In this paper, we give the definition of non-null Mannheim curve and null Mannheim curve in an n-dimensional Lorentz manifold. Furthermore, we give the condition for the non-null Mannheim partner curves and the null Mannhein partner curves

Dae Won Yoon

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International Journal of Pure and Applied Mathematics
Volume 96 No. 2 2014, 165-174
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: http://dx.doi.org/10.12732/ijpam.v96i2.1
PA
ijpam.eu
MANNHEIM CURVES IN
AN N-DIMENSIONAL LORENTZ MANIFOLD
Dae Won Yoon
Department of Mathematics Education and RINS
Gyeongsang National University
Jinju 660-701, REPUBLIC OF KOREA
Abstract: In this paper, we give the definition of non-null Mannheim curve
and null Mannheim curve in an n-dimensional Lorentz manifold. Furthermore,
we give the condition for the non-null Mannheim partner curves and the null
Mannhein partner curves.
AMS Subject Classification: 53B30, 53A35
Key Words: Lorentz manifold, Mannheim partner curves, Cartan frame
1. Introduction
In modern physics(especially general relatively), space-time is represented by a
Lorentz manifold. Lorentz geometry plays an important role in the translation
between modern differential geometry and mathematical physics.
On the other hand, the curves are a fundamental structure of differential
geometry. An increasing interest of the theory of curves makes a development of
special curves to be examined. A way to the characterizations and the classifica-
tions for curves is the relationship between the Frenet vectors of the curves. One
of the curves is the Mannheim curve. Space curves of which principal normals
are the binormals of another curve are called Mannheim curves. The notion of
Received: November 18, 2013 c© 2014 Academic Publications, Ltd.
url: www.acadpubl.eu
166 D.W. Yoon
Mannheim curves was discovered by A. Mannheim in 1878. The articles con-
cerning the Mannheim curves are rather few. In [1], a remarkable class of the
Mannheim curves is studied. O. Tigano [7] obtained general Mannheim curves
in a Euclidean 3-space. Mannheim partner curves in a Euclidean 3-space and a
Minkowski 3-space are studied and the necessary and sufficient conditions for
the Mannheim partner curves are obtained in [3] and [5]. Recently, Mannheim
curves are generalized and some characterizations and examples of generalized
Mannheim curves in a Euclidean 4-space are introduced by [6].
In this paper, we study the Mannheim partner curves in an n-dimensional
Lorentz manifold and give the condition for non-null Mannheim partner curves
and null Mannhein partner curves.
2. Preliminaries
Let V be an n-dimensional real vector space over R. A bilinear form on V
is an R-bilinear function 〈 , 〉 : V × V → R. A scalar product 〈 , 〉 on V is a
non-degenerate symmetric bilinear form on V . An index q of the scalar product
〈 , 〉 of V is the largest integer that is the dimension of a subspace W in V on
which 〈 , 〉|W is negative definite. In particular, if q = 1, it is called a Lorentz
vector space with Lorentz product.
A vector X of V is said to be space-like if 〈X,X〉 > 0 or X = 0, time-like
if 〈X,X〉 < 0 and null if 〈X,X〉 = 0 and X 6= 0. A time-like or null vector in
V is said to be causal [4].
Let M be an n-dimensional smooth connected paracompact Hausdorff man-
ifold and let π : TM → M denote the tangent bundle of M . A Lorentz metric
〈 , 〉 for M is a smooth symmetric tensor field of type (0,2) on M such that
for each p ∈ M , the tensor 〈 , 〉p : TpM × TpM → R is a nondegenerate inner
product of signature (−,+,+, · · · ,+).
A Lorentz manifold (M, 〈 , 〉) is a manifoldM together with a Lorentz metric
〈 , 〉 for M . Let’s denote a Lorentz manifold (M, 〈 , 〉) by Ln.
3. Non-Null Mannheim Curves
In this section, we will define the non-null Mannheim curves in Ln and investi-
gate the properties of the non-null Mannheim curves.
Suppose c is a non-null curve of an n-dimensional Lorentz manifold Ln.
Denote by ∇ the Levi-Civita connection on Ln and dc(s)
ds
= c′(s) = V1(s),
MANNHEIM CURVES IN... 167
where s is the arc-length of c. In this case, {V1, V2, · · · , Vn} is the Frenet frame
of c. Thus, the Frenet formula of a non-null curve in Ln are as follows [4]:
V ′1 = ∇v1V1 = ǫ2k1V2,
V ′2 = ∇v1V2 = −ǫ1k1V1 + ǫ3k2V3,
...
V ′n = ∇v1Vn = −ǫn−1kn−1Vn−1,
(3.1)
where ǫi = 〈Vi, Vi〉. The functions k1, k2, · · · , kn−1 called the curvatures of c.
Definition 1. A non-null curve c in an n-dimensional Lorentz manifold
Ln is a Mannheim curve if there is a non-null curve c̄ in Ln such that the
first normal line with the direction V2 at each of c is included in the subspace
generated by (n− 2)- normal lines with the directions V̄3, V̄4, · · · , V̄n of c̄ at the
corresponding point under a bijective smooth function φ : c → c̄. In this case,
c̄ is called a non-null Mannheim partner curve of c.
Theorem 2. The distance between corresponding points of a non-null
Mannheim curve and of its non-null Mannheim partner curve in Ln is a con-
stant.
Proof. Let c be a non-null Mannheim curve in Ln and c̄ a non-null Mannheim
partner curve of c. c̄ is distinct from c. Let the pair of c(s) and c̄(s̄) = c̄(s̄(s))
be of corresponding points of c and c̄. Then the curve c̄ is given by
c̄(s̄) = c̄(s̄(s)) = c(s) + λ(s)V2(s) (3.2)
for some smooth function λ. Let {V̄1, V̄2, · · · V̄n} and {k̄1, k̄2, · · · , k̄n−1} be the
Frenet frame and the curvature functions of c̄, respectively.
By taking the differentiation of equation (3.2) with respect to s and using
equation (3.1), we obtain
φ(s)V̄1(s̄(s)) = (1− ǫ1λk1(s))V1(s) + λ
′(s)V2(s) + ǫ3λk2(s)V3(s), (3.3)
where φ(s) = ds̄
ds
. By definition 1, V2(s) can be represented as the following
form:
V2(s) = f1(s)V̄3(s̄(s)) + f2(s)V̄4(s̄(s)) + · · ·+ fn−2(s)V̄n(s̄(s))
168 D.W. Yoon
for some smooth function fi(i = 1, · · · , n− 2). If we consider
〈V̄1(s̄(s)), f1(s)V̄3(s̄(s)) + f2(s)V̄4(s̄(s)) + · · ·+ fn−2(s)V̄n(s̄(s))〉 = 0
and equation (3.3), then we have λ′(s) = 0. This means that λ is a nonzero
constant. On the other hand, from the distance function between two points,
we have
d(c̄(s̄), c(s)) = |λ|.
Namely, d(c̄(s̄), c(s)) is a constant. This completes the proof.
Theorem 3. If a non-null curve c in Ln is a Mannheim curve, then the
first curvature function k1 and the second curvature function k2 of c satisfy the
equation
k1(1− ǫ1λk1)− ǫ3λk
2
2 = 0 (3.4)
for nonzero constant λ.
Proof. Considering that λ is nonzero constant in equation (3.3), we have
V̄1(s̄(s)) =
1− ǫ1λk1(s)
φ(s)
V1(s) +
ǫ3λk2(s)
φ(s)
V3(s). (3.5)
By taking differentiation both sides of equation (3.5) with respect to s,
ǭ2k̄1(s̄(s))φ(s)V̄2(s̄(s))
=
(
1− ǫ1λk1(s)
φ(s)
)′
V1(s)
+
(
(1− ǫ1λk1(s))ǫ2k1(s)− ǫ2ǫ3λk
2
2(s)
φ(s)
)
V2(s)
+
(
ǫ3λk2(s)
φ(s)
)′
V3(s)
+
(
ǫ3ǫ4λk2(s)k3(s)
φ(s)
)
V4(s).
(3.6)
On the other hand,
〈V̄2(s̄(s)), f1(s)V̄3(s̄(s)) + f2(s)V̄4(s̄(s)) + · · ·+ fn−2(s)V̄n(s̄(s))〉 = 0
implies from (3.6) the coefficient of V2(s) vanishes, that is,
k1(s)(1− ǫ1λk1(s))− ǫ3λk
2
2(s) = 0.
This completes the proof.
MANNHEIM CURVES IN... 169
Theorem 4. If there is a non-null curve c̄ in Ln such that V2 of a non-null
c is lying in the subspace generated by V̄4, V̄5, · · · , V̄n of c̄ at the corresponding
points c(s) and c̄(s̄), then the curvatures k1 and k2 of c are constant functions.
Proof. Let V2 of c be lying in the subspace generated by V̄4, V̄5, · · · , V̄n of
c̄. Then V2(s) can be written as the following form:
V2(s) = g1(s)V̄4(s̄(s)) + · · ·+ gn−3(s)V̄n(s̄(s))
for some smooth function gi(i = 1, · · · , n− 3). If we take into consideration
〈V̄1(s̄(s)), V2(s)〉 = 0
and equation (3.3), then we have λ′(s) = 0. Therefore, equation (3.3) becomes
V̄1(s̄(s)) =
1− ǫ1λk1(s)
φ(s)
V1(s) +
ǫ3λk2(s)
φ(s)
V3(s). (3.7)
By taking differentiation both sides of equation (3.7) with respect to s, we have
ǭ2k̄1(s̄(s))φ(s)V̄2(s̄(s))
=
(
1− ǫ1λk1(s)
φ(s)
)′
V1(s)
+
(
(1− ǫ1λk1(s))ǫ2k1(s)− ǫ2ǫ3λk
2
2(s)
φ(s)
)
V2(s)
+
(
ǫ3λk2(s)
φ(s)
)′
V3(s)
+
(
ǫ3ǫ4λk2(s)k3(s)
φ(s)
)
V4(s).
(3.8)
Since 〈V2(s), V̄2(s̄(s))〉 = 0, we have
λ =
k1
ǫ1k
2
1 + ǫ3k
2
2
. (3.9)
170 D.W. Yoon
Moreover, the differentiation of equation (3.8) with respect to s is
φ(s)(−ǭ1k̄1(s̄(s))V̄1(s̄(s)) + ǭ3k̄2(s̄(s))V̄3(s̄(s)))
=
(
1
ǭ2k̄1(s̄(s))φ(s)
(
1− ǫ1λk1(s)
φ(s)
)′)′
V1(s)
+
(
ǫ2k1(s)
ǭ2k̄1(s̄(s))φ(s)
(
1− ǫ1λk1(s)
φ(s)
)′
−
ǫ2k2(s)
ǭ2k̄1(s̄(s))φ(s)
(
ǫ3λk2(s)
φ(s)
)′)
V2(s)
+
(
(
1
ǭ2k̄1(s̄(s))φ(s)
(
ǫ3λk2(s)
φ(s)
)′)′
−
1
ǭ2k̄1(s̄(s))φ(s)
(
ǫ4λk2(s)k
2
3(s)
φ(s)
))
V3(s)
+
(
ǫ4k3(s)
ǭ2k̄1(s̄(s))φ(s)
(
ǫ3λk2(s)
φ(s)
)′
+
(
1
ǭ2k̄1(s̄(s))φ(s)
(
ǫ3ǫ4λk2(s)k3(s)
φ(s)
))′)
V4(s)
(3.10)
Since 〈V2(s), V̄1(s̄(s))〉 = 0 and 〈V2(s), V̄3(s̄(s))〉 = 0, from equation (3.10) we
can see that
(−ǫ1λk1(s)k
′
1(s)− ǫ3λk2(s)k
′
2(s))φ(s)
+ (−k1(s) + ǫ1λk
2
1(s) + ǫ2λk
2
2(s))φ
′(s) = 0.
(3.11)
By differentiating equation (3.9) with respect to s, we get
k′1(s)− 2λ(ǫ1k1(s)k
′
1(s) + ǫ3k2(s)k
′
2(s)) = 0, (3.12)
which implies from (3.9) and (3.11) we easily show that k′1(s) = 0, that is, k1(s)
is constant. Also, from (3.12) k2(s) is constant. This completes the proof.
4. Null Mannheim Curves in Ln
In this section, we will define the null Mannheim curves whose Mannheim part-
ner curve is non-null curve in Ln. Furthermore, we will investigate the proper-
ties of the null Mannheim curve.
Let Ln be an n-dimensional Lorentz manifold and let us consider x a smooth
curve in Ln locally parametrized by x : I ⊂ R → Ln. The curve x is said to
MANNHEIM CURVES IN... 171
be null if the tangent vector x′(s) = ξ at any point is null vector. That is,
〈ξ, ξ〉 = 0. The following concepts are taken from Duggal and Bejancu [2].
Let Tx denote the tangent bundle of x and define, as in the non-degenerate
case, the bundle Tx⊥ by:
Tx⊥ =
⋃
p∈x
Tpx
⊥, Tpx
⊥ = {ηp ∈ TpL
n : 〈ηp, ξp〉 = 0, ξp ∈ Tpx},
where ξp is a null vector tangent to x at p. It is well known that Tpξ
⊥ is of rank
n− 1. Since ξp is a null vector, it easily follows that Tx is a vector subbundle
of Tx⊥ of rank 1. Then we may consider a complementary vector subbundle
S(Tx⊥) to Tx in Tx⊥ such that:
Tx⊥ = Tx⊥S(Tx⊥),
where ⊥ means orthogonal direct sum. Is is well known that the subbundle
S(Tx⊥), called the screen vector bundle of x, is non-degenerate. Note that,
in contrast with the non-degenerate case, the tangent bundle is contained in
the normal bundle, and the screen bundle is not unique. These two properties
lead to a much more difficult and also different geometry of null curves with
respect to non-degenerate curves. Since S(Tx⊥) is non-degenerate, we have the
decomposition:
TLn|x = S(Tx
⊥)⊥S(Tx⊥)⊥,
where S(Tx⊥)⊥ is the complementary orthogonal vector bundle to S(Tx⊥).
The following result is well known.
Lamma 5. ([2]) Let x be a null curve of a Lorentz manifold Ln and
consider S(Tx⊥) a screen vector bundle of x. Then there exists a unique vector
bundle E over x, of rank 1, such that on each coordinate neighborhood U there
is a unique section N ∈ Γ(Ex) satisfying
〈ξ,N〉 = 1, 〈N,N〉 = 〈N,X〉 = 0
for all X ∈ Γ(S(Tx⊥).
The above vector bundle E will be denoted by ntr(x) and it is called the
null transversal bundle of x with respect to S(Tx⊥). The vector field N is
called the null transversal vector field of x with respect to x′(s). We define the
transversal vector bundle of x, tr(x), as the vector bundle
tr(x) = ntr(x)⊥S(Tx⊥),
172 D.W. Yoon
and then we have
TLn|x = Tx⊕ tr(x) = (Tx⊕ ntr(x))⊥S(Tx
⊥).
Let x(p) be a smooth null vector, parametrized by the distinguished pa-
rameter p such that ||x′′|| = k1 6= 0. Denote by ∇ the Levi-Civita connection
on Ln. Then we obtain the following Frenet formula ([2])
ξ′ = ∇ξξ = k1W1,
N ′ = ∇ξN = k2W1 + k3W2,
W ′1 = ∇ξW1 = −k2ξ − k1N,
W ′2 = ∇ξW2 = −k3ξ + k4W3,
W ′3 = ∇ξW3 = −k4W2 + k5W4,
...
W ′i = ∇ξWi = −ki+1Wi−1 + ki+2Wi+1, i ∈ {3, · · · , n − 3},
W ′n−2 = ∇ξWn−2 = −kn−1Wn−3,
(4.1)
where k1, k2, · · · , kn−1 are smooth functions and {W1,W2, · · · ,Wn−2} is a cer-
tain orthonormal basis of Γ(S(Tx⊥). In general, for any n > 2, we call
F = {ξ,N,W1, · · · ,Wn−2} a Frenet frame on L
n along x with respect to the
screen vector bundle S(Tx⊥) and the equation (4.1) are called its Frenet for-
mula of x.
Let x(p) be a smooth null curve in Ln, parametrized by a special distin-
guished parameter p such that ||x′′|| = 1. Due to [2] we also obtain the following
Cartan formula:
∇ξξ = W1,
∇ξN = r1W1 + r2W2,
∇ξW1 = −r1ξ −N,
∇ξW2 = −r2ξ + r3W3,
∇ξW3 = −r3W2 + r4W4,
...
∇ξWi = −riWi−1 + ri+1Wi+1, i ∈ {3, · · · , n− 3},
∇ξWn−2 = −rn−2Wn−3.
(4.2)
In the sequel, we call F = {ξ,N,W1, · · · ,Wn−2} the Cartan frame, ri the
curvature function of x with respect to F and x the null Cartan curve in Ln,
respectively.
MANNHEIM CURVES IN... 173
Now, we define a null Mannheim curve in Ln. We notice that a Mannheim
partner curve of a null curve cannot be a null curve, because a null vector and
a non-null vector are linear independent in Ln. Therefore, we define a null
Mannheim curve whose Mannheim partner curve is non-null curve.
Definition 6. A null Cartan curve x in an n-dimensional Lorentz manifold
Ln is a Mannheim curve if there is a non-null curve c in Ln such that the first
normal line with the direction W1 at each of x is included in the subspace
generated by (n − 2)- normal lines with the directions V3, V4, · · · , Vn of c at
the corresponding point. In this case, c is called a non-null Mannheim partner
curve of a null Cartan curve x.
Theorem 7. The distance between the corresponding points of null Car-
tan Mannheim curve and of its non-null Mannheim partner curve in Ln is a
constant.
Proof. Let x(p) be a null Cartan curve parametrized by distinguished pa-
rameter p and c(s) a non-null curve parametrized by arc-length parameter
s. If c(s) is a non-null partner curve of a null Cartan curve x(p), then C is
parametrized as
c(s(p)) = x(p) + λ(p)W1(p) (4.3)
for some smooth function λ 6= 0.
For simplicity, ds
dp
= φ(p). Then, using (4.2), we have
φ(p)V1(s(p)) = (1− λ(p)r1(p))ξ(p)− λ(p)N(p) + λ
′(p)W1(p), (4.4)
where {V1, V2, · · · Vn} is a Frenet frame of c. On the other hand, by definition
4.2, W1(s) can be given by:
W1(p) = f1(p)V3(s(p)) + · · ·+ fn−2(p)Vn(s(p)) (4.5)
for some smooth function fi(i = 1, · · · , n − 2). Taking the scalar product of
(4.5) with V1(s(p)) and using (4.4), we have λ
′ = 0. This means that λ is a
nonzero constant. On the other hand, from the distance function between two
points, we have
d(x(p), c(s(p))) = |λ|.
Namely, d(x(p), c(s(p))) is constant. This completes the proof.
Theorem 8. If a null Cartan curve x in Ln is a Mannheim curve, then
the first curvature function r1 satisfies r1 =
1
2λ , where λ is nonzero constant.
174 D.W. Yoon
Proof. By considering λ is nonzero constant in equation (4.3), we have
V1(s(p)) =
(1− λr1(p))
φ(p)
ξ(p)−
λ
φ(p)
N(p). (4.6)
By taking differentiation both sides of equation (4.6) with respect to p, we
obtain
ǫ2k1(s(p))φ(p)V2(s(p)) =
(
1− λr1(p)
φ(p)
)′
ξ(p)−
(
λ
φ(p)
)′
N(p)
+
(
1− 2λr1(p)
φ(p)
)
W1(p)−
λr2(p)
φ(p)
W2(p).
(4.7)
From equation (4.5), we get 〈W1(p), V2(s(p))〉 = 0. This implies from equation
(4.7) we obtain
r1(p) =
1
2λ
.
This completes the proof.
References
[1] R. Blum, A remarkable class of Mannheim curves, Canad. Math. Bull., 9
(1966), 223-228.
[2] K.L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian
manifolds and applications, Kluwer Acad. Publishers, Dordrecht (1996).
[3] H. Liu and F. Wang, Mannheim partner curves in 3-space, J. Geom., 88
(2008), 120-126.
[4] B. O’Neill, Semi-Riemannian geonetry, Academic Press(1983).
[5] K. Orbay and E. Kasap, On Mannheim partner curves in E3, International
J. Phy. Sci., 4 (2009), 261-264.
[6] H. Matsuda and S. Yorozu, On generalized Mannheim curves in Euclidean
4-space, Nihonkai Math. J., 20 (2009), 33-56.
[7] O. Tigano, Sulla determinazione delle curve di Mannheim, Matematiche
Catania, 3 (1948), 25-29.


MANNHEIM CURVES IN AN N-DIMENSIONAL LORENTZ MANIFOLD

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MANNHEIM CURVES IN AN N-DIMENSIONAL LORENTZ MANIFOLD

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