We study a null Mannheim curve with time-like or space-like Mannheim partner
curve in the Minkowski 3-space 𝔼13.
We get the characterization of a null Mannheim curve. Then, we investigate there
is no null-helix Mannheim curve in 𝔼13.

1. Introduction

In the study of the fundamental theory and the characterizations of space curves, the
related curves for which there exist corresponding relations between the curves are
very interesting and important problems. The most fascinating examples of such curve
are associated curves, the curves for which at the corresponding points of them one
of the Frenet vectors of a curve coincides with the one of Frenet vectors of the
other curve. The well-known associated curve is Bertrand curve which is
characterized as a kind of corresponding relation between the two curves. The
relation is that the principal normal of a curve is the principal normal of the
other curve, that is, the Bertrand curve is a curve which shares the normal line
with the other curve [1].

Furthermore, Bertrand curves are not only the example of associated curves. Recently,
a new definition of the associated curves was given by Liu and Wang [2]. They called these new curves as Mannheim
partner curves. They showed that the curve γ1
is the Mannheim partner of the other curve γ
if and only if the curvature κ1
and τ1
of γ1
satisfy the following equation:τ̇=κ1λ(1+λ2τ12),
for some nonzero constant λ.
They also study the Mannheim curves in Minkowski 3-space. Some different
characterizations of Mannheim partner curves are given by Orbay and Kasap [3]. Another example is null Mannheim curves from
Öztekin
and Ergüt
[4]. Since a null vector and a nonnull
vector are linearly independent in the Minkowski space 𝔼13,
they have noticed that the Mannheim partner curve of a null curve cannot be a null
curve. They defined the null Mannheim curves whose Mannheim partner curves are
either time-like or space-like.

In this paper, we get the necessary and sufficient conditions for the null Mannheim
curves. Then, we investigate there exists no null-helix Mannheim curve in the
Minkowski 3-space 𝔼13.

2. Preliminaries

Let 𝔼13
be a 3-dimensional Lorentzian space and C
a smooth null curve in 𝔼13,
given by γ(t)=(γ1(t),γ2(t),γ3(t)),t∈I⊂R. Then, the tangent vector
field l=γ′
in 𝔼13
satisfies〈l,l〉=0.
Denote by TC the
tangent bundle of C and
TC⊥
the TC
perpendicular. Clearly, TC⊥
is a vector bundle over C
of rank 2. Since ξ
is null, the tangent bundle TC of
C is a
subbundle of TC⊥
of rank 1. This implies that TC⊥
is not complementary of TC in
𝔼13∣C.
Thus, we must find a complementary vector bundle to TC of
C in
𝔼13
which will play the role of the normal bundle TC⊥
consistent with the classical non-degenerate theory.

Suppose S(TC⊥) denotes the
complementary vector subbundle to TC in
TC⊥;
that is, we haveTC⊥=TC⊥S(TC⊥), where
⊥
means the orthogonal direct sum. It follows that S(TC⊥) is a
nondegenerate vector subbundle of 𝔼13,
of rank of 1. We call S(TC⊥) a screen
vector bundle of C,
which being non-degenerate, and we haveE13∣C=S(TC⊥)⊥S(TC⊥)⊥,
where S(TC⊥)⊥
is a complementary orthogonal vector subbundle to S(TC⊥) in
𝔼13|C
of rank 2.

We denote by F(C) the algebra of smooth functions
on C and by
Γ(E) the F(C) module of smooth sections of a
vector bundle F
over C. We use
the same notation for any other vector bundle.

Theorem 2.1 (see [<xref ref-type="bibr" rid="B1">5</xref>, <xref ref-type="bibr" rid="B2">6</xref>]).

Let C be
a null curve of a Lorentzian space 𝔼13
and S(TC⊥) a screen
vector bundle of C.
Then, there exists a unique vector bundle ntr(C) over C of rank
1 such that there is a unique section n∈Γ(ntr(C)) satisfying 〈l,n〉=1,〈n,n〉=〈n,X〉=0,∀X∈Γ(S(TC⊥)).

We call the vector bundle ntr(C) the null transversal bundle of
C with
respect to S(TC⊥). Next consider
the vector bundletr(C)=ntr(C)⊥S(TC⊥), which from
(2.5) is complementary but not
orthogonal to TC in
𝔼13∣C.

More precisely, we haveE13∣C=TC⊕tr(C)=(TC⊕ntr(C))⊥S(TC⊥). One calls
tr(C) the transversal vector bundle of
C with
respect to S(TC⊥). The vector
field n in
Theorem 2.1 is called the null
transversal vector field of C
with respect to ξ.
As {ξ,n} is a null basis
of Γ(TC⊕ntr(C)) satisfying (2.5), any screen vector bundle
S(TC⊥) of
C is
Euclidean.

Note that for any arbitrary parameter t
on C and a
screen vector bundle S(TC⊥) one finds a
distinguished parameter given byp=∫t0*t*exp(∫s0s〈γ′′,n〉dt*)ds,
where n is the
null transversal vector field with respect to S(TC⊥) and
γ′.

Let C=C(p) be a smooth null curve,
parametrized by the distinguished parameter p
instead of t
such that ∥γ′′∥=κ≠0
([6]). Using (2.5) and (2.7) and taking into account that
the screen vector bundle S(TC⊥) is Euclidean of
rank 1, one obtains the following Frenet equations [1]: ddp[lnu]=[00κ00τ-τ-κ0][lnu].

Definition 2.2.

Let γ
be a curve in the Minkowski 3-space 𝔼13
and γ′
a velocity of vector of γ.
The curve γ
is called time-like (or space-like) if
〈γ′,γ′〉<0
(or if 〈γ′,γ′〉>0).

Let T,N,B
be the tangent, the principal normal, and the binormal of γ,
respectively. Then, there are two cases for the Frenet formulae.

Case 1.

T and
B
are space-like vectors, and N
is a time-like vector dds[TNB]=[κ0κ0τ0τ0][TNB].

Case 2.

T is
a time-like vector, and N
and B are
space-like vectors dds̃[TNB]=[κ0-κ0τ0-τ0][TNB],
where κ
and τ
are called the dual curvature and dual torsion of γ,
respectively [5].

3. Null Mannheim Curves in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M116"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔼</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>Definition 3.1.

Let C:γ(p) be a Cartan
framed null curve and C*:γ*(p*) a time-like
or space-like curve in the Minkowski space 𝔼13.
If there exists a corresponding relationship between the space curves
C
and C*
such that the principal normal lines of C
coincides with the binormal lines of C*
at the corresponding points of the curves, then C called
a null Mannheim curve and C*
is called a time-like or space-like Mannheim partner curve of
C.
The pair of {C,C*} is said to
be a null Mannheim pair [2, 4].

Theorem 3.2.

Let C:γ(p) be a null
Mannheim curve with time-like Mannheim partner curve C*:γ*(p*), and let
{l(p),n(p),u(p)} be the Cartan frame field along
C
and {T(p*),N(p*),B(p*)} the Frenet frame field along
C*.
Then, C*
is the time-like Mannheim partner curve of γ
if and only if its torsion τ*
is constant such that τ*=∓(1/μ), where
μ
is nonzero constant.

Proof.

Assume that γ
is a null Mannheim curve with time-like Mannheim partner curve
γ*.
Then, by Definition 3.1, we can
write γ(p(p*))=γ*(p*)+μ(p*)B(p*), for
some function μ(p*). By taking
the derivative of (3.1) with
respect to p*
and applying the Frenet formulae, we have ldpdp*=T+μ′B+μ(-τ*N).
Since u
coincides with B,
we get μ′=0,
which means that μ
is a nonzero constant. Thus, we have ldpdp*=T-μτ*N.
Since l
is null and from (3.4), we
obtain -1+(μτ*)2=0→τ*=∓1μ,
which means that γ*
is a time-like curve with constant torsion.

Conversely, let the torsion τ*of
the time-like curve C*
be a constant with τ*=∓(1/μ) for some nonzero constant
μ.
By considering a null curve C:γ(p) defined by
γ(p*)=γ*(p*)+μ(p*)B(p*),
we prove that γ
is a null Mannheim and γ*
is the time-like Mannheim partner curve of γ.
By differentiating (3.6) with
respect to p*,
we get ldpdp*=T-μτ*N.
If we use τ*=∓1/μ
in (3.7), we obtain
ldpdp*=T∓N,which
means that l
lies in the plane which is spanned by T
and N,
hence l⊥B.
The proof is complete.

Theorem 3.3.

A Cartan framed null curve γ
in 𝔼13
is a null Mannheim curve with time-like Mannheim partner curve
γ*
if and only if the torsion τ
of γ
is nonzero constant.

Proof.

Let γ=γ(p) be a null
Mannheim curve in 𝔼13.
Suppose that γ*=γ*(p*) is a
time-like curve whose binormal direction coincides with the principal normal of
γ.
Then, B(p*)=∓u(p). Therefore,
we can write γ*(p)=γ(p)+μ(p)u(p), for
some function μ(p)≠0.
Differentiating (3.9) with
respect to p,
we obtain Tdp*dp=(1-μτ)l-μκn+μ′u.
Since the binormal direction of γ*
coincides with the principal normal of γ,
we get 〈T,u〉=0.
Therefore, we have μ′=0
and μ
is constant. By taking the derivative of (3.10), we get κ*N(dp*dp)2+Td2p*dp2=-μτ′l-μκ′n+(1-2μτ)κu.
Since u
is in the binormal direction of γ*,
we have (1-2μτ)κ=0,
and hence τ=12μ=const.
Conversely, similar to the proof of Theorem 3.2, we easily get a null Mannheim curve with time-like
Mannheim partner curve.

Proposition 3.4.

If γ=γ(p) be a
generalized null-helix in 𝔼13,
then, the curve can not be a Mannheim curve.

Proof.

Suppose that γ=γ(p) is a
Mannheim curve in 𝔼13.
Then, there exists the Mannheim partner curve γ*=γ*(p*) of
γ=γ(p) in
𝔼13.
From Theorems 3.2 and 3.3, the torsions of the Mannheim
pair {γ,γ*},
τ
and τ*,
are nonzero-constant. Since γ=γ(p) be a
generalized null-helix, κ/τ
is constant, and thus κ
is constant. Using (3.10),
(3.13), and the fact that
T
is time-like, we have (dp*dp)2=μκ=const.
From (3.11), we have
κ*N(dp*dp)2=0,
and thus, we get κ*=0.
This shows that γ*=γ*(p*) is a
straight line with nonzero torsion in 𝔼13,
which is impossible. Therefore, γ=γ(p) cannot be a
dual Mannheim curve in 𝔼13.

Corollary 3.5.

(1) If a Cartan framed null curve γ
in 𝔼13
is a null Mannheim curve with time-like Mannheim partner curve
γ*,
the signs of κ
and τ
are the same.

(2) If a Cartan framed null curve γ
in 𝔼13
is a null Mannheim curve with space-like Mannheim partner curve
γ*,
the signs of κ
and τ
are opposite.

Proof.

From (3.10) and (3.13), (dp*dp)2=κ2τifMannheimpartnercurveistime-like,(dp*dp)2=-κ2τifMannheimpartnercurveisspace-like.
The proof is complete.

Remarks.

(a) Theorems hold for null dual Mannheim curve with space-like dual Mannheim
partner curve.

(b) Some results in [4] unfortunately are
not correct. For example, Theorem 3.3 gave necessary and sufficient conditions for null Mannheim
curve, which implies that the null Mannheim curve should be a null-helix from
Proposition 3.4. Moreover,
Propositions in [4] are related with a
null-helix partner curve.

JinD. H.Natural Frenet equations of null curvesLiuH.WangF.Mannheim partner curves in 3-spaceOrbayK.KasapE.On mannheim partner curves in 𝔼3ÖztekinH. B.ErgütM.Null mannheim curves in the minkowski 3-space
𝔼13DuggalK. L.BejancuA.DuggalK. L.JinD. H.