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 [7] for a report on this subject).

In [9] 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] [10], generalized catenoids and right n-cones in [E.sup.n+1] [11], 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] [16] 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 [16], Kim and Yoon gave the complete classification of ruled surfaces in a 3-dimensional Minkowski space with pointwise 1-type Gauss map; in [18] 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 [17]. Recently, in [13], 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 [13] 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 [12]). 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 [12] 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 [12].

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 [12]. By a direct computation (or following [12]) 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 [12]) 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 [12], 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 ([12]).

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 ([12]).

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 ([12]).

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 [12] 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 ([12]).

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 [8]. 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 [12] 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 ([12]), 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 [12], 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 ([12]) 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