室蘭工業大学学術資源アーカイブ JMAA 437 1 133 159

D

ual i t i es and evol ut es of f r ont s i n hyper bol i c

and de Si t t er s pac e

著者

CH

EN

Li ang, TAKAH

ASH

I M

as at om

o

j our nal or

publ i c at i on t i t l e

J our nal of M

at hem

at i c al Anal ys i s and

Appl i c at i ons

vol um

e

437

num

ber

1

page r ange

133- 159

year

2016- 05- 01

U

RL

ht t p: / / hdl . handl e. net / 10258/ 00008615

D

ual i t i es and evol ut es of f r ont s i n hyper bol i c

and de Si t t er s pac e

著者

CH

EN

Li ang, TAKAH

ASH

I M

as at om

o

j our nal or

publ i c at i on t i t l e

J our nal of M

at hem

at i c al Anal ys i s and

Appl i c at i ons

vol um

e

437

num

ber

1

page r ange

133- 159

year

2016- 05- 01

U

RL

ht t p: / / hdl . handl e. net / 10258/ 00008615

Dualities and evolutes of fronts in hyperbolic

and de Sitter space

Liang Chen

and Masatomo Takahashi

1 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.CHINA

2 Muroran Institute of Technology, Muroran 050-8585, JAPAN

Abstract

We consider the differential geometry of evolutes of singular curves in hyperbolic 2-space and de Sitter 2-2-space. Firstly, as an application of the basic Legendrian duality theorems, we give the definitions of frontals in hyperbolic 2-space or de Sitter 2-space, respectively. We also give the notions of moving frames along the frontals. By using the moving frames, we define the evolutes of spacelike fronts and timelike fronts, and investigate the geometric properties of these evolutes. As results, these evolutes can be viewed as wavefronts from the viewpoint of Legendrian singularity theory. At last, we study the relationships among these evolutes.

Keywords: evolute; spacelike front; timelike front; hyperbolic 2-space; de Sitter 2-space.

2010Mathematics Subject classification: Primary 53A35; Secondary 58K05

1

Introduction

This paper is a part of our research projects about the differential geometry of evolutes of singular curves in different ambient space forms. Notions of evolutes (or, focal sets) of regular curves in Euclidean plane or 3-space are classical topics in differential geometry. As well known, the evolute of a regular plane curve is defined as the locus of the center of osculating circle of the original curve. The radius of the osculating circle of a regular plane curve is 1/κ, whereκ

is the curvature of the curve. Unfortunately, if the curve is not regular at some point, then we can not define the evolute at this point as the classical way. The second author of this paper, however, had presented an alternative method for the studying of evolutes of singular curves in Euclidean plane [4, 5]. They firstly define frontals (or fronts) in Euclidean plane and Legendrian curves (or Legendrian immersions) in the unit tangent bundle ofR2_{. The differential geometric}

properties of the frontal is studied in [3]. The most difference between a regular curve and a frontal is that the frontal might exist singular points. A key tool for studying of the frontal is so called moving frame defined in the unit tangent bundle. By using the moving frame, they defined a pair of smooth functions like as the curvature of a regular curve and called the pair the curvature of the Legendrian curve. As results, the existence and uniqueness for the Legendrian curve which take this curvature as the associated curvature are established. Furthermore, they

∗Corresponding author: [email protected].

†Partially supported by NNSF of China (Grant No. 11101072) and STDP of Jilin Province (Grant No. 20150520052JH).

used the moving frame and the curvature of the Legendrian immersion to give a new definition of an evolute of the front. We remark that this new definition on the evolute is consistent with the classical one when the curve is a regular curve. They also studied the evolutes of smooth curves in sphere 2-space as applications of this method [13]. In this paper, we proceed with this way to investigate the evolutes of smooth curves in hyperbolic 2-space and de Sitter 2-space. As it to be expected, the situation presents certain peculiarities when compared with the Euclidean case and the sphere case. For instance, in our case the evolutes of smooth spacelike curves in hyperbolic 2-space (or, de Sitter 2-space) are split into hyperbolic 2-space and de Sitter 2-space. The organization of this paper is as follows. In_§2, we prepare some basic notions on regular curves in hyperbolic 2-space and de Sitter 2-space, respectively. We first review the properties of the evolutes of regular curves in hyperbolic 2-space which developed by S. Izumiya and his collaborators in [8] (for regular hypersurfaces case please see [9, 10]). Moreover, by using a similar way to that of [8], we study the evolutes of spacelike regular curves and timelike regular curves in de Sitter 2-space, respectively. In_§3, we give a brief review on the basic Legendrian duality theorems appeared in [2, 6, 7]. Especially, ∆1-duality and ∆5-duality are very helpful in

this paper. We define the spacelike frontals (or, spacelike fronts) in hyperbolic 2-space and de Sitter 2-space, and spacelike Legendrian curves (or, spacelike Legendrian immersions) by using the ∆1-duality. We also use the ∆5-duality to define the timelike frontals (or, timelike fronts)

in de Sitter 2-space and timelike Legendrian curves (or, timelike Legendrian immersions). The basic properties of the frontals are discussed. We give the definitions of evolutes of spacelike fronts in hyperbolic 2-space, spacelike and timelike fronts in de Sitter 2-space in_§4, respectively. We also study the geometric properties of these evolutes in this section. In the last section,_§5, we investigate the relationships among the evolutes of these fronts in hyperbolic 2-space and de Sitter 2-space.

We shall assume throughout the whole paper that all maps and manifolds are C∞ _unless

the contrary is explicitly stated.

Acknowledgement. The authors would like to thank the referee for helpful comments to improve the original manuscript.

2

The evolutes of regular curves

In this section, we investigate the basic properties of evolutes of regular curves in hyperbolic 2-space or de Sitter 2-space, respectively. Firstly, we will prepare some notions in Minkowski space. For details of Lorentzian geometry, see [12].

Let R3 _{={(x1, x2, x3)|xi ∈}R_{, i= 1,2,3} be a 3-dimensional vector space. For any vectors}

x = (x1, x2, x3) and y = (y1, y2, y3) in R3, the pseudo scalar product of x and y is defined to

be_hx,y_i=₋x1y1+x2y2+x3y3. We call (R3,h,i) theMinkowski 3-space and writeR31 instead

of (R3_,h,i).

We say that a non-zero vectorxinR3

1isspacelike,lightlikeortimelikeifhx,xi>0,hx,xi=

0 or _hx,x_i<0 respectively. The normof the vector x_∈R3

1 is defined bykxk=

p

|hx,x_i|. For any x= (x1, x2, x3),y= (y1, y2, y3)∈R31, we define a vectorx∧y by

x_∧y=

−e1 e2 e3

x1 x2 x3

y1 y2 y3

,

where_{e1,e2,e3} is the canonical basis of R31. For any w ∈R31, we can easily check that

so thatx_∧y is pseudo-orthogonal to bothx and y. Moreover, ifx is a timelike vector,y is a spacelike vector and x_∧y=z, then by a straightforward calculation we have

z_∧x=y, y_∧z=₋x.

If x is a spacelike vector, y is a timelike vector and x_∧y = z, then by a straightforward calculation we have

z_∧x=₋y, y_∧z=x.

If both x, y are a spacelike vectors and x_∧y = z, then by a straightforward calculation we have

z_∧x=₋y, y_∧z=₋x.

For a vectorv _∈R3

1 and a real number c, we define the plane with the pseudo-normal v by

P(v, c) =_{x_∈R₁3_|hx,v_i=c_}.

We call P(v, c) a timelike plane, spacelike plane or lightlike plane if v is spacelike, timelike or lightlike, respectively.

We define hyperbolic 2-space by

H2(₋1) =_{x_∈R3

1 | hx,xi=−1}, de Sitter 2-space by

S₁2 =_{x_∈R₁3 _{| h}x,x_i= 1_},

(open)lightcone at the origin by

LC∗ =_{x_∈R3_{1\ {}0_{}| h}x,x_i= 0_}.

We consider a curve given by the intersection of H2_{(−1) (or, S}2

1) with the planeP(v, c) as

follows:

HP(v, c) =H2_{(−1)∩P(v, c) (or, DP(v, c) =S}2

1 ∩P(v, c))

and call it thehyperbolic (or, de Sitter) ellipse, hyperbolic (or, de Sitter)parabola orhyperbolic

(or, de Sitter) hyperbolaif v is timelike, lightlike or spacelike, respectively.

We study the evolutes of regular curves in hyperbolic 2-space or de Sitter 2-space, respec-tively, in the following.

2.1

The evolutes of regular curves in hyperbolic 2-space

We firstly give a brief review on differential geometry of regular curves inH2_{(−1). For details}

please see [8]. Let γ_h :I _→ H2_{(−1) be a regular curve, we have ||γ˙}

h(t)|| 6= 0, where ˙γh(t) = (dγ_h/dt)(t). Denoted by th(t) = ˙γ_h(t)/_||γ˙_h(t)_{|| ∈} S2

1 the unit spacelike tangent vector. We

can define a unit spacelike vector e_h by e_h(t) = γ_h(t)_∧t_h(t) and call it the normal vector of γ_h, then we have a pseudo orthonormal frame _{γ_h,th,eh} of R31 along γh. By the standard arguments, we can give the following hyperbolic Frenet-Serret type formula:

  

˙ γ_h(t)

˙ t_h(t) ˙ eh(t)

 

 =||γ˙_h(t)_||

  

0 1 0 1 0 κh(t) 0 ₋κh(t) 0

  

  

γ_h(t) th(t) e_h(t)

whereκh(t) = det(γh(t),γ˙h(t),γ¨h(t))

||γ˙h(t)||

3 , we call it the hyperbolic geodesic curvature. We remark that

since γ_h is a regular curve in H2_{(−1), it may admit the arc length parametrization s =s(t).}

Therefore, we can assume that γ_h(s) is a unit speed curve. To the convenience of calculation, however, we stick to the general parametrization in this paper.

Under the assumption that κh(t)6=±1, we define the evolute of γh as follows:

Ev(γ_h) :I _→R3

1, Ev(γh)(t) =±

1

p

|κ2

h(t)−1|

(κh(t)γ_h(t) +eh(t)).

In the caseκ2

h(t)>1,Ev(γh)(t) is located inH2(−1), we call it thehyperbolic evolute ofγh and denote it by Eh

v(γh)(t). If 0 ≤κ2h(t) <1, it is in S12, we call it the de Sitter evolute of γh and denote it byEd

v(γh)(t). Ifκh(t)2−1 = 0 for allt∈I, thenγh is a part of a hyperbolic parabola (horosphere) in H2_{(−1) (cf. [10]). Moreover, if κ}

h(t0)2−1 = 0, then κh(t0)γh(t0) +eh(t0) is

a lightlike point. Then we can not define the evolute at such points in this way. In order to consider the evolute at such points, we need a theory of type changing curves inR3

1. Then we

have the following proposition ([8], Proposition 4.1).

Proposition 2.1 Suppose that γ_h : I _→ H2_{(−1) is a regular curve with κ}2

h(t) 6= 1. Then ˙

κh(t)_≡0 if and only if Eh

v(γh)(t) or Evd(γh)(t) are constant vectors. Under this condition, γh

is a part of a hyperbolic ellipse or a part of a hyperbolic hyperbola, respectively.

Ifv0 =Evh(γh)(t0) and c0 =∓κh(t0)/

p

|κ2

h(t0)−1|, then we have γh andHP(v0, c0) are at

least 3-point contact atγ_h(t0), see [8]. In this case, we callHP(v0, c0) theosculating hyperbolic ellipse (or, osculating hyperbolic hyperbola). Its center v0 is called the center of hyperbolic geodesic curvature. Therefore, the evolutes ofγ_his the locus of the center of hyperbolic geodesic curvature.

2.2

The evolutes of regular spacelike curves in de Sitter 2-space

We now consider the differential geometry of regular spacelike curves in S2

1. Let γd : I → S12

be a regular curve. The regular curve γ_d is said to be spacelike if ˙γ_d(t) is a spacelike vector at any t _∈ I, where ˙γ_d(t) = (dγ_d/dt)(t). We call such curve a spacelike curve. Denoted by td(t) = ˙γ_d(t)/_||γ˙_d(t)_{|| ∈} S2

1 the unit spacelike tangent vector. We can define a unit timelike

vector ed by ed(t) = γd(t)∧td(t) and call it the normal vector of γd, then we have a pseudo orthonormal frame _{γ_d,td,ed} of R31 along γd. By the standard arguments, we can give the following spacelike de Sitter Frenet-Serret type formula:

  

˙ γ_d(t)

˙ t_d(t) ˙ ed(t)

 

 =_||γ˙_d(t)_||

  

0 1 0 −1 0 κd(t)

0 κd(t) 0

  

  

γ_d(t) td(t) e_d(t)

  ,

whereκd(t) = det(γd(t),γ˙d(t),γ¨d(t))

||γ˙d(t)||

3 , we call it thespacelike de Sitter geodesic curvature.

Under the assumption that κd(t)6=±1, we define the evolute of γd as follows:

Ev(γ_d) :I _→R3

1, Ev(γd)(t) =±

1

p

|κ2

d(t)−1|

(κd(t)γ_d(t)₋ed(t)).

In the case κ2

d(t) > 1, Ev(γh)(t) is located in S12, we call it the de Sitter evolute of γd and denote it by Ed

v(γd)(t). If 0 ≤ κ2d(t) < 1, it is in H2(−1), we call it the hyperbolic evolute of γ_d and denote it byEh

v(γd)(t). We remark that for the case κ2d(t) = 1, it has similar geometric meaning with the caseκ2

Proposition 2.2 Suppose that γ_d :I _→S2

1 be a regular spacelike curve with κ2d(t)6= 1. Then ˙

κd(t)_≡0 if and only if Eh

v(γd)(t) or Evd(γd)(t) are constant vectors. Under this condition, γd

is a part of a de Sitter ellipse or a part of a de Sitter hyperbola, respectively.

The proof of this proposition is similar to that of Proposition 4.1 in [8], so we omit it.

We assume that v0 = Evh(γd)(t0) and c0 = ±κd(t0)/

p

|κ2

d(t0)−1|, then we have γd and

DP(v₀, c0) are at least 3-point contact atγd(t0). In this case, we callDP(v0, c0) theosculating de Sitter ellipse(or, osculating de Sitter hyperbola). Its centerv0 is called thecenter of spacelike de Sitter geodesic curvature. Therefore, the evolutes ofγ_dis the locus of the center of spacelike de Sitter geodesic curvature.

2.3

The evolutes of regular timelike curves in de Sitter 2-space

Finally, we consider the differential geometry of regular timelike curves in S2

1. Let γT :I →S12

be a regular curve. The regular curve γ_T is said to be timelike if ˙γ_T(t) is a timelike vector at any t _∈ I, where ˙γ_T(t) = (dγ_T/dt)(t). We call such curve the timelike curve. Denoted by tT(t) = ˙γT(t)/||γ˙T(t)|| ∈ H2(−1) the unit timelike tangent vector. We can define a unit spacelike normal vector e_T bye_T(t) =γ_T(t)_∧t_T(t) and call it the normal vectorof γ_T. Then we have a pseudo orthonormal frame_{γ_T,tT,eT} ofR31 along γT. By the standard arguments, we can give the followingtimelike de Sitter Frenet-Serret type formula:

  

˙ γ_T(t)

˙ t_T(t) ˙ e_T(t)

 

 =_||γ˙_T(t)_||

  

0 1 0 1 0 κT(t) 0 κT(t) 0

  

  

γ_T(t) tT(t) eT(t)

  ,

whereκT(t) =

det(γT(t),γ˙T(t),γ¨T(t))

||γ˙T(t)||

3 , we call it thetimelike de Sitter geodesic curvature.

We define the evolute of γ_T in de Sitter space as follows:

E_vd(γ_T) :I _→S₁2, E_vd(γ_T)(t) =_±p 1

κ2

T(t) + 1

(κT(t)γT(t)−eT(t)).

We call it the spacelike de Sitter evoluteof γ_T. Then we have the following proposition.

Proposition 2.3 Suppose that γ_T : I _→ S2

1 be a regular timelike curve. Then κ˙T(t) ≡ 0 if

and only if Ed

v(γT)(t) is a constant vector. Under this condition, γT is a part of a de Sitter

hyperbola.

The proof of this proposition is also similar to that of Proposition 4.1 in [8], so we omit it.

We assume that v0 = Evd(γT)(t0) and c0 = ±κT(t0)/

p

κ2

T(t0) + 1 then we have γT and

DP(v₀, c0) are at least 3-point contact atγT(t0). In this case, we callDP(v0, c0) theosculating de Sitter hyperbola. Its center v0 is called the center of timelike de Sitter geodesic curvature.

Therefore, the spacelike de Sitter evolutes ofγ_T is the locus of the center of timelike de Sitter geodesic curvature.

3

The frontals in hyperbolic space and de Sitter

2-space

orthonormal Frenet frame at these singular points. We also can not use the Frenet-Serret type formula to study the properties of the original curve. In order to overcome this difficulty, we take advantage of the way developed by the second author of this paper in [3] instead of the classical way. We give the detailed descriptions about this way as follows.

3.1

The spacelike frontals in hyperbolic 2-space

We firstly consider the differential geometry of curves in H2_{(−1). Let γ}

h : I → H2(−1) be a smooth curve. We call γ_h the spacelike frontal in H2_{(−1), if there exists a smooth mapping}

γd_h : I _→ S2

1, such that the pair (γh,γdh) : I → ∆1 satisfies (γh(t),γdh(t))∗θ = 0 for all t ∈ I. Here

∆1 ={(v,w) | hv,wi= 0 } ⊂H2(−1)×S12

is a 3-dimensional manifold and θ is a canonical contact 1-form on ∆1 (cf. [6, 7]). The

condition (γ_h(t),γ_hd(t))∗_{θ = 0 is equivalent tohγ˙}

h(t),γdh(t)i= 0, for all t ∈I. We call (γh,γdh) the spacelike Legendrian curve in ∆1. Moreover, if (γh,γdh) is an immersion, we call γh the

spacelike front inH2_{(−1) and (γ}

h,γdh) the spacelike Legendrian immersion in ∆1.

Let (γ_h,γd_h) be a spacelike Legendrian curve in ∆1. Ifγhis singular at a pointt0 inH2(−1),

then we can not define the Frenet-Serret formula at this point. By the definition of the spacelike Legendrian curve, however, the γd_h is always well defined even if at a singular point of γ_h. Let γs_h(t) =γ_h(t)_∧γd_h(t) _∈S2

1. We have a moving frame {γh,γdh,γsh} which called the hyperbolic

Legendrian Frenet frame of R3_{1 along} γ_h. By the standard arguments, we have the following

hyperbolic Legendrian Frenet-Serret type formula:

  

˙ γ_h(t)

˙ γd_h(t)

˙ γs_h(t)

 

 =

  

0 0 mh(t) 0 0 nh(t)

mh(t) ₋nh(t) 0

  

  

γ_h(t) γd_h(t) γs_h(t)

  ,

where mh(t) = _hγ˙_h(t),γs_h(t)_i and nh(t) = _hγ˙d_h(t),γ_hs(t)_i. We call the pair (mh, nh) the

space-like hyperbolic Legendrian curvature of spacelike Legendrian curve (γ_h,γd_h). We remark that if (γ_h,γd_h) is a spacelike Legendrian curve (respectively, spacelike Legendrian immersion) with the spacelike hyperbolic Legendrian curvature (mh, nh), then both (γh,−γdh) and (−γh,γdh) are spacelike Legendrian curves (respectively, spacelike Legendrian immersions) with the spacelike hyperbolic Legendrian curvatures (₋mh, nh) and (mh,−nh), respectively.

We will characterize the properties of (mh, nh) as follows.

Proposition 3.1 If (γ_h,γd_h) :I _→∆1 is a spacelike Legendrian curve with the spacelike hyper-bolic Legendrian curvature (mh, nh), then (mh, nh) depends on the parametrization of (γh,γdh).

Proof. Let (¯γ_h,γ¯d_h) : ¯I _→ ∆1 be a spacelike Legendrian curve and ( ¯mh,¯nh) be its spacelike hyperbolic Legendrian curvature. Suppose that t : ¯I _→ I is a (positive) change of param-eter, that is, t is surjective and has a positive derivative at every point. We assume that (¯γ_h(u),γ¯d_h(u)) = (γ_h(t(u)),γd_h(t(u))) for allu_∈I¯, then we have

(

¯

mh(u)¯γs_h(u) = ˙¯γ_h(u) =mh(t(u)) ˙t(u)γs_h(t(u)),

¯

nh(u)¯γsh(u) = ˙¯γdh(u) = nh(t(u)) ˙t(u)γsh(t(u)). It follows from ¯γs_h(u) = γs_h(t(u)), we have

(

¯

mh(u) = mh(t(u)) ˙t(u),

¯

✷

Proposition 3.2 Suppose that (γ_h,γd_h) : I _→ ∆1 is a spacelike Legendrian curve with the spacelike hyperbolic Legendrian curvature (mh, nh). Then (mh(t), nh(t))6= (0,0) if and only if ( ˙γ_h(t),γ˙d_h(t))₆= (0,0), for allt _∈I.

Proof. By the hyperbolic Legendrian Frenet-Serret type formula, this assertion holds. ✷

Example 3.3 Letγ_h be a regular curve inH2_{(−1) with the hyperbolic geodesic curvatureκh.}

If we takeγd_h =eh, then (γ_h,γd_h) is a spacelike Legendrian curve with the spacelike hyperbolic Legendrian curvature (_−||γ˙_h||,_||γ˙_h||κh). In fact, it is a spacelike Legendrian immersion. More-over, by a straightforward calculation, we have nh(t) =_|mh(t)|κh(t) for all t ∈I. In this case, we have nh(t) = 0 if and only if κh(t) = 0.

Example 3.4 Let γ_h : I _→ H2_{(−1) be γ}

h(t) = ( √

1 +t2n_+t2m_{, t}n_{, t}m_{), where m = n+k,}

m, n, k_∈N_{. It is obviously that the origin is the singular point of γ}

h. We assume that

γd_h(t) = _√ 1

k2_t2m_+m2_t2k_+n2(kt

m√_{1 +t}2n_+t2m_{, kt}m+n_+mtk_{, kt}2m_−n).

By a straightforward calculation, we have

γs_h(t) = √

1 +t2n_+t2m √

k2_t2m_+m2_t2k_+n2

mt2k+n_+ntn √

1 +t2n_+t2m, n, mt k

and (γ_h,γd_h) is a spacelike Legendrian curve with the spacelike hyperbolic Legendrian curvature (mh, nh), where

mh(t) =

tn−1√_k2_t2m_+m2_t2k_+n2

√

1 +t2n_+t2m ,

nh(t) =

ktk−1_(k2_t2m+2n_+m(m+n)t2m_+n(m+n)t2n_+mn) (k2_t2m_+m2_t2k_+n2₎√_{1 +t}2n_+t2m .

Moreover, if k= 1, then (γ_h,γd_h) is a spacelike Legendrian immersion. In the case when n= 2 and m= 3, we call γ_h the hyperbolic 3/2-cusp, see Figure 1 (i).

Example 3.5 LetI = [0,2π). We defineγ_h :I _→H2_{(−1) by}

γ_h(t) =p1 + cos6_{t+ sin}6_t,cos3_t,sin3_t

and call it the hyperbolic astroid, see Figure 1 (ii). It is obviously that γ_h is singular at point

t= 0, π/2, π and 3π/2. We assume that

γd_h(t) = _√ 1

1 + sin2_tcos2_t

sintcostp1 + sin6t+ cos6_{t,sint(1 + cos}4_{t),cost(1 + sin}4_t).

By a straightforward calculation, we have

γs_h(t) = √ 1

1 + sin2tcos2_t

sin2t₋cos2t,₋costp1 + sin6t+ cos6_t,sintp_{1 + sin}6_{t+ cos}6_t

and (γ_h,γd_h) is a spacelike Legendrian curve with the spacelike hyperbolic Legendrian curvature (mh, nh), where

mh(t) = 3 sintcost √

1 + sin2tcos2_t

√

1 + sin6t+ cos6_t ,

nh(t) =

3 sin4_tcos4_{t+ 3 sin}2_tcos2_t−1−sin6_t−cos6_t

(1 + sin2tcos2_t)√_{1 + sin}6_{t+ cos}6_t .

Moreover, (γ_h,γd_h) is also a spacelike Legendrian immersion.

Let (γ_h,γd_h) : I _→ ∆1 be a spacelike Legendrian curve in ∆1. We define a mapping

γφ_h :I _→H2_{(−1) by}

γφ_h(t) = coshφγ_h(t) + sinhφγd_h(t),

where φ is any fixed real number. Then γφ_h(t) is the point on the geodesic started from γ_h(t) directed byγd_h(t) with a constant length. We call it the hyperbolic parallelof the frontalγ_h (cf. [10]). Then we have the following assertion.

Proposition 3.6 Let (γ_h,γd_h) : I _→ ∆1 be a spacelike Legendrian curve with the spacelike hyperbolic Legendrian curvature(mh, nh). For any fixed real number φ, (γφh,(γ

φ

h)d) :I →∆1 is a spacelike Legendrian curve with the spacelike hyperbolic Legendrian curvature(mφ_h, nφ_h), where

(γφ_h)d:I _→S₁2, (γφ_h)d(t) = sinhφγ_h(t) + coshφγd_h(t),

mφ_h(t) = coshφmh(t) + sinhφnh(t), nφh(t) = sinhφmh(t) + coshφnh(t).

Moreover, (mφ_h)2_(t)−(nφ

h)2(t) = m2h(t)−n2h(t) for all t∈I.

Proof. It is obviously that (γφ_h(t),(γφ_h)d_{(t)) ∈ ∆}

1 and hγ˙φh(t),(γ φ

h)d(t)i = 0 for all t ∈ I. By the definition of spacelike Legendrian curve, (γφ_h,(γφ_h)d_{) is a spacelike Legendrian curve in ∆}

1.

On the other hand, we have γφ_h(t)_∧(γφ_h)d_{(t) = γ}s

h(t). Hence {γ φ h,(γ

φ

h)d,γsh} is a hyperbolic Legendrian Frenet frame of R3

1 along γ

φ

h. Moreover, ˙

γφ_h(t) = (coshφmh(t) + sinhφnh(t))γs_h(t), (γ˙φ_h)d_{(t) = (sinhφmh(t) + coshφnh(t))γ}s h(t). According to the hyperbolic Legendrian Frenet-Serret type formula, we have

mφ_h(t) = coshφmh(t) + sinhφnh(t), nφ_h(t) = sinhφmh(t) + coshφnh(t)

are the spacelike hyperbolic Legendrian curvature. By a direct calculation, we have (mφ_h)2_(t)−

(nφ_h)2_{(t) = m}2

3.2

The spacelike frontals in de Sitter 2-space

We now consider the differential geometry of spacelike curves in S2

1. Suppose γd : I → S12

is a spacelike curve at regular points t _∈ I, namely, ˙γ_d(t) is a spacelike vector at the regular points. We callγ_dthespacelike frontalinS2

1 if there exists a smooth mappingγhd :I →H2(−1) such that the pair (γh_d,γ_d) : I _→ ∆1 satisfies (γhd(t),γd(t))∗θ = 0 for all t ∈ I. The condition (γh_d(t),γ_d(t))∗_{θ = 0 is equivalent to hγ˙}

d(t),γhd(t)i = 0, for all t ∈ I. We call (γd,γhd) the

spacelike Legendrian curvein ∆1. Moreover, if (γd,γhd) is an immersion, we callγdthespacelike

frontin S2

1 and (γd,γhd) the spacelike Legendrian immersion in ∆1.

Let (γ_d,γh_d) be a spacelike Legendrian curve in ∆1 andγsd(t) = γd(t)∧γhd(t)∈S12. We have

a moving frame _{γ_d,γh_d,γs_d} which called the spacelike de Sitter Legendrian Frenet frame of

R3

1 alongγd. By the standard arguments, we have the following spacelike de Sitter Legendrian

Frenet-Serret type formula:

  

˙ γ_d(t)

˙ γh_d(t)

˙ γs_d(t)

 

 =

  

0 0 md(t) 0 0 nd(t) −md(t) nd(t) 0

  

  

γ_d(t) γh_d(t) γs_d(t)

  ,

wheremd(t) = _hγ˙_d(t),γs_d(t)_i and nd(t) = _hγ˙h_d(t),γs_d(t)_i. We call the pair (md, nd) the spacelike

de Sitter Legendrian curvature of the spacelike Legendrian curve (γ_d,γh_d). We remark that if (γ_d,γh_d) is a spacelike Legendrian curve (respectively, spacelike Legendrian immersion) with the spacelike de Sitter Legendrian curvature (md, nd), then both (γd,−γhd) and (−γd,γhd) are spacelike Legendrian curves (respectively, spacelike Legendrian immersions) with the spacelike de Sitter Legendrian curvatures (₋md, nd) and (md,−nd), respectively.

We can also characterize the properties of the spacelike de Sitter Legendrian curvature (md, nd) by the similar arguments with the spacelike hyperbolic Legendrian curvature (mh, nh). We summarize here as follows.

Proposition 3.7 If (γ_d,γh_d) is a spacelike Legendrian curve with the spacelike de Sitter Leg-endrian curvature (md, nd), then (md, nd) depends on the parametrization of (γd,γhd).

The proof is almost the same with Proposition 3.1, so that we omit it.

Proposition 3.8 Suppose that (γ_d,γh_d) be a spacelike Legendrian curve with the spacelike de Sitter Legendrian curvature(md, nd). Then(md(t), nd(t))6= (0,0)if and only if ( ˙γd(t),γ˙hd(t))6= (0,0), for all t_∈I.

Example 3.9 Letγ_d be a regular spacelike curve in S2

1 with the spacelike de Sitter geodesic

curvatureκd. If we take γhd =ed, then (γd,γhd) is a spacelike Legendrian curve in ∆1 with the

spacelike de Sitter Legendrian curvature (_kγ˙_dk,_kγ˙_dkκd). In fact, it is a spacelike Legendrian immersion in ∆1. Moreover, it follows from the spacelike de Sitter Legendrian Frenet-Serret

type formula, we have nd(t) =|md(t)|κd(t) for all t∈ I. Then we have nd(t) = 0 if and only if

κd(t) = 0.

Let (γ_d,γh_d) be a spacelike Legendrian curve in ∆1. We define a mappingγφd :I →S12 by

γφ_d(t) = coshφγ_d(t) + sinhφγh_d(t),

Proposition 3.10 Let (γ_d,γh_d) be a spacelike Legendrian curve in ∆1 with the spacelike de Sitter Legendrian curvature (md, nd). For any fixed real number φ, (γφ_d,(γφ_d)h) is a spacelike

Legendrian curve in ∆1 with the spacelike de Sitter Legendrian curvature (mφd, n φ

d), where (γφ_d)h :I _→H2(₋1), (γφ_d)h(t) = sinhφγ_d(t) + coshφγh_d(t),

mφ_d(t) = coshφmd(t) + sinhφnd(t), nφ_d(t) = sinhφmd(t) + coshφnd(t).

Moreover, (mφ_d)2_(t)−(nφ

d)2(t) = m2d(t)−n2d(t) for all t ∈I.

Proof. It is obviously that ((γφ_d)h_(t),γφ

d(t))∈∆1 and hγ˙φd(t),(γ φ

d)h(t)i= 0. By the definition of spacelike Legendrian curve, (γφ_d,(γφ_d)h_{) is a spacelike Legendrian curve in ∆}

1. On the other

hand, we haveγφ_d(t)_∧(γφ_d)h_{(t) =γ}s

d(t). Hence{γ φ d,(γ

φ

d)h,γsd}is a spacelike de Sitter Legendrian Frenet frame of R3

1 along γ

φ

d. Moreover,

˙

γφ_d(t) = (coshφmd(t) + sinhφnd(t))γs_d(t), (γ˙φ_d)h_{(t) = (sinhφmd(t) + coshφnd(t))γ}s d(t). According to the spacelike de Sitter Legendrian Frenet-Serret type formula, we have

mφ_d(t) = coshφmd(t) + sinhφnd(t), nφ_d(t) = sinhφmd(t) + coshφnd(t)

are the spacelike de Sitter Legendrian curvature. By a direct calculation, we have (mφ_d)2_(t)−

(nφ_d)2_{(t) = m}2

d(t)−n2d(t) for all t ∈I. ✷

3.3

The timelike frontals in de Sitter 2-space

We now consider the differential geometry of timelike curves in S2

1. Let γT : I → S12 be a

timelike curve at regular points t _∈ I, namely, ˙γ_T(t) is a timelike vector at the regular point. We callγ_T the timelike frontal in S2

1 if there exists a smooth mapping γdT :I →S12, such that

the pair (γ_T,γd_T) :I _→∆5 satisfies (γT(t),γdT(t))∗α= 0 for all t∈I. Here

∆5 ={(v,w)| hv,wi= 0 } ⊂S12×S12

is a 3-dimensional manifold andαis a canonical contact 1-form on ∆5 (cf. [2, 6]). The condition

(γ_T(t),γ_Td(t))∗_{α = 0 is equivalent to hγ˙}

T(t),γdT(t)i = 0, for all t ∈ I. We call (γT,γdT) the

timelike Legendrian curvein ∆5. Moreover, if (γT,γdT) is an immersion, we callγT thetimelike

frontin S2

1 and (γT,γdT) thetimelike Legendrian immersion in ∆5.

Let (γ_T,γd_T) be a timelike Legendrian curve in ∆5 and γhT(t) = γT(t)∧γdT(t) ∈ H2(−1). We have a moving frame _{γ_T,γd_T,γh_T} which called the timelike de Sitter Legendrian Frenet frame of R3

1 along γT. By the standard arguments, we have the following timelike de Sitter

Legendrian Frenet-Serret type formula:

   

˙ γ_T(t)

˙ γd_T(t)

˙ γh_T(t)

  

 =

  

0 0 mT(t) 0 0 nT(t)

mT(t) nT(t) 0

  

  

γ_T(t) γd_T(t) γh_T(t)

  ,

where mT(t) = −hγ˙T(t),γhT(t)i and nT(t) = −hγ˙dT(t),γhT(t)i. We call the pair (mT, nT) the

timelike de Sitter Legendrian curvatureof timelike Legendrian curve (γ_T,γd_T). We remark that if (γ_T,γd_T) is a timelike Legendrian curve (respectively, timelike Legendrian immersion) in ∆5

are timelike Legendrian curves (respectively, timelike Legendrian immersions) in ∆5 with the

timelike de Sitter Legendrian curvatures (₋mT, nT) and (mT,−nT), respectively.

We can also characterize the properties of the timelike de Sitter Legendrian curvature (mT, nT) by the similar arguments with the spacelike de Sitter Legendrian curvature (md, nd). We summarize them and omit the proofs as follows.

Proposition 3.11 If(γ_T,γd_T) :I _→∆5 is a timelike Legendrian curve with the timelike de Sit-ter Legendrian curvature (mT, nT), then (mT, nT) depends on the parametrization of (γT,γdT).

Proposition 3.12 Suppose that (γ_T,γd_T) : I _→ ∆5 is a timelike Legendrian curve with the timelike de Sitter Legendrian curvature (mT, nT). Then we have (mT(t), nT(t))6= (0,0) if and

only if ( ˙γ_T(t),γ˙d_T(t))₆= (0,0), for all t_∈I.

Example 3.13 Let γ_T be a regular timelike curve in S2

1 with the timelike de Sitter geodesic

curvatureκT. If we take γdT =eT, then (γT,γdT) is a timelike Legendrian curve in ∆5 with the

timelike de Sitter Legendrian curvature (_kγ˙_Tk,_kγ˙_TkκT). In fact, it is a timelike Legendrian immersion in ∆5. Moreover, we have nT(t) = |mT(t)|κT(t) for all t ∈ I. Therefore, we have

nT(t) = 0 if and only if κT(t) = 0.

Let (γ_T,γd_T) :I _→∆5 be a timelike Legendrian curve in ∆5. We defineγθT :I →S12 by

γθ_T(t) = cosθγ_T(t) + sinθγd_T(t),

whereθ_∈[0,2π) is a fixed number. Thenγθ_T(t) is the point on the geodesic started from γ_T(t) directed by γd_T(t) with a constant length. We call it the timelike parallel of the timelike frontal γ_T. Then we have the following assertion.

Proposition 3.14 Let (γ_T,γd_T) :I _→∆5 be a timelike Legendrian curve with the timelike de Sitter Legendrian curvature (mT, nT). For any fixed θ ∈ [0,2π), (γθT,(γθT)d) : I → ∆5 is a timelike Legendrian curve with the timelike de Sitter Legendrian curvature (mθ

T, nθT), where (γθ_T)d:I _→S₁2, (γθ_T)d(t) =₋sinθγ_T(t) + cosθγd_T(t),

mθ_T(t) = cosθmT(t) + sinθnT(t), nθT(t) = −sinθmT(t) + cosθnT(t).

Moreover, (mθ

T)2(t) + (nθT)2(t) = m2T(t) +n2T(t) for all t ∈I.

Proof. It is obviously that (γθ_T(t),(γθ_T)d_(t))∈∆

5 andhγ˙θT(t),(γθT)d(t)i= 0. By the definition of timelike Legendrian curve, (γθ_T,(γθ_T)d_{) is a timelike Legendrian curve in ∆}

5. On the other hand,

we have γθ_T(t)_∧(γθ_T)d_{(t) = γ}h

T(t). Hence {γθT,(γθT)d,γhT} is a timelike de Sitter Legendrian Frenet frame of R3_{1 along} γθ_T. Moreover,

˙

γθ_T(t) = (cosθmT(t) + sinθnT(t))γhT(t), (γTθ˙)d(t) = (−sinθmT(t) + cosθnT(t))γhT(t).

Therefore

mθ_T(t) = cosθmT(t) + sinθnT(t), nθT(t) =−sinθmT(t) + cosθnT(t)

are the timelike de Sitter Legendrian curvature and (mθ

T)2(t) + (nθT)2(t) =m2T(t) +n2T(t) for all

t_∈I. ✷

4

The evolutes of fronts in hyperbolic 2-space and de

Sitter 2-space

4.1

The evolutes of spacelike fronts in hyperbolic 2-space

We firstly consider the geometric meanings of evolutes of spacelike fronts in H2_{(−1). Let}

(γ_h,γd_h) : I _→ ∆1 be a spacelike Legendrian immersion with the spacelike hyperbolic

Legen-drian curvature (mh, nh) which satisfies n2h(t) 6= m2h(t) for all t ∈ I. We define a mapping Ev(γh) :I →R31 by

Ev(γh)(t) =±

1

p

|n2

h(t)−m2h(t)|

nh(t)γh(t)−mh(t)γdh(t)

and call it the totally evolute of γ_h inR3

1. We remark that if n2h(t)> m2h(t), then Ev(γh)(t) ∈

H2_{(−1). In this case, we denote it byE}h

v(γh) and call it thehyperbolic evoluteofγh. Moreover, if n2

h(t)< m2h(t), thenEv(γh)(t)∈S12. We rewrite it as Evd(γh) and call it the de Sitter evolute of γ_h. Furthermore, if nh(t)2 −mh(t)2 = 0 for all t ∈ I, then γh is a part of a hyperbolic parabola (horosphere) in H2_{(−1). If n}

h(t0)2 −mh(t0)2 = 0, then nh(t0)γh(t0)−mh(t0)γdh(t0)

is a lightlike point. Then we can not define the evolute at such points in this way. By a direct calculation, we have the following properties about the totally evolutes of γ_h.

Proposition 4.1 Suppose that (γ_h,γd_h) : I _→ ∆1 is a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mh, nh). Then the totally evolute Ev(γh) of γh

is independent on the parametrization of (γ_h,γd_h).

Proof. According to the proof of Proposition 3.1, if we take t : ¯I _→I as a (positive) change of parameter, that is, t is surjective and has a positive derivative at every point. Then we have

(

¯

mh(u) = mh(t(u)) ˙t(u),

¯

nh(u) =nh(t(u)) ˙t(u).

Therefore,

Ev(γh)(u) = ±

1

p

|n¯2

h(u)−m¯2h(u)| ¯

nh(u)γ_h(u)₋m¯h(u)γd_h(u)

=_±q 1

|n2

h(t(u))−m2h(t(u))|t˙2(u)

nh(t(u))γ_h(t(u))₋mh(t(u))γd_h(t(u))t˙(u)

=_Ev(γh)(t). ✷

Remark 4.2 Let (γ_h,γd_h) :I _→∆1 be a spacelike Legendrian immersion.

(i) If we take ₋γ_h instead of γ_h, then the totally evolute of γ_h does not change.

(ii) If we take ₋γd_h instead of γd_h, then the totally evolute of γ_h does not change.

Proposition 4.3 Let γ_h : I _→ H2_{(−1) be a regular curve in H}2_{(−1) with the hyperbolic} geodesic curvature κh which satisfies κh 6=±1. Then we have the following assertions:

(i) If κ2

h(t)>1, then Evh(γh)(t) =Evh(γh)(t). (ii) If κ2

Proof. Without loss of generality, by Example 3.3, we takeγd_h =e_h, then (γ_h,γd_h) is a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (_−||γ˙_h||,_||γ˙_h||κh). It follows from the definition of the totally evolute γ_h, we have

Ev(γ_h)(t) =_±p 1

|n2

h(t)−m2h(t)|

nh(t)γ_h(t)₋mh(t)γd_h(t)

=_±p 1

|κ2

h(t)−1|

(κh(t)γh(t) +eh(t)).

Since κ2

h(t) > 1 if and only if n2h(t) > m2h(t), we have Evh(γh)(t) = Evh(γh)(t). Moreover,

κ2

h(t)<1 if and only ifn2h(t)< m2h(t), we have Evd(γh)(t) =Evd(γh)(t). ✷ According to the above proposition, we have shown that the definition of the totally evolute ofγ_h is consistent with the definition of the evolute ofγ_h whenγ_h is a regular curve inH2_(−1).

Proposition 4.4 Suppose that (γ_h,γd_h) : I _→ ∆1 be a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mh, nh). Then we have the following assertions: (i) If t0 is a singular point of γh, then Evh(γh)(t0) = ±γh(t0).

(ii) If t0 is a singular point of γdh, then Evd(γh)(t0) = ±γdh(t0).

(iii) (a) If n2

h(t)> m2h(t), then (Evh(γh),γsh) :I →∆1 is a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature

˙

mhnh−mhn˙h

n2

h−m2h

,_±

q

n2

h−m2h

.

(b) If n2

h(t)< m2h(t), then (Evd(γh),γsh) :I →∆5 is a timelike Legendrian immersion with the timelike de Sitter Legendrian curvature

mhn˙h−m˙hnh

m2

h−n2h

,_±

q

m2

h−n2h

.

Proof. (i) Since t0 is a singular point of γh, we have mh(t0) = 0. It follows that

Evh(γh)(t0) =±

1

p

n2

h(t0)

nh(t0)γh(t0) = ±γh(t0).

(ii) Since t0 is a singular point of γdh, we have nh(t0) = 0. It follows that

Evd(γh)(t0) =±

1

p

m2

h(t0)

mh(t0)γdh(t0) =±γdh(t0).

(iii) We firstly suppose that n2

h(t)> m2h(t) and denote that

e

γ_h =_Evh(γ_h), γed_h =γ_h∧γd_h =γs_h, γes_h =eγ_h∧γ_ed_h.

By a straightforward calculation, we have

˙

e

γ_h(t) = m˙ h(t)nh(t)−mh(t) ˙nh(t)

n2

h(t)−m2h(t)

±1

p

n2

h(t)−m2h(t)

mh(t)γ_h(t)₋nh(t)γd_h(t)

!

˙

e

γd_h(t) = ˙γs_h(t) = _±

q

n2

h(t)−m2h(t) ± 1

p

n2

h(t)−m2h(t)

mh(t)γ_h(t)₋nh(t)γd_h(t)

!

and

e

γs_h(t) = p ±1

n2

h(t)−m2h(t)

mh(t)γ_h(t)₋nh(t)γd_h(t).

It follows that

hγe˙_h(t),γe˙_h(t)_i=

˙

mh(t)nh(t)−mh(t) ˙nh(t)

n2

h(t)−m2h(t)

2

≥0,

heγ˙_h(t),γed_h(t)_i=_hE˙_vh(γ_h)(t),γs_h(t)_i= 0, ( ˙γe_h(t),γe˙d_h(t))₆= (0,0) and

e

mh(t) =_hγe˙_h(t),γes_h(t)_i= m˙h(t)nh(t)−mh(t) ˙nh(t)

n2

h(t)−m2h(t)

,

e

nh(t) = _hγe˙_d(t),γes_h(t)_i=_±

q

n2

h(t)−m2h(t). This means that (γe_h,γed_h) = (_Eh

v(γh),γsh) : I → ∆1 is a spacelike Legendrian immersion with

the spacelike hyperbolic Legendrian curvature

˙

mhnh−mhn˙h

n2

h−m2h

,_±

q

n2

h−m2h

. Therefore the

assertion (a) of (iii) holds. Moreover, if n2

h(t)< m2h(t), then

˙

Evd(γh)(t) =

mh(t) ˙nh(t)₋m˙h(t)nh(t)

m2

h(t)−n2h(t)

±1

p

m2

h(t)−n2h(t)

mh(t)γh(t)−nh(t)γdh(t)

!

.

We have

hE˙vd(γh)(t),E˙vd(γh)(t)i=−

mh(t) ˙nh(t)−m˙h(t)nh(t)

m2

h(t)−n2h(t)

2

≤0.

Thus ˙_Ed

v(γh)(t) is a timelike vector at a regular point of Evd(γh) in S12. We denote that

e

γ_T =_Evd(γ_h), eγd_T =γs_h, γeh_T =eγ_{T ∧}γ_ed_T.

By using almost the same arguments as above, we have

˙

e

γd_T(t) = ˙γs_h(t) = _±

q

m2

h(t)−n2h(t)

±1

p

m2

h(t)−n2h(t)

mh(t)γh(t)−nh(t)γdh(t)

!

,

e

γh_T(t) = p ±1

m2

h(t)−n2h(t)

mh(t)γh(t)−nh(t)γdh(t)

,

hγe˙_T(t),γed_T(t)_i=_hE˙_vd(γ_h)(t),γs_h(t)_i= 0, ( ˙γe_T(t),γe˙d_T(t))₆= (0,0) and

e

mT(t) =hγe˙T(t),eγ h T(t)i=

mh(t) ˙nh(t)−m˙h(t)nh(t)

m2

h(t)−n2h(t)

,

e

nT(t) =hγe˙T(t),γe h

T(t)i=±

q

m2

h(t)−n2h(t). This means that (γe_T,γed_T) = (_Ed

v(γh),γsh) : I → ∆5 is a timelike Legendrian immersion with

the timelike de Sitter Legendrian curvature

mhn˙h−m˙hnh

m2

h−n2h

,_±

q

m2

h−n2h

Let (γ_h,γd_h) : I _→ ∆1 be a spacelike Legendrian immersion with the spacelike hyperbolic

Legendrian curvature (mh, nh) which satisfies n2h 6=m2h. We now explain the hyperbolic or de Sitter evolute of the spacelike front γ_h : I _→ H2_{(−1) as a wavefront from the viewpoint of}

Legendrian singularity theory [1, 11, 14] as follows. We define a functionHT _:I×H2_(−1)→R

by HT_{(t,v) = hγ}s

h(t),vi and call it the hyperbolic timelike height function on the space-like Legendrian immersion (γ_h,γd_h). We also define another function HS _{: I ×S}2

1 → R by

HS_{(t,v) =hγ}s

h(t),viand call it thehyperbolic spacelike height functionon the spacelike Legen-drian immersion (γ_h,γd_h). By a straightforward calculation, we have the following proposition.

Proposition 4.5 Let(γ_h,γd_h) :I _→∆1 be a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mh, nh).

(i) Suppose that v _∈H2_{(−1) and n}2

h(t)> m2h(t) for all t∈I:

(a)HT_{(t,v) = 0 if and only if there exist real numbersa andb such thatv =aγ}

h(t) +bγdh(t). (b) HT_{(t,v) = (∂H}T_{/∂t)(t,v) = 0 if and only if v =E}h

v(γh)(t). (ii) Suppose that v _∈S2

1 and n2h(t)< m2h(t) for all t∈I:

(a) HS_{(t,v) = 0 if and only if there exist real numbers a andb such thatv =aγ}

h(t) +bγdh(t). (b) HS_{(t,v) = (∂H}S_{/∂t)(t,v) = 0 if and only if v =E}d

v(γh)(t).

Proof. (i) Taking_{γ_h,γd_h,γs_h}as the moving frame ofR3_{1 along}γ_h. For all (t,v)_∈I_×H2_(−1),

HT_{(t,v) = 0 if and only if hγ}s

h(t),vi = 0. This means that there exist real numbers a and b withb2_−a2 _{=−1 such thatv =aγ}

h(t) +bγdh(t). Thus, the assertion (a) of (i) holds. Moreover,

HT_{(t,v) = (∂H}T_{/∂t)(t,v) = 0 if and only ifhm}

h(t)γh(t)−nh(t)γdh(t), aγh(t) +bγdh(t)i= 0. By a direct calculation, we can show that v =_Eh

v(γh)(t). Therefore, the assertion (b) of (i) holds. (ii) By almost the same arguments as the assertion (i), we can show the assertion (ii). ✷

One can show that (HT_{, ∂H}T_{/∂t) is non-singular at (t,v)∈ D(H}T_{), where}

D(HT) =_{(t,v)_|HT(t,v) = ∂H T

∂t (t,v) = 0}.

This means thatHT _{is a Morse family. Therefore, the hyperbolic evoluteE}h

v(γh) of the spacelike frontγ_h is a wavefront of a Legendrian immersion generated byHT_{. Moreover, the hyperbolic} spacelike height function HS _{is also a Morse family. Hence the de Sitter evoluteE}d

v(γh) of the spacelike front γ_h is also a wavefront of a Legendrian immersion generated byHS_.

Proposition 4.6 Let(γ_h,γd_h) :I _→∆1 be a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mh, nh). If t0 is a singular point of γh, then we have the

following assertions:

(i) t0 is a regular point of Evh(γh) if and only if m˙h(t0)6= 0.

(ii) t0 is a singular point of Evh(γh) if and only if γ¨h(t0) = 0. Proof. By a direct calculation,

˙

Evh(γh)(t) = ± ˙

mh(t)nh(t)₋mh(t) ˙nh(t)

n2

h(t)−m2h(t)

1

p

n2

h(t)−m2h(t)

mh(t)γ_h(t)₋nh(t)γd_h(t)

!

.

Since mh(t0) = 0, we have ˙Evh(γh)(t0) = ±m˙

h(t0)

|nh(t0)|γ

d

h(t0). Therefore, t0 is a regular point

(re-spectively, a singular point) of _Eh

v(γh) if and only if ˙mh(t0) 6= 0 (respectively, ˙mh(t0) = 0).

Furthermore, since ¨γ_h(t) = ˙mh(t)γs_h(t) +mh(t) ˙γ_hs(t),we have ¨γ_h(t0) = ˙mh(t0)γsh(t). Therefore, ˙

Remark 4.7 If t0 is a singular point of γh, then mh(t0) = 0 and hence we can not define

Ed

v(γh) at this point t0. This is the reason why we don’t consider the properties of Evd(γh) at

singular point of γ_h.

Proposition 4.8 Let(γ_h,γd_h) :I _→∆1 be a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mh, nh).

(i) n2

h(t)> m2h(t) and E˙vh(γh)(t) = 0 for all t ∈ I if and only if γh(t) is a point or there exist

a constant timelike vector v and a constant real number C with C2 _{< 1, such that γ}

h(t) ∈

HP(v,₋1) and γd_h(t)_∈DP(v,₋C) for all t_∈I.

(ii) m2

h(t) > n2h(t) and E˙vd(γh)(t) = 0 for all t ∈ I if and only if γdh(t) is a point or there

exist a constant spacelike vector w and a constant real number C with C2 _{< 1, such that}

γ_h(t)_∈HP(w, C) and γ_hd(t)_∈DP(w,1) for all t_∈I.

Proof. (i) Since

˙

Evh(γh)(t) = ± ˙

mh(t)nh(t)₋mh(t) ˙nh(t)

n2

h(t)−m2h(t)

1

p

n2

h(t)−m2h(t)

mh(t)γh(t)−nh(t)γdh(t)

!

,

we have ˙_Eh

v(γh)(t) = 0 if and only if ˙mh(t)nh(t)−mh(t) ˙nh(t) = 0 for all t ∈ I. Therefore, there exists a constant real number C with C2 _{< 1 such that mh(t) = Cnh(t). In the case}

when C _≡ 0, we have ˙γ_h(t) _≡ 0. This means that γ_h(t) is a point. Moreover, if C _6≡ 0, then ˙

γ_h(t) =Cγ˙d_h(t). It follows that γ_h(t) = Cγd_h(t) +v, where v is a constant timelike vector. It is obviously that _hγ_h(t),v_i=₋1 and _hγd_h(t),v_i=₋C for all t_∈I.

Conversely, if γ_h(t) is a point for all t _∈ I, then mh(t) _≡ m˙h(t) _≡ 0. It follows that ˙

Eh

v(γh)(t) ≡ 0. If v is a constant timelike vector and C is a constant real number, then γ_h(t) _∈ HP(v,₋1) and γd_h(t) _∈ HP(v,₋C). Since _hγ_h(t),v_i = ₋1 and _hγd_h(t),v_i = ₋C, we have _hγ˙_h(t),v_i = 0 and _hγ˙d_h(t),v_i = 0. Therefore, v = γ_h(t) ₋ Cγd_h(t). Then ˙v =

mh(t)γs_h(t)₋Cnh(t)γs_h(t) = 0. It follows that mh(t) = Cnh(t) and ˙mh(t) = Cn˙h(t). This means that ˙mh(t)nh(t)−mh(t) ˙nh(t) = 0 for all t∈I. Therefore, ˙Evh(γh)(t) = 0 for all t ∈I.

(ii) By almost the same arguments as the proof of the assertion (i), we can show that the

assertion (ii) holds. ✷

Proposition 4.9 Suppose that (γ_h,γd_h) :I _→∆1 is a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mh, nh) and (γφh,(γ

φ

h)d) :I →∆1 is a spacelike par-allel Legendrian immersion with the spacelike hyperbolic Legendrian curvature (mφ_h, nφ_h). Then

Ev(γh) = Ev(γ φ

h) for any φ∈R.

Proof. According to the definition of the totally evolute of the spacelike front in H2_{(−1) and}

Proposition 3.6, we have

Ev(γφh)(t) = ±

1

q

|(nφ_h)2_(t)−(mφ

h)2(t)|

nφ_h(t)γφ_h(t)₋mφ_h(t)(γφ_h)d(t)

=_±p 1

|nh2(t)−mh2(t)|

nh(t)γ_h(t)₋mh(t)(γ_h)d(t)=_Ev(γh)(t)

for any φ_∈R_{. ✷}

Example 4.10 Letγ_h :I _→H2_{(−1) beγ}

h(t) = ( √

1 +t4_+t6_{, t}2_{, t}3_{) andγ}d

h :I →S12 be

γd_h(t) = √ 1 4 + 9t2_+t6

t3√1 +t4_+t6_{, t}5_{+ 3t, t}6

−2.

By Example 3.4, we have

mh(t) = t √

4 + 9t2_+t6

√

1 +t4_+t6 , nh(t) =

t10_{+ 15t}6_{+ 10t}4_{+ 6}

(4 + 9t2_+t6₎√_{1 +t}4_+t6.

Therefore, we have

Ev(γ_h)(t) =_± 6(1 +t

4 _+t6₎3/2_{, 3t}8_{+ 6t}6_−27t4_−6t2_{, 6t}9_{+ 8t}7_{+ 24t}3_{+ 8t}

p

|(t10_{+ 15t}6_{+ 10t}4_{+ 6)}2_−t2_{(4 + 9t}2_+t6₎3_| ,

see Figure 2.

Example 4.11 Letγ_h :I _→H2(₋1) be

γ_h(t) = (p1 + cos6_{t+ sin}6_t,cos3_t,sin3_t)

and γd_h :I _→S2 1 be

γd_h(t) = √ 1

1 + sin2tcos2_t

sintcostp1 + sin6t+ cos6_{t,sint(1 + cos}4_{t),cost(1 + sin}4_t),

whereI = [0,2π). By Example 3.5, we have

mh(t) = 3 sintcost √

1 + sin2tcos2_t

√

1 + sin6_{t+ cos}6_t ,

nh(t) = 3 sin

4_tcos4_{t+ 3 sin}2_tcos2_t

−1₋sin6t₋cos6t

(1 + sin2tcos2_t)√_{1 + sin}6_{t+ cos}6_t .

Thus, we have

Ev(γh)(t) = ±

(a√a, acos3_{t+b/cost, asin}3_t+b/sint)

p

|(b₋a)2_{−3b(1 + sin}2_tcos2_t)2_| ,

wherea = 1 + cos6_{t+ sin}6_{t, b= 3 sin}4_tcos4_{t+ 3 sin}2_tcos2_{t, see Figure 3.}

hyperbolic evolute of the hyperbolic 3/2-cusp

de Sitter evolute of the hyperbolic 3/2-cusp

hyperbolic evolute of the hyperbolic astroid

de Sitter evolute of the hyperbolic astroid

Figure 3

4.2

The evolutes of spacelike fronts in de Sitter 2-space

We now consider the geometric meanings of evolutes of spacelike fronts in S2

1. Let (γd,γhd) be a spacelike Legendrian immersion in ∆1 with the spacelike de Sitter Legendrian curvature

(md, nd) which satisfiesn2d(t)6=m2d(t) for all t∈I. We define a mappingEv(γd) :I →R31 by

Ev(γ_d)(t) = _±p 1

|n2

d(t)−m2d(t)|

nd(t)γ_d(t)₋md(t)γh_d(t)

and call it the totally evolute of γ_d in R3

1. We remark that if n2d(t)< m2d(t), then Ev(γd)(t) ∈

H2_{(−1). In this case, we denote it byE}h

v(γd) and call it thehyperbolic evoluteofγd. Moreover, if n2

d(t)> m2d(t), then Ev(γd)(t) ∈S12. We rewrite it asEvd(γd) and call it thede Sitter evolute of γ_d. Furthermore, for the case n2

d(t)−m2d(t) = 0, it has similar geometric meaning with the case n2

h(t)−m2h(t) = 0. By a direct calculation, we have the following properties about the totally evolutes ofγ_d.

Proposition 4.12 Suppose that (γh_d,γ_d) : I _→ ∆1 is a spacelike Legendrian immersion with the spacelike de Sitter Legendrian curvature (md, nd). Then the totally evolute Ev(γd) of γd is

independent on the parametrization of (γ_d,γh_d).

Proof. If we take t: ¯I _→I as a (positive) change of parameter, that is, t is surjective and has a positive derivative at every point. Then we have

(

¯

md(u) = md(t(u)) ˙t(u),

¯

nd(u) =nd(t(u)) ˙t(u).

Therefore,

Ev(γd)(u) = ±

1

p

|n¯2

d(u)−m¯2d(u)| ¯

nd(u)γ_d(u)₋m¯d(u)γh_d(u)

=_±q 1

|n2

d(t(u))−m2d(t(u))|t˙2(u)

nd(t(u))γ_d(t(u))₋md(t(u))γh_d(t(u))t˙(u)

Remark 4.13 Let (γh_d,γ_d) :I _→∆1 be a spacelike Legendrian immersion.

(i) If we take ₋γ_d instead of γ_d, then the totally evolute of γ_d does not change.

(ii) If we take ₋γh_d instead of γh_d, then the totally evolute of γ_d does not change.

Proposition 4.14 Let γ_d : I _→ S2

1 be a regular spacelike curve in S12 with the spacelike de Sitter geodesic curvature κd which satisfies κd6=±1. Then we have the following assertions: (i) If κ2

d(t)>1, then Evd(γd)(t) =Evd(γd)(t). (ii) If κ2

d(t)<1, then Evh(γd)(t) = Evh(γd)(t).

Proof. Without loss of generality, by Example 3.9, we takeγh_d =ed, then (γ_d,γh_d) is a spacelike Legendrian immersion with the spacelike de Sitter Legendrian curvature (_||γ˙_d||,_||γ˙_d||κd). It follows from the definition of the totally evolute γ_d, we have

Ev(γ_d)(t) = _±p 1

|n2

d(t)−m2d(t)|

nd(t)γ_d(t)₋md(t)γh_d(t)

=_±p 1

|κ2

d(t)−1|

(κd(t)γ_d(t)₋ed(t)).

Since κ2

d(t) > 1 if and only if n2d(t) > m2d(t), we have Evd(γd)(t) = Evd(γd)(t). Moreover,

κ2

d(t)<1 if and only ifn2d(t)< m2d(t), we have Evh(γd)(t) = Evh(γd)(t). ✷ According to the above proposition, we have shown that the definition of the totally evolute of γ_d is consistent with the definition of the evolute of γ_d when γ_d is a regular spacelike curve inS2

1.

Proposition 4.15 If (γh_d,γ_d) :I _→∆1 be a spacelike Legendrian immersion with the spacelike de Sitter Legendrian curvature (md, nd), then we have the following assertions:

(i) If t0 is a singular point of γd, then Evd(γd)(t0) =±γd(t0).

(ii) If t0 is a singular point of γhd, then Evh(γd)(t0) = ±γhd(t0).

(iii) (a) If n2

d(t)> m2d(t), then (Evd(γd),γsd) : I → ∆5 is a timelike Legendrian immersion with the timelike de Sitter Legendrian curvature

mdn˙d−m˙ dnd

n2

d−m2d

,_±

q

n2

d−m2d

.

(b) If n2

d(t)< m2d(t), then (Evh(γd),γsd) :I →∆1 is a spacelike Legendrian immersion with the spacelike hyperbolic Legendrian curvature

˙

mdnd−mdn˙d

m2

d−n2d

,_±

q

m2

d−n2d

.

Proof. (i) Since t0 is a singular point of γd, we have md(t0) = 0. It follows that

Evd(γd)(t0) =±

1

p

n2

d(t0)

nd(t0)γd(t0) =±γd(t0).

(ii) Since t0 is a singular point of γhd, we have nd(t0) = 0. It follows that

Evh(γd)(t0) = ±

1

p

m2

d(t0)