On Finite Type Closed Curves on the Pseudo-Hyperbolic Space H^{3}(−c^{2})

Yong-Soo Pyo and Joo-Yeon Lee

Abstract

We obtain some nonexistence theorems of certain finite type closed curves on the pseudo-hyperbolic space H^{3}(−c^{2}) in the Minkowski spacetime E_{1}^{4}.

1991 Mathematics Subject Classification : 53C50 Key words: pseudo-Riemannian manifold, pseudo-Euclidean space, Minkowski spacetime, pseudo-hyperbolic space, finite type closed curve

1 Introduction

First, we will survey briefly the fundamental concepts and properties in the pseudo-Riemannian geometry. We refer mainly to O’Neill([9]) and Chen([3],[4]). For the general concepts in the Riemannian geometry, refer to the book of Kobayashi and Nomizu([8]).

Let M be a C^{∞}-class differentiable manifold of dimension n and g a C^{∞}-class differentiable symmetric nondegenerate tensor field of type (0, 2) on M. The pseudo-Riemannian metric g_{p} at every point p of M defines the scalar product on the tangent space T_{p}(M) of M at p. The index of g_{p} is not necessarily constant in general. If the index of g_{p} is constant t(0 ≤ t ≤ n) on M, then we call g a pseudo-Riemannian metric of signature (t, n − t). And a C^{∞}-class differentiable manifold (M, g) furnished with a pseudo-Riemannian metric g is called a pseudo-Riemannian manifold. A pseudo-Riemannian manifold of signature (0, n) means a Riemannian manifold. Let v be a tangent vector to a pseudo-Riemannian manifold M with a pseudo-Riemannian metric g. Then v is said to be

spacelike if g(v, v) > 0 or v = 0; lightlike if g(v, v) = 0 and v ≠ 0, timelike if g(v, v) < 0.

The simplest example of pseudo-Riemannian manifold is a pseudo-Euclidean space. Let (x^{1}, x^{2}, ⋯, x^{m}) be a point in the set R^{m} of all ordered m-tuples of real num-

bers. For each t(0 ≤ t ≤ m), we define a scalar product g_{0} on T_{p}(R^{m}) at the point p of R^{m} by

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 111-120 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

g_{0}(v_{p}, w_{p}) = − ∑_{i=1}^{t} v^{i}w^{i} + ∑_{i=t+1}^{m} v^{i}w^{i},

where v_{p} = ∑_{i=1}^{m} v^{i}∂/∂x^{i} and w_{p} = ∑_{i=1}^{m} w^{i}∂/∂x^{i}. E_{t}^{m} denotes a R^{m} with a canonical

pseudo-Riemannian metric g_{0}. In this case, g_{0} is called a pseudo-Euclidean metric of signature (t, m − t) and E_{t}^{m} is called a pseudo-Euclidean space of signature (t, m − t). In particular, E_{1}^{m} is called a Minkowski spacetime.

From now on, we will use <, > instead of a pseudo-Euclidean metric g_{0}. And we denote by H_{t}^{m}(−c^{2}) = {p ∈ E_{t+1}^{m+1}| < p, p >= −c^{2}}. In this case, it is called the pseudo-hyperbolic space of radius c > 0 and center 0 in E_{t+1}^{m+1}. For a vector a_{0} = (a_{1}, a_{2}, ⋯, a_{t}, ⋯, a_{m}) in E_{t}^{m},

ā_{0} = (−a_{1}, −a_{2}, ⋯, −a_{t}, a_{t+1}, a_{t+2}, ⋯, a_{m})

is called the conjugate vector of a_{0}. In [7] and [10], the authors proved the following Theorem A. Only 1-type closed curve γ(s) on H_{t}^{m}(−c^{2}) is an intersection of H_{t}^{m}(−c^{2}) and a 2-plane P lying in Π_{a_{0}}, where P is determined by two spacelike vectors and Π_{a_{0}} denotes a hyperplane through a_{0} which is orthogonal to the conjugate vector ā_{0} in the sense of Euclidean scalar product.

Ishikawa([7]), and Shin and Pyo([11]) also proved some nonexistence theorems concerning finite type closed curves on pseudo-hyperbolic spaces H^{2}(−c^{2}) and H^{4}(−c^{2}). For instance, Theorem B. There exists neither 2-type closed curves nor 3-type closed curves on H^{2}(−c^{2}). Remark. Finite type curves in a Euclidean space were investigated in [1], [2], [5],

[6] etc.

The purpose of this article is to prove some theorems on nonexistence of certain finite type closed curves on the pseudo-hyperbolic space H^{3}(−c^{2}) in the Minkowski spacetime E_{1}^{4}.

2 Preliminaries

Every closed curve γ : [0, 2πr] → E_{t}^{m} of the length 2πr in E_{t}^{m} may be regarded as an isometric immersion of a circle of radius r into E_{t}^{m}. We use the arc length s as a parameter of γ. Then the Laplacian Δ on the circle is given by Δ = −d^{2}/ds^{2} and the eigenvalues are {(l/r)^{2}; l = 1, 2, ⋯}. The corresponding eigenspace V_{l} is constructed by using cos(ls/r) and sin(ls/r). Hence, every closed curve γ : [0, 2πr] → E_{t}^{m} has the spectral decomposition

γ(s) = a_{0} + ∑_{l=1}^{∞}{a_{l} cos(ls/r) + b_{l} sin(ls/r)},

where a_{l}, b_{l} are some vectors in E_{t}^{m} (see [2],[5]). In particular, if γ is a k-type closed curve of the length 2π on H_{t}^{m}(−c^{2}), then γ can be expressed as

(2.1)  γ(s) = a_{0} + ∑_{i=1}^{k}{a_{i} cos(p_{i}s) + b_{i} sin(p_{i}s)},

where a_{i} or b_{i} is nonzero vector in E_{t}^{m} for each i = 1, 2, ⋯, k, p_{i} are the positive integers with p_{1} < p_{2} < ⋯ < p_{k} and s is the arc length parameter of γ. Because of γ(s) being on H_{t}^{m}(−c^{2}) and a_{0} the center of mass of γ, a_{0} is a timelike vector in E_{t+1}^{m+1} (see [7]). Furthermore, from < γ(s), γ(s) >= −c^{2}, we have the following

(2.2)  2 < a_{0}, a_{0} > +2c^{2} + ∑_{i=1}^{k} D_{ii} = 0,

(2.3)  ∑_{p_{i}=l} M_{i} + ∑_{2p_{i}=l} A_{ii} + 2 ∑_{{p_{i}+p_{j} =l \atop i>j}} A_{ij} + 2 ∑_{{p_{i}−p_{j} =l \atop i>j}} D_{ij} = 0,

(2.4)  ∑_{p_{i}=l} M̄_{i} + ∑_{2p_{i}=l} Ā_{ii} + 2 ∑_{{p_{i}+p_{j} =l \atop i>j}} Ā_{ij} − 2 ∑_{{p_{i}−p_{j} =l \atop i>j}} D̄_{ij} = 0,

for each l ∈ {p_{i}, 2p_{i}, p_{i} + p_{j}, p_{i} − p_{j} ; 1 ≤ j < i ≤ k}, where

\begin{matrix}M_{i} = 4 < a_{0}, a_{i} >, & M̄_{i} = 4 < a_{0}, b_{i} >, \\ A_{ij} =< a_{i}, a_{j} > − < b_{i}, b_{j} >, & Ā_{ij} =< a_{i}, b_{j} > + < b_{i}, a_{j} >, \\ D_{ij} =< a_{i}, a_{j} > + < b_{i}, b_{j} >, & D̄_{ij} =< a_{i}, b_{j} > − < b_{i}, a_{j} > . \end{matrix}

From now on, we call the real numbers M_{i} and M̄_{i} (resp. A_{ii} and Ā_{ii}, A_{ij} and Ā_{ij}, or D_{ij} and D̄_{ij}) to be corresponding to the integer p_{i}(resp. 2p_{i}, p_{i} + p_{j}, or p_{i} − p_{j}). Since s is the arc length parameter of γ(s), we have

(2.5)  2 = ∑_{i=1}^{k} p_{i}^{2}D_{ii},

(2.6)  ∑_{2p_{i}=l} p_{i}^{2}A_{ii} + 2 ∑_{{p_{i}+p_{j} =l \atop i>j}} p_{i}p_{j}A_{ij} − 2 ∑_{{p_{i}−p_{j} =l \atop i>j}} p_{i}p_{j}D_{ij} = 0,

(2.7)  ∑_{2p_{i}=l} p_{i}^{2}Ā_{ii} + 2 ∑_{{p_{i}+p_{j} =l \atop i>j}} p_{i}p_{j}Ā_{ij} + 2 ∑_{{p_{i}−p_{j} =l \atop i>j}} p_{i}p_{j}D̄_{ij} = 0.

Moreover, if < γ^{(r)}(s), γ^{(r)}(s) > is constant (r = 1, 2, ⋯), then we have

(2.8)  ∑_{2p_{i}=l} p_{i}^{2r}A_{ii} + 2 ∑_{{p_{i}+p_{j} =l \atop i>j}}(p_{i}p_{j})^{r}A_{ij} + (−1)^{r}2 ∑_{{p_{i}−p_{j} =l \atop i>j}}(p_{i}p_{j})^{r}D_{ij} = 0,

(2.9)  ∑_{2p_{i}=l} p_{i}^{2r}Ā_{ii} + 2 ∑_{{p_{i}+p_{j} =l \atop i>j}}(p_{i}p_{j})^{r}Ā_{ij} − (−1)^{r}2 ∑_{{p_{i}−p_{j} =l \atop i>j}}(p_{i}p_{j})^{r}D̄_{ij} = 0.

Next, let γ be a k-type closed curve on H_{t}^{m}(−c^{2}) given in (2.1). Divide the set A = {p_{i}, 2p_{i}, p_{i} + p_{j}, p_{i} − p_{j} ; 1 ≤ j < i ≤ k} as the union of the subsets as follows:

(2.10)  A = A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{N},

where all elements in each subset A_{n}(n = 1, 2, ⋯, N) are equal to each other and if n_{1} ≠ n_{2}, then every element in A_{n_{1}} is not equal to any element in A_{n_{2}}.

3 Main Results

Let γ be a closed k-type curve on H_{t}^{m}(−c^{2}) in E_{t+1}^{m+1}. Then γ is expressed as γ(s) =

a_{0} + ∑_{i=1}^{k}{a_{i} cos(p_{i}s/r) + b_{i} sin(p_{i}s/r)}, where a_{i} or b_{i} is nonzero vector in E_{t+1}^{m+1} for

each i = 1, 2, ⋯, k and p_{i} are the positive integers satisfying p_{1} < p_{2} < ⋯ < p_{k}. Here s is the arc length parameter of γ and the length of γ is 2πr. Therefore every k-type closed curve γ(s) of the length 2π may be described as

(3.1)  γ(s) = a_{0} + ∑_{i=1}^{k}{a_{i} cos(p_{i}s) + b_{i} sin(p_{i}s)},

where a_{i} ≠ 0 or b_{i} ≠ 0 for each i. We prove our results for r = 1, because the proof for case r ≠ 1 is the same as one for case of r = 1. Lemma 3.1([7]). (1) If < γ^{(r)}(s), γ^{(r)}(s) > is constant (r = 1, 2, ⋯, l) and the number of members in A_{n} is less than or equal to l + 1, then M_{i} and M̄_{i} (resp. A_{ii} and Ā_{ii}, A_{ij} and Ā_{ij}, or D_{ij} and D̄_{ij}) of corresponding to the integer p_{i} (resp. 2p_{i}, p_{i} + p_{j}, or p_{i} − p_{j}) in A_{n} vanish.

(2) In particular, for every k-type closed curve γ(s) on H_{t}^{m}(−c^{2}) in E_{t+1}^{m+1}, we have

A_{kk} = Ā_{kk} = 0,
A_{k(k−1)} = Ā_{k(k−1)} = 0,
A_{(k−1)(k−1)} = Ā_{(k−1)(k−1)} = 0.

Now, let γ(s) be a k-type closed curve on H^{3}(−c^{2}) in a Minkowski spacetime E_{1}^{4} as (3.1). Then we can obtain the following lemmas. Lemma 3.2. If γ(s) satisfies the following conditions

(3.2)  M_{k} = M̄_{k} = M_{k−1} = 0 and D_{k(k−1)} = D̄_{k(k−1)} = 0,

then either (1) {a_{0}, a_{k−1}, a_{k}, b_{k}} forms a basis for E_{1}^{4}, or (2) b_{k−1} is a lightlike vector and {a_{0}, b_{k−1}, a_{k}, b_{k}} is a basis of E_{1}^{4}. Proof. Since a_{0} is a timelike vector in E_{1}^{4}, from the first equation of (3.2), we know that a_{k−1}, a_{k} and b_{k} are spacelike vectors. Hence a_{k} and b_{k} are nonzero vectors because < a_{k}, a_{k} >=< b_{k}, b_{k} > and γ(s) is of k-type. From Lemma 3.1(2) and the second equation of (3.2), we have

\begin{aligned}< a_{k−1}, a_{k−1} > & = & < b_{k−1}, b_{k−1} >, \\ < a_{k}, b_{k} > & = & < a_{k−1}, b_{k−1} >= 0, \\ < a_{k}, a_{k−1} > & = & < b_{k}, b_{k−1} >= 0, \end{aligned}

and

< a_{k}, b_{k−1} >=< b_{k}, a_{k−1} >= 0.

If a_{k−1} ≠ 0, then a_{0}, a_{k−1}, a_{k}, b_{k} are linearly independent vectors in E_{1}^{4} and hence {a_{0}, a_{k−1}, a_{k}, b_{k}} is a basis of E_{1}^{4}.

Suppose a_{k−1} = 0. Since < a_{k−1}, a_{k−1} >=< b_{k−1}, b_{k−1} >= 0 and γ(s) is of k-type, b_{k−1} is a lightlike vector. If we put Aa_{0} + Bb_{k−1} + Ca_{k} + Db_{k} = 0, then we can obtain A = B = C = D = 0 because < a_{0}, b_{k−1} >≠ 0(see [11]). Therefore we complete the proof. Remark. If the k-type closed curve γ on H^{3}(−c^{2}) satisfies M_{k−1} = 0 and b_{k−1} is a lightlike vector in E_{1}^{4}, then a_{k−1} = 0 because < a_{k−1}, a_{k−1} >=< b_{k−1}, b_{k−1} >= 0. Lemma 3.3. Suppose that {a_{0}, a_{k−1}, a_{k}, b_{k}} is a basis of E_{1}^{4} satisfying (3.2). Then b_{k−1} is a parallel nonzero vector to a_{0}. Proof. Put b_{k−1} = Aa_{0} + Ba_{k−1} + Ca_{k} + Db_{k}. Combining Lemma 3.1(2) and (3.2), we have B = C = D = 0 because a_{k−1}, a_{k} and b_{k} are nonzero spacelike vectors. Since < a_{0}, b_{k−1} >≠ 0, b_{k−1} = Aa_{0} ≠ 0 for a constant A. Next, we can obtain the following Lemma 3.4. Suppose that {a_{0}, a_{k−1}, a_{k}, b_{k}} is a basis of E_{1}^{4} satisfying (3.2). If a pair {a_{i}, b_{i}}(i = 1, 2, ⋯, k − 2) satisfies

A_{ki} = Ā_{ki} = 0

and

< a_{k−1}, a_{i} >=< a_{k−1}, b_{i} >= 0,

then A_{ii} = Ā_{ii} = 0 if and only if M_{i} = M̄_{i} = 0. Proof. Put a_{i} = Aa_{0} + Ba_{k−1} + Ca_{k} + Db_{k} and b_{i} = Ea_{0} + F a_{k−1} + Ga_{k} + Hb_{k}. Combining Lemma 3.1(2), (3.2) and our assumptions, we have B = F = 0, C = H and D = −G. Hence a_{i} = Aa_{0} + Ca_{k} + Db_{k} and b_{i} = Ea_{0} − Da_{k} + Cb_{k} for some constants A, C, D and E.

Suppose A_{ii} = Ā_{ii} = 0. Then we get AE = 0 and A^{2} − E^{2} = 0 because < a_{k}, a_{k} >=< b_{k}, b_{k} > and < a_{k}, b_{k} >= 0, and hence A = E = 0. Therefore M_{i} = 4 < a_{0}, a_{i} >= 0 and M̄_{i} = 4 < a_{0}, b_{i} >= 0.

Conversely, if M_{i} = M̄_{i} = 0, then we have a_{i} = Ca_{k} + Db_{k} and b_{i} = −Da_{k} + Cb_{k} for some constants C and D. Hence < a_{i}, a_{i} >=< b_{i}, b_{i} > and < a_{i}, b_{i} >= 0. Lemma 3.5. Suppose that {a_{0}, a_{k−1}, a_{k}, b_{k}} is a basis of E_{1}^{4} satisfying (3.2). If a pair {a_{i}, b_{i}}(i = 1, 2, ⋯, k − 2) is satisfying

A_{ki} = Ā_{ki} = 0, D_{ki} = D̄_{ki} = 0

and

< a_{k−1}, a_{i} >=< a_{k−1}, b_{i} >= 0,

then a_{i} and b_{i} are parallel to a_{0}.

Proof. If we put a_{i} = Aa_{0} +Ba_{k−1} +Ca_{k} +Db_{k} and b_{i} = Ea_{0} +F a_{k−1} +Ga_{k} +Hb_{k}, then we have, from Lemma 3.1(2), (3.2) and our assumptions, B = C = D = 0, F = G = H = 0. It follows that a_{i} = Aa_{0} and b_{i} = Ea_{0} for some constants A and B. Finally, we have the following lemma. Lemma 3.6. Suppose that {a_{0}, b_{k−1}, a_{k}, b_{k}} is a basis of E_{1}^{4} satisfying (3.2). If a pair {a_{i}, b_{i}}(i = 1, 2, ⋯, k − 2) is satisfying

A_{ki} = Ā_{ki} = 0

and

< b_{k−1}, a_{i} >=< b_{k−1}, b_{i} >= 0,

then A_{ii} = Ā_{ii} = 0.

Proof. If we put a_{i} = Aa_{0}+Bb_{k−1}+Ca_{k}+Db_{k} and b_{i} = Ea_{0}+F b_{k−1}+Ga_{k}+Hb_{k}, Combining Lemma 3.1(2), (3.2) and our assumptions, we have C = H and D = −G. Since b_{k−1} is lightlike and < a_{0}, b_{k−1} >≠ 0, A = E = 0. Hence a_{i} = Bb_{k−1}+Ca_{k}+Db_{k} and b_{i} = F b_{k−1} − Da_{k} + Cb_{k} for some constants B, C, D and F. Therefore < a_{i}, a_{i} >=< b_{i}, b_{i} > and < a_{i}, b_{i} >= 0.

From now on, we prove the following nonexistence theorems for a k-type(k ≥ 2) closed curve γ(s) on H^{3}(−c^{2}). Theorem 3.1. There exists no 2-type closed curve γ(s) on H^{3}(−c^{2}). Proof. We assume the existence of the 2-type closed curve

γ(s) = a_{0} + a_{1} cos(p_{1}s) + b_{1} sin(p_{1}s) + a_{2} cos(p_{2}s) + b_{2} sin(p_{2}s)

on H^{3}(−c^{2}). From Lemma 3.1, we see

M_{1} = M̄_{1} = 0, M_{2} = M̄_{2} = 0.

Hence a_{1}, b_{1}, a_{2} and b_{2} are spacelike vectors in E_{1}^{4}. Furthermore a_{1}, b_{1}, a_{2} and b_{2} are nonzero vectors because A_{11} = A_{22} = 0 and γ(s) is of 2-type. We also have

Ā_{11} = Ā_{22} = 0, A_{21} = Ā_{21} = 0, D_{21} = D̄_{21} = 0.

Therefore a_{0}, a_{1}, b_{1}, a_{2}, b_{2} are linearly independent vectors in E_{1}^{4}. It contradicts. Theorem 3.2. There exists no 3-type closed curve γ(s) on H^{3}(−c^{2}) satisfying M_{2} = 0 and D_{32} = D̄_{32} = 0. Proof. We assume the existence of the 3-type closed curve

γ(s) = a_{0} + a_{1} cos(p_{1}s) + b_{1} sin(p_{1}s) + a_{2} cos(p_{2}s)
+ b_{2} sin(p_{2}s) + a_{3} cos(p_{3}s) + b_{3} sin(p_{3}s)

on H^{3}(−c^{2}) satisfying the assumptions M_{2} = 0 and D_{32} = D̄_{32} = 0.

First, if we assume that a_{2} ≠ 0, then {a_{0}, a_{2}, a_{3}, b_{3}} is a basis of E_{1}^{4} satisfying (3.2) by Lemmas 3.1 and 3.2. Case 1. In case of {p_{1}, p_{2}, p_{3}} = {p_{1}, 2p_{1}, 3p_{1}}, it follows that A = {p_{1}, p_{2} − p_{1}, p_{3} − p_{2}} ∪ {p_{2}, 2p_{1}, p_{3} − p_{1}} ∪ {p_{1} + p_{2}, p_{3}} ∪ {p_{1} + p_{3}, 2p_{2}} ∪ {p_{2} + p_{3}} ∪ {2p_{3}}. Applying (2.3), (2.4), (2.6) and (2.7) for the subclasses {p_{1}, p_{2} − p_{1}, p_{3} − p_{2}} and {2p_{1}, p_{2}, p_{3} − p_{1}} of A, and combining Lemmas 3.1 and 3.4, we obtain

\begin{matrix}A_{21} = Ā_{21} = 0, & D_{21} = D̄_{21} = 0, \\ A_{31} = Ā_{31} = 0, & M_{1} = M̄_{1} = 0, \\ A_{11} = Ā_{11} = 0, & D_{31} = D̄_{31} = 0 \end{matrix}

by our assumptions. Furthermore, we have M̄_{2} = 0. Hence b_{2} is a spacelike vector in E_{1}^{4}. It is a contradiction to Lemma 3.3. Case 2. In case of {p_{1}, p_{2}, p_{3}} = {p_{1}, 2p_{1}, 4p_{1}}, it follows that A = {p_{1}, p_{2} − p_{1}} ∪ {p_{2}, 2p_{1}, p_{3} − p_{2}} ∪ {p_{1} + p_{2}, p_{3} − p_{1}} ∪ {p_{3}, 2p_{2}} ∪ {p_{1} + p_{3}} ∪ {p_{2} + p_{3}} ∪ {2p_{3}}. Applying (2.3), (2.4), (2.6) and (2.7) for the subclass {2p_{1}, p_{2}, p_{3} − p_{2}} of A, we get M̄_{2} = 0 by the assumption D_{32} = D̄_{32} = 0. Hence Lemma 3.3 leads a contradiction. Case 3. In case of {p_{1}, p_{2}, p_{3}} = {p_{1}, 3p_{1}, 5p_{1}}, A = {p_{1}} ∪ {2p_{1}, p_{2} − p_{1}, p_{3} − p_{2}} ∪ {p_{2}} ∪ {p_{1} + p_{2}, p_{3} − p_{1}} ∪ {p_{3}} ∪ {p_{1} + p_{3}, 2p_{2}} ∪ {p_{2} + p_{3}} ∪ {2p_{3}}. From Lemma 3.1(1), we obtain M̄_{2} = 0. This is a contradiction. Case 4. Let {p_{1}, p_{2}, p_{3}} ≠ {p_{1}, 2p_{1}, 3p_{1}}, {p_{1}, 2p_{1}, 4p_{1}} or {p_{1}, 3p_{1}, 5p_{1}}. In this case, each subset A_{n} of A consists of at most two elements. Hence, we have M̄_{2} = 0 by Lemma 3.1(1). It contradicts.

Summarizing all cases, we complete the proof of this theorem in the case of a_{2} ≠ 0.

Now, let a_{2} = 0. Then, by Lemmas 3.1(1) and 3.2, {a_{0}, b_{2}, a_{3}, b_{3}} forms a basis for E_{1}^{4} satisfying (3.2). In Case 1, applying (2.3), (2.4), (2.6) and (2.7) for the subclass {p_{1}, p_{2} − p_{1}, p_{3} − p_{2}} of A, and combining the condition D_{32} = D̄_{32} = 0 and Lemma 3.1(1), we have

A_{31} = Ā_{31} = 0, A_{21} = Ā_{21} = 0, D_{21} = D̄_{21} = 0.

Hence A_{11} = Ā_{11} = 0 by Lemma 3.6. Applying (2.3), (2.4), (2.6), (2.7) and the above equation for the subclass {2p_{1}, p_{2}, p_{3} −p_{1}} of A, we get M̄_{2} = 0. Since b_{2} is a lightlike vector by Lemma 3.2, it contradicts.

The other cases are also impossible.

Therefore we complete the proof of this theorem.

For a 3-type closed curve γ(s) = a_{0} + ∑_{t=1}^{3}{a_{t} cos(p_{t}s) + b_{t} sin(p_{t}s)} on H^{3}(−c^{2}), if

a_{2} = 0, then we have M_{2} = 0 and < b_{2}, a_{3} >=< b_{2}, b_{3} >= 0 by Lemma 3.1(2). Hence D_{32} = D̄_{32} = 0. Therefore, from Theorem 3.2, we have the following corollary. Corollary 3.1. There exists no 3-type closed curve

γ(s) = a_{0} + ∑_{t=1}^{3}{a_{t} cos(p_{t}s) + b_{t} sin(p_{t}s)}

on H^{3}(−c^{2}) satisfying a_{2} = 0. Corollary 3.2. There exists no 3-type closed curve with constant curvature on H^{3}(−c^{2}). Proof. Let

γ(s) = a_{0} + a_{1} cos(p_{1}s) + b_{1} sin(p_{1}s) + a_{2} cos(p_{2}s)
+ b_{2} sin(p_{2}s) + a_{3} cos(p_{3}s) + b_{3} sin(p_{3}s)

be a 3-type closed curve with constant curvature on H^{3}(−c^{2}). Then each subclass of A consists of at most three elements. From Lemma 3.1(1), we get

M_{2} = M̄_{2} = 0, M_{3} = M̄_{3} = 0.

Hence a_{2}, b_{2}, a_{3} and b_{3} are spacelike vectors. Furthermore, they are nonzero vector because A_{22} = A_{33} = 0 and γ(s) is of 3-type. Therefore a_{0}, a_{2}, b_{2}, a_{3}, b_{3} are linearly independent vectors in E_{1}^{4} by Lemma 3.1. This implies a contradiction.

Next, we get the following Theorem 3.3. There exists no 4-type closed curve with constant curvature on H^{3}(−c^{2}) satisfying D_{43} = D̄_{43} = 0. Proof. Assume the existence of the 4-type closed curve

γ(s) = a_{0} + ∑_{t=1}^{4}{a_{t} cos(p_{t}s) + b_{t} sin(p_{t}s)}

satisfying our assumptions. If {p_{1}, p_{2}, p_{3}, p_{4}} = {p_{1}, 2p_{1}, 3p_{1}, 4p_{1}}, then A = {p_{1}, p_{2} − p_{1}, p_{3} − p_{2}, p_{4} − p_{3}} ∪ {p_{2}, 2p_{1}, p_{3} − p_{1}, p_{4} − p_{2}} ∪ {p_{3}, p_{1} + p_{2}, p_{4} − p_{1}} ∪ {p_{4}, 2p_{2}, p_{1} + p_{3}} ∪ {p_{1} + p_{4}, p_{2} + p_{3}} ∪ {2p_{3}, p_{2} + p_{4}} ∪ {p_{4} + p_{3}} ∪ {2p_{4}}. Let A_{i} be the subclass consisting of all elements in A to be equal to p_{i}. Then the number of elements in A_{3} (and A_{4}) is less than or equal to three in this case. Hence, from Lemma 3.1, we have

\begin{matrix}M_{3} = M̄_{3} = 0, & M_{4} = M̄_{4} = 0, \\ A_{33} = A_{44} = 0, & A_{43} = Ā_{43} = 0. \end{matrix}

Since D_{43} = D̄_{43} = 0, we get a_{0}, a_{3}, b_{3}, a_{4}, b_{4} are linearly independent vectors in E_{1}^{4} by the same way as the proof Corollary 3.2. It contradicts.

In case of {p_{1}, p_{2}, p_{3}, p_{4}} ≠ {p_{1}, 2p_{1}, 3p_{1}, 4p_{1}}, we can also imply a contradiction by the same way.

From Theorem 3.3, we can obtain the following corollary. Corollary 3.3. There exists no 4-type closed curve

γ(s) = a_{0} + ∑_{t=1}^{4}{a_{t} cos(p_{t}s) + b_{t} sin(p_{t}s)}

on H^{3}(−c^{2}) satisfying a_{3} = 0. Theorem 3.4. There exists no 5-type closed curve γ(s) on H^{3}(−c^{2}) with D_{54} = D̄_{54} = 0 satisfying < γ^{(l)}(s), γ^{(l)}(s) > is constant (l = 2, 3). Proof. Assume the existence of the 5-type closed curve

γ(s) = a_{0} + ∑_{t=1}^{5}{a_{t} cos(p_{t}s) + b_{t} sin(p_{t}s)}

satisfying our conditions. Let {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}} = {p_{1}, 2p_{1}, 3p_{1}, 4p_{1}, 5p_{1}}, it follows that A = {p_{1}, p_{2}−p_{1}, p_{3}−p_{2}, p_{4}−p_{3}, p_{5}−p_{4}}∪{p_{2}, 2p_{1}, p_{3}−p_{1}, p_{4}−p_{2}, p_{5}− p_{3}}∪{p_{3}, p_{1}+p_{2}, p_{4}−p_{1}, p_{5}−p_{2}}∪{p_{4}, 2p_{2}, p_{1}+p_{3}, p_{5}−p_{1}}∪{p_{5}, p_{1}+p_{4}, p_{2}+p_{3}}∪ {2p_{3}, p_{1}+p_{5}, p_{2}+p_{4}}∪{p_{2}+p_{5}, p_{3}+p_{4}}∪{2p_{4}, p_{3}+p_{5}}∪{p_{4}+p_{5}}∪{2p_{5}}. Applying Lemma 3.1(1) for the subclasses {p_{4}, 2p_{2}, p_{1} +p_{3}, p_{5} −p_{1}} and {p_{5}, p_{1} +p_{4}, p_{2} +p_{3}} of A, we obtain

M_{4} = M̄_{4} = 0, M_{5} = M̄_{5} = 0.

Hence a_{4}, b_{4}, a_{5} and b_{5} are spacelike vectors in E_{1}^{4}. Furthermore, from Lemma 3.1(2), we have

A_{44} = A_{55} = 0, A_{54} = Ā_{54} = 0.

Therefore a_{0}, a_{4}, b_{4}, a_{5}, b_{5} are linearly independent vectors in E_{1}^{4} because D_{54} = D̄_{54} = 0 and γ(s) is of 5-type. It contradicts.

By the same way, in case of {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}} ≠ {p_{1}, 2p_{1}, 3p_{1}, 4p_{1}, 5p_{1}}, we can also imply a contradiction.

Finally, we get the following theorem. Theorem 3.5. There exists no 6-type closed curve γ(s) on H^{3}(−c^{2}) with D_{65} = D̄_{65} = 0 satisfying < γ^{(l)}(s), γ^{(l)}(s) > is constant (l = 2, 3). Proof. Assume the existence of the 6-type closed curve

γ(s) = a_{0} + ∑_{t=1}^{6}{a_{t} cos(p_{t}s) + b_{t} sin(p_{t}s)}

satisfying our conditions. Case 1. Let {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}} = {p_{1}, 2p_{1}, 3p_{1}, 4p_{1}, 5p_{1}, 6p_{1}}, it follows that A = {p_{1}, p_{2} − p_{1}, p_{3} − p_{2}, p_{4} − p_{3}, p_{5} − p_{4}, p_{6} − p_{5}} ∪ {p_{2}, 2p_{1}, p_{3} − p_{1}, p_{4} − p_{2}, p_{5} − p_{3}, p_{6} −p_{4}}∪{p_{3}, p_{1} +p_{2}, p_{4} −p_{1}, p_{5} −p_{2}, p_{6} −p_{3}}∪{p_{4}, 2p_{2}, p_{1} +p_{3}, p_{5} −p_{1}, p_{6} − p_{2}} ∪ {p_{5}, p_{1} + p_{4}, p_{2} + p_{3}, p_{6} − p_{1}} ∪ {2p_{3}, p_{1} + p_{5}, p_{2} + p_{4}, p_{6}} ∪ {p_{2} + p_{5}, p_{3} + p_{4}, p_{1} + p_{6}} ∪ {2p_{4}, p_{3} + p_{5}, p_{2} + p_{6}} ∪ {p_{4} + p_{5}, p_{3} + p_{6}} ∪ {2p_{5}, p_{4} + p_{6}} ∪ {p_{5} + p_{6}} ∪ {2p_{6}}. Applying Lemma 3.1(1) for the subclasses {p_{5}, p_{1} + p_{4}, p_{2} + p_{3}, p_{6} − p_{1}} and {p_{6}, 2p_{3}, p_{1} + p_{5}, p_{2} + p_{4}} of A, we obtain

M_{5} = M̄_{5} = 0, M_{6} = M̄_{6} = 0.

And, from Lemma 3.1(2), we have

A_{55} = A_{66} = 0, A_{65} = Ā_{65} = 0.

Hence a_{0}, a_{5}, b_{5}, a_{6}, b_{6} are linearly independent vectors in E_{1}^{4} by our assumptions. It contradicts.

In case of {p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}} ≠ {p_{1}, 2p_{1}, 3p_{1}, 4p_{1}, 5p_{1}, 6p_{1}}, we can also imply a contradiction.

References

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Division of Mathematical Sciences Pukyong National University Pusan 608-737, Korea email : yspyo@dolphin.pknu.ac.kr