Mathematical Combinatorics (International Book Series), vol. II, 2020

2020·4.1 MB·English

by  Mao, Linfan

ISBN 978-1-59973-661-7
VOLUME 2, 2020
MATHEMATICAL COMBINATORICS
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Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES AND
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Vol.2, 2020 ISBN 978-1-59973-661-7
MATHEMATICAL COMBINATORICS
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Edited By Linfan MAO
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Math.Combin.Book Ser. Vol.2(2020), 1-17
Mannheim Partner D-Curves in Minkowski 3-Space
Tanju Kahraman1, Mehmet O¨nder3, Mustafa Kazaz1, H. Hu¨seyin U˘gurlu2
1. CelalBayarUniversity,DepartmentofMathematics,FacultyofArtsandSciences,Manisa,Turkey
2. GaziUniversity,GaziFacultyofEducation,DepartmentofSecondaryEducationScienceand
MathematicsTeaching,MathematicsTeachingProgram,Ankara,Turkey
3. DelibekirliVillage,31440,Kırıkhan,Hatay,Turkey
E-mail: [email protected],[email protected]
[email protected],[email protected]
Abstract: In this paper, we give the definition, different types and characterizations of
Mannheim partner D-curves in Minkowski 3-space E3. We find the relations between the
1
geodesic curvatures, the normal curvatures and the geodesic torsions of these associated
curves. Furthermore, we show that the definition and the characterizations of Mannheim
partnerD-curvesincludethoseofMannheimpartnercurvesinsomespecialcasesinMinkows-
ki 3-space E3.
1
Key Words: Minkowski 3-space, Mannheim partner D-curves, Darboux frame.
AMS(2010): 53A35, 53B30, 53C50.
§1. Introduction
In the study of the fundamental theory and the characterizations of space curves, the related
curvesforwhichthereexistcorrespondingrelationsbetweenthecurvesareveryinterestingand
an important problem. The most fascinating examples of such curves are associated curves,
the curves for which at the corresponding points of them one of the Frenet vectors of a curve
coincideswiththeoneoftheFrenetvectorsoftheothercurve. Thewellknownoftheassociated
curvesisBertrandcurvewhichischaracterizedasakindofcorrespondingrelationbetweenthe
two curves. The relation is that the principal normal of a curve is the principal normal of
another curve i.e, the Bertrand curve is a curve which shares the normal line with another
curve. Over years many mathematicians have studied on Bertrand curves in different spaces
and consider the properties of these curves [1-6].
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 [9,17]. They called these
new curves as Mannheim partner curves: Let x and x be two curves in the three dimensional
1
Euclidean E3. If there exists a corresponding relationship between the space curves x and x
1
such that, at the corresponding points of the curves, the principal normal lines of x coincides
with the binormal lines of x , then x is called a Mannheim curve, and x is called a Mannheim
1 1
partnercurveofx. Thepair{x,x }issaidtobeaMannheimpair. Theyshowedthatthecurve
1
1ReceivedMarch13,2020,AcceptedJune2,2020.
2 TanjuKahraman,MehmetO¨nder,MustafaKazaz,H.Hu¨seyinUg˘urlu
x (s ) is the Mannheim partner curve of the curve x(s) if and only if the curvature κ and the
1 1 1
torsion τ of x (s ) satisfy the following equation
1 1 1
dτ κ
τ˙ = = 1(1+λ2τ2)
ds λ 1
1
forsomenon-zeroconstantλ. TheyalsostudytheMannheimcurvesinMinkowski3-space[9,16].
Some different characterizations of Mannheim partner curves are given by Orbay and others
[12]. The differential geometry of the curves fully lying on a surface in Minkowski 3-spaceE3
1
is given by Ugurlu, Kocayigit and Topal [8,14,15]. They have given the Darboux frame of
the curves according to the Lorentzian characters of surfaces and the curves. Finally, in the
Euclidean 3-space, Mannheim partner D -curves is defined by Kazaz, M. and others [7]
InthispaperweconsiderthenotionoftheMannheimpartnercurveforthecurveslyingon
the surfaces. We call these new associated curves as Mannheim partner D-curves and by using
theDarbouxframeofthecurveswegivethedefinition,differenttypesandthecharacterizations
of these curves in Minkowski 3-spaceE3.
1
§2. Preliminaries
The Minkowski 3-space E3 is the real vector space IR3 provided with the standard flat metric
1
given by
(cid:104) , (cid:105)=−dx2+dx2+dx2
1 2 3
where(x ,x ,x )isarectangularcoordinatesystemofE3. Anarbitraryvector(cid:126)v =(v ,v ,v )
1 2 3 1 1 2 3
in E3 can have one of three Lorentzian causal characters; it can be spacelike if (cid:104)(cid:126)v,(cid:126)v(cid:105) > 0 or
1
(cid:126)v = 0, timelike if (cid:104)(cid:126)v,(cid:126)v(cid:105) < 0 and null (lightlike) if (cid:104)(cid:126)v,(cid:126)v(cid:105) = 0 and (cid:126)v (cid:54)= 0. Similarly, an arbitrary
curve α(cid:126) =α(cid:126)(s) can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors
α(cid:48)(s) are respectively spacelike, timelike or null (lightlike)[11]. We say that a timelike vector
is future pointing or past pointing if the first compound of the vector is positive or negative,
respectively. For any vectors(cid:126)x=(x ,x ,x ) and(cid:126)y =(y ,y ,y ) in E3, Lorentz vector product
1 2 3 1 2 3 1
of (cid:126)x and (cid:126)y is defined by
(cid:12) (cid:12)
(cid:12) e −e −e (cid:12)
(cid:12) 1 2 3 (cid:12)
(cid:12) (cid:12)
(cid:126)x×(cid:126)y =(cid:12)(cid:12) x1 x2 x3 (cid:12)(cid:12)=(x2y3−x3y2, x1y3−x3y1, x2y1−x1y2)
(cid:12) (cid:12)
(cid:12) y1 y2 y3 (cid:12)
where

 1 i=j,
δ =
ij
 0 i(cid:54)=j,
e = (δ ,δ ,δ ) and e ×e =−e , e ×e =e , e ×e =−e .
i i1 i2 i3 1 2 3 2 3 1 3 1 2
MannheimPartnerD-CurvesinMinkowski3-Space 3
(cid:110) (cid:111)
Denote by T(cid:126), N(cid:126), B(cid:126) the moving Frenet frame along the curve α(s) in the Minkowski
space E3. For an arbitrary spacelike curve α(s) in the space E3, the following Frenet formulae
1 1
are given,
    
T(cid:126)(cid:48) 0 k 0 T(cid:126)
1
 N(cid:126)(cid:48) = −εk 0 k  N(cid:126) ,
1 2
    
B(cid:126)(cid:48) 0 k 0 B(cid:126)
2
(cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69)
where T(cid:126),T(cid:126) = 1, N(cid:126),N(cid:126) = ε = ±1, B(cid:126),B(cid:126) = −ε, T(cid:126),N(cid:126) = T(cid:126),B(cid:126) = N(cid:126),B(cid:126) = 0 and
k andk arecurvatureandtorsionofthespacelikecurveα(s)respectively. Here,εdetermines
1 2
the kind of spacelike curve α(s). If ε = 1, then α(s) is a spacelike curve with spacelike first
principal normal N(cid:126) and timelike binormal B(cid:126). If ε = −1, then α(s) is a spacelike curve with
timelike principal normal N(cid:126) and spacelike binormal B(cid:126). Furthermore, for a timelike curve α(s)
in the space E3, the following Frenet formulae are given in as follows,
1
    
T(cid:126)(cid:48) 0 k 0 T(cid:126)
1
 N(cid:126)(cid:48) = k 0 k  N(cid:126) .
1 2
    
B(cid:126)(cid:48) 0 −k 0 B(cid:126)
2
(cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69)
where T(cid:126),T(cid:126) =−1, N(cid:126),N(cid:126) = B(cid:126),B(cid:126) =1, T(cid:126),N(cid:126) = T(cid:126),B(cid:126) = N(cid:126),B(cid:126) =0 and k and k
1 2
are curvature and torsion of the timelike curve α(s) respectively [14,15].
Definition 2.1([11]) (i) (Hyperbolic angle) Let (cid:126)x and (cid:126)y be future pointing (or past pointing)
timelike vectors in IR3. Then there is a unique real number θ (cid:62) 0 such that < (cid:126)x, (cid:126)y >=
1
−|(cid:126)x| |(cid:126)y| coshθ. This number is called the hyperbolic angle between the vectors (cid:126)x and (cid:126)y.
(ii) (Central angle) Let (cid:126)x and (cid:126)y be spacelike vectors in IR3 that span a timelike vector
1
subspace. Then there is a unique real number θ (cid:62) 0 such that < (cid:126)x, (cid:126)y >= |(cid:126)x| |(cid:126)y| coshθ. This
number is called the central angle between the vectors (cid:126)x and (cid:126)y.
(iii) (Spacelike angle) Let (cid:126)x and (cid:126)y be spacelike vectors in IR3 that span a spacelike vector
1
subspace. Then there is a unique real number θ (cid:62) 0 such that < (cid:126)x, (cid:126)y >= |(cid:126)x| |(cid:126)y| cosθ. This
number is called the spacelike angle between the vectors (cid:126)x and (cid:126)y.
(iv) (Lorentzian timelike angle) Let (cid:126)x be a spacelike vector and (cid:126)y be a timelike vector in
IR3. Then there is a unique real number θ (cid:62) 0 such that <(cid:126)x, (cid:126)y >=|(cid:126)x| |(cid:126)y| sinhθ. This number
1
is called the Lorentzian timelike angle between the vectors (cid:126)x and (cid:126)y.
Definition 2.2([11]) A surface in the Minkowski 3-space IR3 is called a timelike surface if
1
the induced metric on the surface is a Lorentz metric and it is called a spacelike surface if the
induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector
on the spacelike (timelike) surface is a timelike (spacelike) vector, respectively.
Lemma 2.1([11]) In the Minkowski 3-space IR3, the following properties are satisfied:
1
(i) Two timelike vectors are never orthogonal;
(ii) Two null vectors are orthogonal if and only if they are linearly dependent;
(iii) A timelike vector is never orthogonal to a null (lightlike) vector .
4 TanjuKahraman,MehmetO¨nder,MustafaKazaz,H.Hu¨seyinUg˘urlu
§3. Darboux Frame of a Curve Lying on a Surface in Minkowski 3-space E3
1
Let S be an oriented surface in three-dimensional Minkowski space E3 and let consider a non-
1
null curve x(s) lying on S fully. Since the curve x(s) is also in space, there exists Frenet frame
(cid:110) (cid:111)
T(cid:126),N(cid:126),B(cid:126) at each points of the curve where T(cid:126) is unit tangent vector, N(cid:126) is principal normal
vector and B(cid:126) is binormal vector, respectively.
Sincethecurvex(s)liesonthesurfaceS thereexistsanotherframeofthecurvex(s)which
(cid:110) (cid:111)
is called Darboux frame and denoted by T(cid:126),(cid:126)g,(cid:126)n . In this frame T(cid:126) is the unit tangent of the
curve, (cid:126)n is the unit normal of the surface S and (cid:126)g is a unit vector given by (cid:126)g =(cid:126)n×T(cid:126). Since
the unit tangent T(cid:126) is common in both Frenet frame and Darboux frame, the vectors N(cid:126), B(cid:126), (cid:126)g
and(cid:126)nlieonthesameplane. Then, ifthesurfaceS isanorientedtimelikesurface, therelations
between these frames can be given as follows:
If the curve x(s) is timelike, If the curve x(s) is spacelike
         
T(cid:126) 1 0 0 T(cid:126) T(cid:126) 1 0 0 T(cid:126)
 (cid:126)g = 0 cosϕ sinϕ  N(cid:126) , (cid:126)g = 0 coshϕ sinhϕ  N(cid:126) .
         
(cid:126)n 0 −sinϕ cosϕ B(cid:126) (cid:126)n 0 sinhϕ coshϕ B(cid:126)
IfthesurfaceS isanorientedspacelikesurface,thenthecurvex(s)lyingonS isaspacelike
curve. So, the relations between the frames can be given as follows
    
T(cid:126) 1 0 0 T(cid:126)
 (cid:126)g = 0 coshϕ sinhϕ  N(cid:126) .
    
(cid:126)n 0 sinhϕ coshϕ B(cid:126)
In all cases, ϕ is the angle between the vectors(cid:126)g and N(cid:126).
According to the Lorentzian causal characters of the surface S and the curve x(s) lying on
S, the derivative formulae of the Darboux frame can be changed as follows:
(i) If the surface S is a timelike surface, then the curve x(s) lying on S can be a spacelike
or a timelike curve. Thus, the derivative formulae of the Darboux frame of x(s) is given by
 T(cid:126)˙   0 k −εk  T(cid:126) 
g n
 (cid:126)g˙ = k 0 ετ  (cid:126)g ,(cid:68)T(cid:126),T(cid:126)(cid:69)=ε=±1, (cid:104)(cid:126)g,(cid:126)g(cid:105)=−ε, (cid:104)(cid:126)n,(cid:126)n(cid:105)=1. (1)
g g
    
(cid:126)n˙ k τ 0 (cid:126)n
n g
(ii) If the surface S is a spacelike surface, then the curve x(s) lying on S is a spacelike
MannheimPartnerD-CurvesinMinkowski3-Space 5
curve. Thus, the derivative formulae of the Darboux frame of x(s) is given by
 T(cid:126)˙   0 k k  T(cid:126) 
g n
 (cid:126)g˙ = −k 0 τ  (cid:126)g ,(cid:68)T(cid:126),T(cid:126)(cid:69)=1, (cid:104)(cid:126)g,(cid:126)g(cid:105)=1, (cid:104)(cid:126)n,(cid:126)n(cid:105)=−1. (2)
g g
    
(cid:126)n˙ k τ 0 (cid:126)n
n g
In these formulae k ,k and τ are called the geodesic curvature, the normal curvature
g n g
and the geodesic torsion, respectively. Here and in the following, we use “dot” to denote the
derivative with respect to the arc length parameter of a curve.
The relations between geodesic curvature, normal curvature, geodesic torsion and κ, τ are
given as follows (See [9,14,15]):
• if both S and x(s) are timelike or spacelike,
dϕ
k =κcosϕ,k =κsinϕ,τ =τ + ; (3)
g n g ds
• if S is timelike and x(s) is spacelike
dϕ
k =κcoshϕ,k =κsinhϕ,τ =τ + . (4)
g n g ds
Furthermore, the geodesic curvature k and geodesic torsion τ of the curve x(s) can be calcu-
g g
lated as follows:
(cid:28)dx d2x (cid:29) (cid:28)dx dn(cid:29)
k =− , ×n ,τ =−ε , n× , (5)
g ds ds2 g ds ds
(cid:28)dx d2x (cid:29) (cid:28)dx dn(cid:29)
k =− , ×n ,τ = , n× . (6)
g ds ds2 g ds ds
(cid:68) (cid:69)
where ε= T(cid:126),T(cid:126) =±1.
In the differential geometry of surfaces, for a curve x(s)lying on a surface S the followings
are well-known
• x(s)is a geodesic curve ⇔ k =0,
g
• x(s)is an asymptotic line ⇔ k =0,
n
• x(s)is a principal line ⇔τ =0 [10].
g
Alongeverypointofthesurfacepassesageodesicineverydirection. Ageodesicisuniquely
determined by an initial point and tangent at that point. All straight lines on a surface are
geodesics. Along all curved geodesics the principal normal coincides with the surface normal.
Along asymptotic lines osculating planes and tangent planes coincide, along geodesics they are
normal. Through a point of a nondevelopable surface pass two asymptotic lines which can be
real or imaginary [13].