1] = 1/[square root of (2)] (0, ..., 0,1,1). We may suppose that l is the line spanned by the null vector [[??].sub.n+1] and [PI] is the [x.sub.n][x.sub.n+1]-plane without loss of generality. Let [gamma](t) = [square root of (2[phi])](t) [[??].sub.n] + [square root of (2[psi])](t) [[??].sub.n+1] be a parametrization of [gamma] in the plane [PI] with [x.sub.n] = [psi](t) > 0, t [member of] I [subset] R. Let [[THETA].sub.1]([u.sub.1], ..., [u.sub.n-2]), ..., [[THETA].sub.n-1]([u.sub.1], ..., [u.sub.n-2]) be the components of the orthogonal parametrization [THETA]([u.sub.1], ..., [u.sub.n-2]) given by (2.1) of the unit sphere [S.sup.n-2](1) in the basis {[[??].sub.1], ..., [[??].sub.n-1]}.

Then a parametrization of a rotation hypersurface [M.sub.q,L] of [L.sup.n+1] with a space-like axis is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

The subgroup of Lorentz group which fixes the direction [[??].sub.n+1] of the light-like axis l can be seen in .

Note that in the third case if [phi](t) = [[phi].sub.0] or [psi](t) = [[psi].sub.0] is a constant, the profile curve is degenerate. However, in the other cases if [phi](t) = [[phi].sub.0] > 0 is a constant and [phi](t) = t, the rotation hypersurface [M.sub.1,T] is the Lorentzian cylinder [S.sup.n-1](0,[[phi].sub.0]) x [L.sup.1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the hyperbolic cylinder [H.sup.n-1] (0,-[[phi].sub.0]) x R, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the pseudo-spherical cylinder [S.sub.1.sup.n-1] (0,[[phi].sub.o])x R. If [phi] (t) = t and [psi](t) = [[psi].sub.0] is a constant, then [M.sub.0,T] is a space-like hyperplane of [L.sup.n+1], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are time-like hyperplanes of [L.sup.n+1]. Therefore all these are rotation hypersurfaces of [L.sup.n+1] with constant mean curvature.

Let [nabla] and [nabla]' denote the Riemannian connection on [M.sub.q] and [L.sup.n+1], respectively. Then, for any vector fields X,Y tangent to [M.sub.q], we have the Gauss formula

[nabla]'xY = [nabla]xY + h(X,Y), (2.6)

where h is the second fundamental form which is symmetric in X and Y. For a unit normal vector field [xi], the Weingarten formula is given by

[nabla]'x[xi] = -[A.sub.[xi]]X, (2.7)

where [A.sub.[xi]] is the Weingarten map or the shape operator with respect to [xi]. The Weingarten map [A.sub.[xi]] is a self-adjoint endomorphism of TM which cannot be diagonalized in general. It is known that h and [A.sub.[xi]] are related by

<h(X,Y), [xi]> = < [A.sub.[xi]]X,Y>. (2.8)

The covariant derivative of the secon d fundamental form h is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

where [nabla][perpendicular to] denotes the linear connection induced on the normal bundle T[perpendicular to]M. Then the Codazzi equation is given by

([??][[x.sup.h])(Y,Z) = ([[??].sub.Y]h)(X,Z). (2.10)

Also, from (2.9) we have

([??].sub.x]h)(Y,Z) = ([??].sub.X]h)(Z,Y). (2.11)

For any normal vector [xi] the covariant derivative [nabla][A.sub.xi] of [A.sub.xi] is defined by

([[nabla].sub.x][A.sub.[xi]])Y = [[nabla.sub.x]([A.sub.[xi]]Y) - [A.sub.[xi]]([[nabla].sub.[xi]]Y). (2.12)

Let [xi] be a unit normal vector. Since [nabla][[perpendicular to].sub.X] [xi] = 0, we have by (2.9)

<([[nabla].sub.X][A.sub.[xi]])Y,Z> = <([[??].sub.X]h)(Y,Z), [xi]>. (2.13)

Let [M.sub.q] be a hypersurface with index q in [L.sup.n+1]. The map G : [M.sup.n] [right arrow] [Q.sup.n]([[epsilon].sub.G]) [subset] [L.sup.n+1] which sends each point of [M.sub.q] to the unit normal vector to [M.sub.q] at the point is called the Gauss map of the hypersurface [M.sub.q], where [[epsilon].sub.G](= [+ or -]1) denotes the signature of the vector G and [Q.sup.n]([[epsilon].sub.G]) is an n-dimensional space form given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let [e.sub.1], ..., [e.sub.n] be an orthonormal local tangent frame on a hypersurface [M.sub.q] of [L.sup.n+1] with signatures [[epsilon].sub.i] = <[e.sub.i], [e.sub.i]> = [??]1, and [A.sub.G] denote the shape operator of [M.sub.q] in the unit normal direction G. Then the mean curvature H of [M.sub.q] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

A space-like hypersurface of [L.sup.n+1] with vanishing mean curvature is called maximal.

3. HYPERSURFACES WITH POINTWISE 1-TYPE GAUSS MAP

In this section we give a characterization of hypersurfaces of Lorentz-Minkowski space with pointwise 1-type Gauss map.

Lemma 3.1. Let [M.sub.q] be a hypersurface with index q in a Lorentz-Minkowski space [L.sup.n+1]. Then we have

trace([nabla][A.sub.G]) = n[nabla][alpha], (3.1)

where [alpha] = [square root of ([[epsilon].sub.G] <H,H>] and [[epsilon].sub.G] = <G,G>.

Proof. Let [e.sub.1], ... , [e.sub.n] be a local orthonormal tangent basis on [M.sub.q] with [[epsilon].sub.i] = < [e.sub.i], [e.sub.i] >, i = 1, ... , n. For any vector X tangent to [M.sub.q] we have by using (2.9)-(2.11) and (2.13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

because [[epsilon].sub.i] [[omega].sub.i.sup.j] + [[epsilon].sub.j] [[omega].sub.j.sup.i] = 0, where [[omega].sub.i.sup.j], i,j = 1, ..., n, are the connection forms associated to [e.sub.1], ..., [e.sub.n], and [nabla][alpha] is the gradient of the mean curvature. Therefore we obtain (3.1).

Lemma 3.2. Let [M.sub.q] be a hypersurface with index q in a Lorentz-Minkowski space [L.sup.n+1]. Then the Laplacian of the Gauss map G is given as

[DELTA]G = [[epsilon].sub.G][[parallel][A.sub.G][parallel].sup.2]G+n[nabla][alpha], (3.2)

where [[parallel][A.sub.G][parallel].sup.2] = tr([A.sub.G][A.sub.G]), [[epsilon].sub.G] = < G,G >, and [alpha] = [square root of ([[epsilon].sub.G] < H,H >)].

Proof. Let [C.sub.0] be a fixed vector in [L.sup.n+1]. For any vectors X,Y tangent to M using the Gauss and Weingarten formulas we have

YX < G, [C.sub.0] = -< [[nabla].sub.Y] ([A.sub.G](X)) + h([A.sub.G](X),Y),[C.sub.0] >. (3.3)

Let [e.sub.i], ..., [e.sub.n] be a local orthonormal tangent basis on [M.sub.q] with [[epsilon].sub.i] = < [e.sub.i], [e.sub.i] >. By using (2.12), (3.3), and Lemma 3.1, we calculate the Laplacian of < G, [C.sub.0] > as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

as [[SIGMA].sub.i=1.sup.n] [[epsilon].sub.i]h ([A.sub.G]([e.sub.i],[e.sub.i]) = [[epsilon].sub.G] [[parallel][A.sub.G][parallel].sup.2]G Since (3.4) holds for any [C.sub.0] [member of] [L.sup.n+1], the proof is complete.

Now, from definition (1.1) and equation (3.2) we state the following theorem which characterizes the hypersurfaces of Lorentz-Minkowski spaces with pointwise 1-type Gauss map of the first kind.

Theorem 3.3. Let [M.sub.q] be an oriented hypersurface with index q in a Lorentz-Minkowski space [L.sup.n+1]. Then [M.sub.q] has proper pointwise 1-type Gauss map of the first kind if and only if [M.sub.q] has constant mean curvature and [[parallale][A.sub.G][parallel].sup.2] is non-constant.

Hence we have

Corollary 3.4. All oriented isoparametric hypersurfaces of a Lorentz-Minkowski space [L.sup.n+1] have 1-type Gauss map.

For example, space-like hyperplanes, Lorentzian hyperplanes, hyperbolic spaces [H.sup.n](0,-c), de Sitter spaces [S.sub.1.sup.n](0,c), Lorentzian cylinders [S.sup.n-1](0,c) x [L.sup.1], hyperbolic cylinders [H.sup.n-1](0,-c) x R, and the pseudo-spherical cylinders [S.sub.1.sup.n-1] (0,c) x R of [L.sup.n+1] have 1-type Gauss map.

From Lemma 3.2 we can also state

Theorem 3.5. If an oriented hypersurface [M.sub.q] with index q in a Lorentz- Minkowski space [L.sup.n+1] has proper pointwise 1-type Gauss map of the second kind, then the mean curvature of M is a non-constant function on [M.sub.q].

4. ROTATION HYPERSURFACES WITH POINTWISE 1-TYPE GAUSS MAP OF THE FIRST AND THE SECOND KIND

In this section we obtain a classification of rotation hypersurfaces of [L.sup.n+1] with pointwise 1-type Gauss map of the first and the second kind, and give some examples.

Lemma 4.1. Let [M.sub.q] be one of the rotation hypersurfaces [M.sub.q,T], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1]. If [M.sub.q] has pointwise 1-type Gauss map in [L.sup.n+1], then either the Gauss map is harmonic, that is, [DELTA]G = 0 or the function f defined in (1.1) depends only on t and the vector C in (1.1) is parallel to the axis of the rotation of [M.sub.q].

Proof. Let [M.sub.q] = [M.sub.q,T], which is defined by (2.2). The Gauss map of [M.sub.q,T] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.1)

with [[epsilon].sub.G] = < G,G > = -[epsilon], where [epsilon] = sgn ([[phi].sup.'2] - [[psi].sup.'2]) = [+ or -]1.

The principal curvature of the shape operator [A.sub.G] of [M.sub.q,T] in the direction G was obtained in . By a direct computation (or following ) we have the mean curvature [alpha] of [M.sub.q,T] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.2)

which is the function of t, and the square of the length of the shape operator as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

Since the mean curvature [alpha] is the function of t, by a direct computation we obtain the gradient of [alpha] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Also, by (4.1) we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

Using (3.2) and (4.4), the Laplacian of the Gauss map (4.1) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.5)

If M has pointwise 1-type Gauss map, then (1.1) holds for some function f and some vector C. When the Gauss map is not harmonic, equations (1.1), (2.1), (4.1), and (4.5) imply that C = c[[eta].sub.n+1] which is the rotation axis of [M.sub.q,T] for some nonzero constant c [member of] R, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.6)

from which the function f is independent of [u.sub.1], ..., [u.sub.n-1].

In the case [M.sub.q] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or [M.sub.q] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we obtain the same result by a similar discussion.

Theorem 4.2. There do not exist rotation hypersurfaces [M.sub.q] in [L.sup.n+1] with a light-like rotation axis and harmonic Gauss map.

Proof. Without losing generality we may parametrize [M.sub.q] by (2.5), that is, [M.sub.q] = [M.sub.q,L]. Then the Gauss map [??] of [M.sub.q,L] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.7)

with [[epsilon].sub.[??]] <[??], [??]> = [??], where [??] = sgn ([phi]' [psi]') = [+ or -]1.

By a direct computation (or see ) we have the mean curvature [??] of [M.sub.q,L] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.8)

which is the function of t, and the square of the length of the shape operator [A.sub.[??]] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.9)

Since the mean curvature [??] is the function of t, we then obtain the gradient of [??] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Also, by using (4.7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.10)

Using (3.2) and (4.10), the Laplacian of the Gauss map (4.7) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.11)

Suppose that the Gauss map is harmonic, that is, [DELTA][??] = 0: Then, considering (4.7), we have [arallel][A.sub.G][parallel] = 0 from (4.11), which implies [phi]' = 0 because of (4.9). This is not possible as the hypersurface is nondegenerate, that is, [phi]' [psi] [not equal to] 0: Therefore the Gauss map of [M.sub.q,L] is not harmonic.

Lemma 4.3. Let [M.sub.q,L] be a rotation hypersurface of [L.sup.n+1] with a light-like rotation axis parametrized by (2.5). If [M.sub.q,L] has pointwise 1-type Gauss map in [L.sup.n+1], then the function f in (1.1) depends only on t, and the vector C in (1.1) is parallel to the rotation axis.

Proof. The Gauss map of [M.sub.q,L] and its Laplacian are given by (4.7) and (4.11), respectively. Suppose that [M.sub.q,L] has pointwise 1-type Gauss map in [L.sup.n+1]. By (1.1), (2.1), (4.7), and (4.11) we see that C = c[[eta].sub.n+1] which is the rotation axis of [M.sub.q,L] for some nonzero constant c [member of] R, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.12)

from which the function f is independent of [u.sub.1], ..., [u.sub.n-1].

Here we give some examples for later use. Let [phi](t) = t, t > 0 and [psi](t) = g(t) in the definitions of rotation hypersurfaces [M.sub.q,T], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and [M.sub.q,L], where g(t) is a differentiable function. In , the following results were obtained for the rotation hypersurfaces of [L.sup.n+1] with constant mean curvature:

1) The rotation hypersurface [M.sub.q,T] of [L.sup.n+1] has constant mean curvature [alpha] if and only if the function g(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.13)

where a is an arbitrary constant, [epsilon] = sgn(1 - [g.sup.'2]) = [+ or -]1, and q = 0 for [epsilon] = 1 and q = 1 for [epsilon] = -1.

2) The rotation hypersurface of the first kind [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] has constant mean curvature [??] if and only if the function g(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.14)

where a is an arbitrary constant, [??] = sgn([g.sup.'2] - 1) = [+ or -]1, and q = 0 for [??] = 1 and q = 1 for [??] = -1.

3) The Lorentzian rotation hypersurface of the second kind [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] has constant mean curvature [??] if and only if the function g(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.15)

where a is an arbitrary constant.

4) The rotation hypersurface [M.sub.q,L] of [L.sup.n+1] has constant mean curvature [??] if and only if the function g(t) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.16)

where a is an arbitrary constant, [??] = sgn(g') = [+ or -]1, and q = 0 for [??] = 1 and q = 1 for [??] = -1.

Example 4.4. The rotation hypersurface [M.sub.q,T] of [L.sup.n+1] defined by (2.2) for the function g(t) given by (4.13) has the Gauss map from (4.1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.17)

with [[epsilon].sub.G] = <G,G> = -[epsilon]. Since [M.sub.q,T] has constant mean curvature, we have the Laplacian of the Gauss map by using (3.2) and (4.3) as

[DELTA]G = -[epsilon] (n[[alpha].sup.2] + n(n-1)[a.sup.2]/[t.sup.2n)G,

which implies that the rotation hypersurface [M.sub.q,T] for the function (4.13) has proper pointwise 1-type Gauss map of the first kind if a [not equal to] 0. For instance, when [alpha] = 0, the generalized catenoids of the first and the third kind have proper pointwise 1-type Gauss map of the first kind. If a = 0 and [alpha] [not equal to] 0, then [M.sub.q,T] has 1-type Gauss map. In this case, [M.sub.0,T] is a part of a hyperbolic n-space [H.sup.n]([c.sub.0] [[eta].sub.n+1],-1/[absolute value of [alpha]]) when [epsilon] = 1, and the Lorentzian rotation hypersurface [M.sub.1,T] of [L.sup.n+1] is a part of the de Sitter n-space [S.sub.1.sup.n]([c.sub.0][[eta].sub.n+1],1/[absolute value of [alpha]]) when [epsilon] = -1 for some [c.sub.0] [member of] R ().

Example 4.5. The Gauss map of the rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] defined by (2.3) for the function g(t) given by (4.14) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.18)

with [[??].sub.[??] = <[??], [??]>= - [??]. By a direct calculation from (3.2) we have the Laplacian of the Gauss map as

[DELTA][??] = -[??] ([n[??].sup.2] + n(n - 1)[a.sup.2]/[t.sup.2n])[??],

which implies that the rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for the function (4.14) has proper pointwise 1-type Gauss map of the first kind if a [not equal to] 0. For instance, when [??] = 0, the generalized catenoids of the second and the fourth kind have proper pointwise 1-type Gauss map of the first kind. If a = 0 and [??] [not equal to] 0, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] has 1-type Gauss map. In this case, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is a part of a hyperbolic n-space [H.sup.n]([c.sub.0] [[eta].sub.n+1], - 1/[absolute value of [??]]) when the Lorentzian rotation hypersurface M1,S1 of [L.sup.n+1] is a part of the de Sitter n-space Sn [??] = -1 for some [c.sub.0] [member of] R ().

Example 4.6. Now we consider the rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] defined by (2.4) for the function g(t) given by (4.15). Then the Gauss map [??] of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.19)

with [[??].sub.[??] = <[??],[??]> 1. By a direct calculation from (3.2) we have the Laplacian of the Gauss map as

[DELTA][??] (n[[??].sup.2] + n(n - 1)[a.sup.2]/[t.sup.2n])[??],

which implies that the rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for the function (4.15) has proper pointwise 1-type Gauss map of the first kind if a [not equal to] 0. For instance, when [??] = 0, the generalized catenoids of the fifth kind have proper pointwise 1-type Gauss map of the first kind. If a = 0 and [??] [not equal to] 0, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]2 has 1- type Gauss map, and it is a part of the de Sitter n-space [S.sub.1.sup.n] ([c.sub.0][[eta].sub.n+1], 1/[absolute value of [??]]) for some [c.sub.0] [member of] R ().

Example 4.7. The rotation hypersurface [M.sub.q,L] of [L.sup.n+1] defined by (2.5) for the function g(t) given by (4.16) has the Gauss map as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.20)

with a - [??][t.sup.n] > 0 and [[epsilon].sub.[??]] = <[??], [??]> = -[??], where [[??].sub.n] are [[??].sub.n+1] vectors in the pseudo-orthonormal basis given in the definition of [M.sub.q,L]. By a direct calculation from (3.2) we have the Laplacian of the Gauss map as

[DELTA][??] = -[??](n[[??].sup.2] + n(n - 1)[a.sup.2]/4[t.sup.2n])[??],

which implies that the rotation hypersurface [M.sub.q,L] for the function (4.16) has proper pointwise 1-type Gauss map of the first kind if a [not equal to] 0. For instance, when [??] = 0, the Enneper's hypersurfaces of the second and the third kind  have proper pointwise 1-type Gauss map of the first kind. If a = 0 and [??] [not equal to] 0, then [M.sub.q,L] has 1-type Gauss map. In this case, [M.sub.0,L] is a part of a hyperbolic n-space [H.sup.n] ([c.sub.0][??].sub.n+1], 1/[absolute value of [??]]) when [??] = 1, and the Lorentzian rotation hypersurface [M.sub.1,L] of [L.sup.n+1] is a part of the de Sitter n-space [S.sub.1.sup.n] ([c.sub.0][??].sub.n+1], 1/[absolute value of [??]]) when [??] = -1 for some [c.sub.0] [member of] R ().

Example 4.8. (Spherical n-cone) Consider the rotation hypersurface [M.sub.q,T] of [L.sup.n+1] parametrized by (2.2) for the functions [phi](t) = t, (t > 0) and [psi](t) = at, a > 0. It is a right n- cone [C.sub.a,]T with a time-like rotation axis based on a sphere [S.sup.n-1], which is space-like if 0 < a < 1 and time-like if [absolute value of [a]] > 1. The Gauss map G of [C.sub.a,T] and its Laplacian [DELTA]G are, respectively, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [epsilon] = sgn(1 - [a.sup.2]). Therefore the spherical n-cone [C.sub.a,T] has proper pointwise 1-type Gauss map of the second kind.

Example 4.9. (Hyperbolic n-cone) Now we consider the rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] parametrized by (2.3) for the functions [phi](t) = t, (t > 0) and [psi](t) = at, a > 0. It is a hyperbolic n-cone [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of the first kind with a space-like rotation axis based on a hyperbolic space [H.sup.n-1], which is space-like if [absolute value of [a]] > 1 and time-like if 0 < a < 1. The Gauss map [??] of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and its Laplacian [DELTA][??] are, respectively, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [??] = sgn([a.sup.2]-1). Therefore the hyperbolic n-cone [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of the first kind has proper pointwise 1- type Gauss map of the second kind.

Example 4.10. (Pseudo-spherical n-cone) The rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] parametrized by (2.4) for the functions [phi](t) = t, (t > 0) and [psi](t) = at, a > 0 is a hyperbolic n-cone [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of the second kind with a space-like rotation axis. It is a time-like (Lorentzian) n-cone based on a pseudo-sphere [S.sub.1.sup.n-1] which has the Gauss map [??] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the Laplacian [DELTA][??] of Gauss map [??] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [??] = sgn([a.sup.2]-1). Therefore the pseudo-spherical n-cone [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of the second kind has proper pointwise 1-type Gauss map of the second kind.

The notion of rotation surfaces of polynomial and rational kinds was introduced by Chen and Ishikawa in . A rotation hypersurface in [L.sup.n+1] is said to be of polynomial kind if the functions [phi](t) and [psi](t) in the parametrization of the rotation hypersurfaces given in the first section are polynomials, and it is said to be of rational kind if [phi](t) and [psi](t) are rational functions. A rotation hypersurface of rational kind is simply called rational rotation hypersurface. Without loss of generality we consider rotation hypersurfaces [M.sub.q,T], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], or in [L.sup.n+1] given by (2.2), (2.3), and (2.4), respectively, for [phi](t) = t, t > 0 and [psi](t) = g(t), where g(t) is a function of class [C.sup.3].

By the following theorem we classify rational rotation hypersurfaces of [L.sup.n+1] in terms of pointwise 1-type Gauss map of the first kind.

Theorem 4.11.

(1) A rational rotation hypersurface [M.sub.q,T] of [L.sup.n+1] parametrized by (2.2) has pointwise 1-type Gauss map of the first kind if and only if it is an open portion of a space-like hyperplane or a Lorentzian cylinder [S.sup.n-1] x [L.sup.1] of [L.sup.n+1].

(2) A rational rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] parametrized by (2.3) has pointwise 1-type Gauss map of the first kind if and only if it is an open portion of a time-like hyperplane or a hyperbolic cylinder [H.sup.n-1] x R of [L.sup.n+1].

(3) A rational rotation hypersurface [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [L.sup.n+1] parametrized by (2.4) has pointwise 1-type Gauss map of the first kind if and only if it is an open portion of a time-like hyperplane or a pseudo-spherical cylinder [S.sub.1.sup.n-1] x R of [L.sup.n+1]

(4) A rational rotation hypersurface [M.sub.q,L] of [L.sup.n+1] parametrized by (2.5) has pointwise 1-type Gauss map of the first kind if and only if it is an open portion of hyperbolic n-space [H.sup.n], de Sitter n-space [S.sub.1.sup.n] or Enneper's hypersurface of the second kind or the third kind.

Moreover, the Enneper's hypersurfaces of the second kind and the third kind of [L.sup.n+1] are the only polynomial rotation hypersurfaces of [L.sup.n+1] with proper pointwise 1-type Gauss map of the first kind.

Proof. In the parametrization (2.2) of [M.sub.q,T], if [phi] is a constant, the hypersurface [M.sub.q,T] is an open portion of a Lorentzian cylinder [S.sup.n-1] x [L.sup.1] of [L.sup.n+1]. If [phi] is not a constant, we put [phi] = t, t > 0 and [psi] (t) = g(t) in the parametrization (2.2) of [M.sub.q,T]. In  it was shown that [M.sub.q,T] has constant mean curvature [alpha] if and only if the function g(t) is given by (4.13). Now, if a = [alpha] = 0 in (4.13), then g(t) is a constant. In this case, the hypersurface [M.sub.q,T] is an open portion of a space-like hyperplane.

If a [not equal to] 0 and [alpha] = 0, that is, [M.sub.q,T] is the generalized catenoid of the first or the third kind (), then (4.13) implies that the function g(t) can be expressed in terms of elliptic functions and g(t) is not a rational function of t.

If a = 0 and [alpha] [not equal to] 0, then from (4.13), we get g(t) = [[alpha].sup.-1] [square root of ([[alpha].sup.2][a.sup.2] + [epsilon] + c)], which is not rational, where c is an arbitrary constant and t > 1/[absolute value of (a)] when [epsilon] = -1. Therefore [M.sub.q,T] is not rational kind. In this case, the hypersurface [M.sub.q,T] is an open portion of a hyperbolic n-space [H.sup.n-1] when [epsilon] = 1 or an open portion of a de Sitter n-space [S.sub.1.sup.n-1] when [epsilon] = -1.

If a[alpha] [not equal to] 0, then g(t) given by (4.13) cannot be rational even if n = 2. If it were rational, its derivative would be rational, which contradicts the integrand in (4.13). The converse is trivial.

Parts 2 and 3 can similarly be proved by using (2.3), (2.4), (4.14), and (4.15).

For the proof of part 4, we put [phi] = t, t > 0 and [psi] (t) = g(t) in the parametrization (2.5) of [M.sub.q,L]. In , it was proved that [M.sub.q,L] has constant mean curvature [??] if and only if the function g(t) is given by (4.16). Now, if a = 0 and [??] [not equal to] 0 in (4.16), then we obtain g(t) = c - [??]/4[[??].sup.2]t which is a rational function, and [M.sub.q,L] is an open part of a hyperbolic n-space [H.sup.n] when q = 0 ([??] = 1) and [M.sub.q,L] is an open part of a de Sitter n-space [S.sub.1.sup.n] when q = 1 ([??] = -1).

If a [not equal to] 0 and [??] = 0, then we have g(t) = [??][t.sup.2n- 1]/[a.sup.2](2n-1) + c which is a polynomial. In this case, [M.sub.q,L] is an open portion of Enneper's hypersurface () of the second or the third kind according to [??] = 1 or [??] = -1. From Example 4.7 it is seen that Enneper's hypersurfaces are the only polynomial (rational) rotation hypersurfaces of [L.sup.n+1] with proper pointwise 1-type Gauss map of the first kind.

If a [??] [not equal to[ 0, then the function g(t) given by (4.16) is not rational for n [greater than or equal to] 2 because the integration of [t.sup.2(n-1)]/[(a-2[??][t.sup.n]).sup.2] contains at least one term involving a logarithmic or arctangent function. The converse of part 4 follows from Corollary 3.4 and Example 4.7.

Corollary 4.12. The rotation hypersurface [M.sub.q,L] of [L.sup.n+1] parametrized by (2.5) for the function g(t) = c- [??]/4[[??].sup.a]t is the only non-polynomial rational rotation hypersurface of [L.sup.n+1] with pointwise 1-type Gauss map.

The proof follows from the proof of Theorem 4.11 and Example 4.7.

Theorem 4.13. Let [M.sub.q] be one of the rotation hypersurfaces [M.sub.q,T], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or in [L.sup.n+1] parametrized by (2.2), (2.3), and (2.4), respectively. If [M.sub.q] is a polynomial kind rotation hypersurface, then it has proper pointwise 1-type Gauss map of the second kind if and only if it is an open portion of a spherical n-cone, hyperbolic n-cone, or pseudo-spherical n-cone.

Proof. Let [M.sub.q] = [M.sub.q,T]. In the parametrization (2.2) of [M.sub.q,T] we take [phi](t) = t, t > 0 and [psi](t) = g(t), where g(t) is a polynomial. Then we have the Gauss map G from (4.1) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.21)

with [[epsilon].sub.G] = <G,G> = -[epsilon], where [epsilon] = sgn(1-[g.sup.'2]) = [+ or -]1 and [absolute value of (g')] [not equal to] 1. Also, from (4.5) the Laplacian of the Gauss map G is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.22)

where [[paralle][A.sub.G][parallel].sup.2] is given by (4.3) for [phi](t) = t and [psi](t) = g(t) and the derivative of [alpha] from (4.2) is evaluated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Suppose that M has pointwise 1-type Gauss map of the second kind. Then, by definition, the vector C in (1.1) is nonzero and by Lemma 4.1 C = c [[eta].sub.n+1] for some nonzero constant c. Thus, (1.1) and (4.22) imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Eliminating f in the above equations and using (4.3) and (4.23), we obtain

P(t) = c [square root of ([epsilon](1 - [g.sup.'2])Q(t))], (4.24)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If degg(t) [greater than or equal to] 2, then degP(t) = degQ(t) [greater than or equal to] 7, which is a contradiction. Consequently, degg(t) = 1. That is, g'(t) = a for some nonzero constant a with [absolute value of (a)] [not equa to] 1. Hence, we get c = -1 [square root of ([epsilon](1-[a.sup.2))]. Therefore, the rotation hypersurface [M.sub.q,T] with the parametrization (2.2) for [phi](t) = t, t > 0 and [psi](t) = at + b is an open portion of a spherical n-cone. The proof of the converse for [M.sub.q] = [M.sub.q,T] follows from Example 4.8.

By a similar discussion as above it can be shown that if [M.sub.q] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or [M.sub.q] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then it is an open portion of a hyperbolic n-cone or an open portion of a pseudo-spherical n-cone, respectively.

Theorem 4.14. There do not exist rational rotation hypersurfaces [M.sub.q,T], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], or in [L.sup.n+1], except polynomial hypersurfaces, with pointwise 1-type Gauss map of the second kind.

Proof. Let [M.sub.q] = [M.sub.q,T]. Assume that [M.sub.q,T] is a rational rotation hypersurface in [L.sup.n+1], except polynomial hypersurface, with pointwise 1-type Gauss map of the second kind. In the parametrization (2.2) of [M.sub.q,T], we take [phi](t) = t, t > 0 and [psi](t) = g(t), where g(t) is a rational function. The derivatives of g(t) are also rational functions in t. We may put g'(t) = r(t)/q(t), where r(t) and q(t) are relative prime polynomials. Let degq(t) = k.

From (4.24) we know that [square root of ([psilon](1-[g.sup.'2]) is also a rational function. Hence there exists a polynomial p(t) satisfying [q.sup.2](t) - [r.sup.2](t) = [epsilon][p.sup.2](t), where r(t), q(t), and p(t) are relatively prime. Put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then these functions are also rational.

Suppose that k [greater than or equal to] 1. Then, for each i = 1, ..., 4, we see that [q.sup.7] (t)[P.sub.i](t) is a polynomial. Similarly, we see that for each i = 1, ..., 3, [q.sup.6](t)Qi(t) is a polynomial. However, we have

[q.sup.6](t)[Q.sub.4](t) = [epsilon]r(r)[p.sup.6](t)/q(t)

As (4.24) gives

P(t) = c p(t)/q(t) Q(t) (4.25)

it follows that [q.sup.6](t)[Q.sub.4](t) is a polynomial. This is a contradiction because r(t), q(t), and p(t) are relatively prime. Therefore g'(t) is not rational, so is g(t). Hence k = 0, that is, g(t) is a polynomial, and by Theorem 4.13 [M.sub.q] = [M.sub.q,T] is nothing but a spherical n-cone.

By a similar discussion, when [M.sub.q] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] or [M.sub.q] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], we have the same result.

Theorem 4.15. There do not exist rational rotation hypersurfaces [M.sub.q,L] in [L.sup.n+1] with a light-like axis and pointwise 1-type Gauss map of the second kind.

Proof. Suppose that [M.sub.q,L] given by (2.5) is a rational rotation hypersurface with pointwise 1-type Gauss map of the second kind. Then we put [phi](t) = t, t > 0 and [psi](t) = g(t) in (2.5), where g(t) is a rational function. From (4.7) and (4.11) the Gauss map [??] G of [M.sub.q,L] and its Laplacian [DELTA][??] are, respectively, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.26)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.27)

where [??] = sgn(g') = [+ or -]1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.28)

from (4.8), and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.29)

Since [M.sub.q,L] has pointwise 1-type Gauss map of the second kind, by Lemma 4.3 the vector C in the definition (1.1) is parallel to [[eta].sub.n+1], that is, C = c [[eta].sub.n+1], and (1.1) and (4.27) imply that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Eliminating f in the above equations, and using (4.28) and (4.29), we obtain

P(t) = c [square root of (2[??]g'Q(t), (4.30)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which are rational functions as g(t) is rational. The function [square root of ([??]g')] in (4.30) is also a rational function. Then we may put g' = [??][r.sup.2](t)/[q.sup.2](t), where r(t) and q(t) are relatively prime polynomials. Taking derivative, we have

g"(t) = [??][R.sub.1](t)/[q.sup.3] and g"'(t) = [??][R.sub.2](t)/[q.sup.4],

where

[R.sub.2](t) = 2([q.sub.2][r'.sub.2] + [q.sub.2] r" - 4rqr' q' - [r.sub.2] + 3[r.sub.2][q'.sub.2]

which are polynomials in t. Hence,

P(t) ] [??][r.sub.2][??](t)/[q.sub.8] and Q(t) = [??](t)/[q.sub.6],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore equation (4.30) becomes

r(t)[??](t) = c[??][square root of (2q)](t)[??](t). (4.31)

Let degr(t) = m and degq(t) = k. We may write r(t) = [[SIGMA].sub.s=0.sup.m] [a.sub.s][t.sup.s] and q(t) [[SIGMA].sub.s=0.sup.k][b.sub.s][t.sup.s] such that [a.sub.m] [not equal to] 0 and [b.sub.k] [not equal to] 0. Then, by a straightforward computation we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.32)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.33)

where [d.sub.0] = 2([a.sub.1.sup.2][b.sub.0.sup.2] + 3[a.sub.0.sup.2][b.sub.1.sup.2] + 2[b.sub.0.sup.2][a.sub.0][a.sub.2] - 2[a.sub.0.sup.2][b.sub.0][b.sub.2] - 4[a.sub.0][b.sub.0][a.sub.1][b.sub.1]. Using (4.32) and (4.33), we get deg [??](t) = 4m+2k and deg[??](t) = 4m+2k if m [not equal to] k, and deg[??](t) [less than or equal to] 4m+2k-1 if m = k:

Now, if m [not equal to] k, then deg(r(t)[??](t)) = 5m+2k and deg(q(t)[??](t)) = 4m+3k. Hence, by comparing the degree of the polynomials r(t)[??](t) and q(t)[??](t), from (4.31) we have 5m+2k = 4m+3k, which implies that m = k, which is a contradiction. If m = k, then deg(r(t)[??](t)) = 7m and deg(q(t)[??](t)) [less than or equal to] 7m-1, which is also a contradiction because of (4.31). Therefore [square root of (2[??]g')] is not a rational function, and so is g(t).

Corollary 4.16. There do not exist polynomial rotation hypersurfaces [M.sub.q,L] in [L.sup.n+1] with a light-like axis and pointwise 1-type Gauss map of the second kind.

Considering Theorem 4.13, Theorem 4.14, and Theorem 4.15, we have the following classification theorem for rational rotation hypersurfaces of [L.sup.n+1] with pointwise 1-type Gauss map of the second kind.

Theorem 4.17. Let M be a rational rotation hypersurface of [L.sup.n+1]. Then M has pointwise 1-type Gauss map of the second kind in [L.sup.n+1] if and only if it is an open portion of a spherical n-cone, hyperbolic n-cone, or pseudo-spherical n-cone.

Received 17 September 2008, accepted 3 November 2008

doi: 10.3176/proc.2009.3.01

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Ugur Dirsun

Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey; udursun@itu.edu.tr
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