Table Of ContentHELICOIDAL FLAT SURFACES IN THE 3-SPHERE
F.MANFIO1 ANDJ.P.DOSSANTOS2
Abstract. Inthispaper, helicoidalflatsurfacesinthe3-dimensionalsphere
S3 areconsidered. Acompleteclassificationofsuchsurfacesisgiveninterms
oftheirfirstandsecondfundamentalformsandbylinearsolutionsofthecor-
respondinganglefunction. TheclassificationisobtainedbyusingtheBianchi-
6 Spivak construction for flat surfaces and a representation for constant angle
1 surfacesinS3.
0
2
n
a 1. Introduction
J
Helicoidalsurfacesin3-dimensionalspaceformsariseasanaturalgeneralization
9
of rotational surfaces in such spaces. These surfaces are invariant by a subgroup of
1
thegroupofisometriesoftheambientspace,calledhelicoidalgroup,whoseelements
] can be seen as a composition of a translation with a rotation for a given axis.
G
In the Euclidean space R3, do Carmo and Dajczer [5] describe the space of all
D
helicoidal surfaces that have constant mean curvature or constant Gaussian curva-
. ture. Thisspacebehavesasacircularcylinder,whereagivengeneratorcorresponds
h
t totherotationalsurfacesandeachparallelcorrespondstoaperiodicfamilyofheli-
a
coidalsurfaces. HelicoidalsurfaceswithprescribedmeanorGaussiancurvatureare
m
obtained by Baikoussis and Koufogiorgos [2]. More precisely, they obtain a closed
[
formofsuchasurfacebyintegratingthesecond-orderordinarydifferentialequation
1 satisfied by the generating curve of the surface. Helicoidal surfaces in R3 are also
v considered by Perdomo [19] in the context of minimal surfaces, and by Palmer and
0
Perdomo [18] where the mean curvature is related with the distance to the z-axis.
3
In the context of constant mean curvature, helicoidal surfaces are considered by
9
Solomon and Edelen in [8].
4
0 In the 3-dimensional hyperbolic space H3, Mart´ınez, the second author and
. Tenenblat [16] give a complete classification of the helicoidal flat surfaces in terms
1
0 of meromorphic data, which extends the results obtained by Kokubu, Umehara
6 and Yamada [13] for rotational flat surfaces. Moreover, the classification is also
1 given by means of linear harmonic functions, characterizing the flat fronts in H3
:
v thatcorrespondtolinearharmonicfunctions. Namely,itiswellknownthatforflat
i surfaces in H3, on a neighbourhood of a non-umbilical point, there is a curvature
X
lineparametrizationsuchthatthefirstandsecondfundamentalformsaregivenby
r
a
I = cosh2φ(u,v)(du)2+sinh2φ(u,v)(dv)2,
(1) II = sinhφ(u,v)coshφ(u,v)(cid:0)(du)2+(dv)2(cid:1),
whereφisaharmonicfunction,i.e.,φ +φ =0. Inthiscontext,themainresult
uu vv
states that a surface in H3, parametrized by curvature lines, with fundamental
formsasin(1)andφ(u,v)linear, i.e, φ(u,v)=au+bv+c, isflatifandonlyif, the
surface is a helicoidal surface or a peach front, where the second one is associated
2010 Mathematics Subject Classification. Primary53A35,53B20,53C42.
Key words and phrases. Helicoidalsurfaces,flatsurfaces,3-sphere.
1ResearchpartiallysupportedbyFAPESP/Brazil,grant2014/01989-9.
2ResearchpartiallysupportedbyFEMAT/UnBandFAPDF.
1
2 F.MANFIOANDJ.P.DOSSANTOS
to the case (a,b,c)=(0,±1,0). Helicoidal minimal surfaces were studied by Ripoll
[20]andhelicoidalconstantmeancurvaturesurfacesinH3 areconsideredbyEdelen
[7], as well as the cases where such invariant surfaces belong to R3 and S3.
Similarly to the hyperbolic space, for a given flat surface in the 3-dimensional
sphere S3, there exists a parametrization by asymptotic lines, where the first and
the second fundamental forms are given by
I = du2+2cosωdudv+dv2,
(2)
II = 2sinωdudv
for a smooth function ω, called the angle function, that satisfy the homogeneous
wave equation ω =0. Therefore, one can ask which surfaces are related to linear
uv
solutions of such equation.
Theaimofthispaperistogiveacompleteclassificationofhelicoidalflatsurfaces
in S3, established in Theorems 1 and 2, by means of asymptotic lines coordinates,
with first and second fundamental forms given by (2), where the angle function is
linear. Inordertodothis,oneusestheBianchi-Spivakconstructionforflatsurfaces
in S3. This construction and the Kitagawa representation [12], are important tools
used in the recent developments of flat surface theory. Examples of applications of
suchrepresentationscanbeseenin[9]and[1]. Ourclassificationalsomakesuseofa
representationforconstantanglesurfacesinS3,whocomesfromacharacterization
of constant angle surfaces in the Berger spheres obtained by Montaldo and Onnis
[17].
This paper is organized as follows. In Section 2 we give a brief description of
helicoidalsurfacesinS3,aswellasaordinarydifferentialequationthatcharacterizes
those one that has zero intrinsic curvature.
In Section 3, the Bianchi-Spivak construction is introduced. It will be used to
proveTheorem1,whichstatesthataflatsurfaceinS3,withasymptoticparameters
and linear angle function, is invariant under helicoidal motions.
In Section 4, Theorem 2 establishes the converse of Theorem 1, that is, a heli-
coidal flat surface admits a local parametrization, given by asymptotic parameters
where the angle function is linear. Such local parametrization is obtained by using
a characterization of constant angle surfaces in Berger spheres, which is a conse-
quence of the fact that a helicoidal flat surface is a constant angle surface in S3,
i.e., it has a unit normal that makes a constant angle with the Hopf vector field.
In section 5 we present an application for conformally flat hypersurfaces in R4.
The classification result obtained is used to give a geometric characterization for
special conformally flat surfaces in 4−dimensional space forms. It is known that
conformally flat hypersurfaces in 4-dimensional space forms are associated with
solutions of a system of equations, known as Lam´e’s system (see [11] and [6] for
details). In[6],Tenenblatandthesecondauthorobtainedinvariantsolutionsunder
symmetry groups of Lam´e’s system. A class of those solutions is related to flat
surfacesinS3, parametrizedbyasymptoticlineswithlinearanglefunction. Thusa
geometric description of the correspondent conformally flat hypersurfaces is given
in terms of helicoidal flat surfaces in S3.
2. Helicoidal flat surfaces
Given any β ∈R, let {ϕ (t)} be the one-parameter subgroup of isometries of S3
β
given by
1 0 0 0
0 1 0 0
ϕβ(t)= 0 0 cosβt −sinβt .
0 0 sinβt cosβt
HELICOIDAL FLAT SURFACES IN THE 3-SPHERE 3
Whenβ (cid:54)=0, thisgroupfixesthesetl={(z,0)∈S3}, whichisagreatcircleandit
is called the axis of rotation. In this case, the orbits are circles centered on l, i.e.,
{ϕ (t)} consists of rotations around l. Given another number α∈R, consider now
β
the translations {ψ (t)} along l,
α
cosαt −sinαt 0 0
sinαt cosαt 0 0
ψα(t)= 0 0 1 0 .
0 0 0 1
Definition 1. A helicoidal surface in S3 is a surface invariant under the action of
the helicoidal 1-parameter group of isometries
cosαt −sinαt 0 0
sinαt cosαt 0 0
(3) φα,β(t)=ψα(t)◦ϕβ(t)= 0 0 cosβt −sinβt ,
0 0 sinβt cosβt
given by a composition of a translation ψ (t) and a rotation ϕ (t) in S3.
α β
Remark 1. When α=β, these isometries are usually called Clifford translations.
In this case, the orbits are all great circles, and they are equidistant from each
other. In fact, the orbits of the action of G coincide with the fibers of the Hopf
fibration h : S3 → S2. We note that, when α = −β, these isometries are also, up
to a rotation in S3, Clifford translations. For this reason we will consider in this
paper only the cases α(cid:54)=±β.
Withthesebasicpropertiesinmind,ahelicoidalsurfacecanbelocallyparametrized
by
(4) X(t,s)=φ (t)·γ(s),
α,β
where γ :I ⊂R→S2 is a curve parametrized by the arc length, called the profile
+
curve of the parametrization X. Here, S2 is the half totally geodesic sphere of S3
+
given by
S2 =(cid:8)(x ,x ,x ,0)∈S3 :x >0(cid:9).
+ 1 2 3 3
Then we have
X = φ (t)·(−αx ,αx ,0,βx ),
t α,β 2 1 3
X = φ (t)·γ(cid:48)(s).
s α,β
Moreover, a unit normal vector field associated to the parametrization X is given
by N =N˜/(cid:107)N˜(cid:107), where N˜ is explicitly given by
(5) N˜ =φ (t)·(cid:0)βx (x(cid:48)x −x x(cid:48),βx (x x(cid:48) −x(cid:48)x ),βx (x(cid:48)x −x x(cid:48)),−αx(cid:48)(cid:1).
α,β 3 2 3 2 3 3 1 3 1 3 3 1 2 1 2 3
Let us now consider a parametrization by the arc length of γ given by
(cid:0) (cid:1)
(6) γ(s)= cosϕ(s)cosθ(s),cosϕ(s)sinθ(s),sinϕ(s),0 .
We will finish this section discussing the flatness of helicoidal surfaces in S3.
Recall that a simple way to obtain flat surfaces in S3 is by means of the Hopf
fibration h:S3 →S2. More precisely, if c is a regular curve in S2, then h−1(c) is a
flatsurfaceinS3 (cf. [21]). SuchsurfacesarecalledHopf cylinders. Thenextresult
provides a necessary and sufficient condition for a helicoidal surface, parametrized
as in (4), to be flat.
Proposition1. Ahelicoidalsurfacelocallyparametrizedasin (4),whereγ isgiven
by (6), is a flat surface if and only if the following equation
(7) β2ϕ(cid:48)(cid:48)sin3ϕcosϕ−β2(ϕ(cid:48))2sin4ϕ+α2(ϕ(cid:48))4cos4ϕ=0
is satisfied.
4 F.MANFIOANDJ.P.DOSSANTOS
Proof. Sinceφ (t)∈O(4)andγ isparametrizedbythearclength,thecoefficients
α,β
of the first fundamental form are given by
E = α2cos2ϕ+β2sin2ϕ,
F = αθ(cid:48)cos2ϕ,
G = (ϕ(cid:48))2+(θ(cid:48))2cos2ϕ=1.
Moreover, the Gauss curvature K is given by
4(EG−F2)2K = E(cid:2)E G −2F G +(G )2(cid:3)+G(cid:2)E G −2E F +(E )2(cid:3)
s s t s t t t t s s
+F(E G −E G −2E F +4F F −2F G )
t s s t s s t s t t
−2(EG−F2)(E −2F +G ).
ss st tt
Thus, it follows from the expression of K and from the coefficients of the first
fundamental form that the surface is flat if, and only if,
(8) E (EG−F2) −2(EG−F2)E =0.
s s ss
When α=±β, the equation (8) is trivially satisfied, regardless of the chosen curve
γ. For the case α(cid:54)=±β, since
EG−F2 =β2sin2ϕ+α2(ϕ(cid:48))2cos2ϕ,
a straightforward computation shows that the equation (8) is equivalent to
(β2−α2)(cid:0)β2ϕ(cid:48)(cid:48)sin3ϕcosϕ−β2(ϕ(cid:48))2sin4ϕ+α2(ϕ(cid:48))4cos4ϕ(cid:1)=0,
and this concludes the proof. (cid:3)
3. The Bianchi-Spivak construction
A nice way to understand the fundamental equations of a flat surface M in S3
isbyparameterswhosecoordinatecurvesareasymptoticcurvesonthesurface. As
M is flat, its intrinsic curvature vanishes identically. Thus, by the Gauss equa-
tion, the extrinsic curvature of M is constant and equal to −1. In this case, as
the extrinsic curvature is negative, it is well known that there exist Tschebycheff
coordinates around every point. This means that we can choose local coordinates
(u,v)suchthatthecoordinatescurvesareasymptoticcurvesofM andthesecurves
are parametrized by the arc length. In this case, the first and second fundamental
forms are given by
I =du2+2cosωdudv+dv2,
(9)
II =2sinωdudv,
for a certain smooth function ω, usually called the angle function. This function
ω has two basic properties. The first one is that as I is regular, we must have
0 < ω < π. Secondly, it follows from the Gauss equation that ω = 0. In
uv
other words, ω satisfies the homogeneous wave equation, and thus it can be locally
decomposedasω(u,v)=ω (u)+ω (v), whereω andω aresmoothrealfunctions
1 2 1 2
(cf. [10] and [21] for further details).
Given a flat isometric immersion f : M → S3 and a local smooth unit normal
vector field N along f, let us consider coordinates (u,v) such that the first and
the second fundamental forms of M are given as in (9). The aim of this work
is to characterize the flat surfaces when the angle function ω is linear, i.e., when
ω =ω +ω is given by
1 2
(10) ω (u)+ω (v)=λ u+λ v+λ
1 2 1 2 3
where λ ,λ ,λ ∈ R. I order to do this, let us first construct flat surfaces in S3
1 2 3
whose first and second fundamental forms are given by (9) and with linear angle
function. This construction is due to Bianchi [3] and Spivak [21].
HELICOIDAL FLAT SURFACES IN THE 3-SPHERE 5
We will use here the division algebra of the quaternions, a very useful approach
to describe explicitly flat surfaces in S3. More precisely, we identify the sphere S3
with the set of the unit quaternions {q ∈ H : qq = 1} and S2 with the unit sphere
in the subspace of H spanned by 1, i and j.
Proposition 2 (Bianchi-Spivak representation). Let c ,c : I ⊂ R → S3 be two
a b
curvesparametrizedbythearclength,withcurvaturesκ andκ ,andwhosetorsions
a b
are given by τ = 1 and τ = −1. Suppose that 0 ∈ I, c (0) = c (0) = (1,0,0,0) e
a b a b
c(cid:48)(0)∧c(cid:48)(0)(cid:54)=0. Then the map
a b
X(u,v)=c (u)·c (v)
a b
isalocalparametrizationofaflatsurfaceinS3,whosefirstandsecondfundamental
forms are given as in (9), where the angle function satisfies ω(cid:48)(u) = −κ (u) and
1 a
ω(cid:48)(v)=κ (v).
2 b
Since the goal here is to find a parametrization such that ω can be written as
in (10), it follows from Theorem 2 that the curves of the representation must have
constant curvatures. Therefore, we will use the Frenet-Serret formulas in order to
obtain curves with torsion ±1 and with constant curvatures.
Given a real number r >1, let us consider the curve γ :R→S3 given by
r
1 (cid:16) u u (cid:17)
(11) γ (u)= √ rcos ,rsin ,cosru,sinru .
r 1+r2 r r
Astraightforwardcomputationshowsthatγ (u)isparametrizedbythearclength,
r
has constant curvature κ = r2−1 and its torsion τ satisfies τ2 = 1. Observe that
r
γ (u) is periodic if and only if r2 ∈ Q. When r is a positive integer, γ (u) is a
r r
closed curve of period 2πr. A curve γ as in (11) will be called a base curve.
Now we just have to apply rigid motions to a base curve in order to satisfy the
remainingrequirementsoftheBianchi-Spivakconstruction. Itiseasytoverifythat
the curves
1
c (u) = √ (a,0,−1,0)·γ (u),
a a
1+a2
(12)
1
c (v) = √ T(γ (v))·(b,0,0,−1),
b b
1+b2
arebasecurves, andsatisfyc (0)=c (0)=(1,0,0,0)andc(cid:48)(0)∧c(cid:48)(0)(cid:54)=0, where
a b a b
1 0 0 0
0 1 0 0
(13) T = .
0 0 0 1
0 0 1 0
Therefore we can establish our first main result:
Theorem 1. The map X :U ⊂R2 →S3 given by
X(u,v)=c (u)·c (v),
a b
where c and c are the curves given in (12), is a parametrization of a flat surface
a b
in S3, whose first and second fundamental forms are given by
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
I = du2+2cos 1−a2 u+ b2−1 v+c dudv+dv2,
a b
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
II = 2sin 1−a2 u+ b2−1 v+c dudv,
a b
wherecisaconstant. Moreover,uptorigidmotions,X isinvariantunderhelicoidal
motions.
6 F.MANFIOANDJ.P.DOSSANTOS
Proof. ThestatementaboutthefundamentalformsfollowsdirectlyfromtheBianchi-
Spivak construction. For the second statement, note that the parametrization
X(u,v) can be written as
X(u,v)=g ·Y(u,v)·g ,
a b
where
1
g = √ (a,0,−1,0),
a
1+a2
1
g = √ (b,0,0,−1),
b
1+b2
and
Y(u,v)=γ (u)·T(γ (v)).
a b
To conclude the proof, it suffices to show that Y(u,v) is invariant by helicoidal
motions. To do this, we have to find α and β such that
(cid:0) (cid:1)
φ (t)·Y(u,v)=Y u(t),v(t) ,
α,β
where u(t) and v(t) are smooth functions. Observe that Y(u,v) can be written as
1
(14) Y(u,v)= (y ,y ,y ,y ),
(cid:112) 1 2 3 4
(1+a2)(1+b2)
where
(cid:16)u v(cid:17)
y (u,v) = abcos + −sin(au+bv),
1 a b
(cid:16)u v(cid:17)
y (u,v) = absin + +cos(au+bv),
2 a b
(cid:16) v(cid:17) (cid:16)u (cid:17)
y (u,v) = bcos au− −asin −bv ,
3 b a
(cid:16) v(cid:17) (cid:16)u (cid:17)
y (u,v) = bsin au− +acos −bv .
4 b a
A straightforward computation shows that if φ (t) is given by (3), we have
α,β
u(t)=u+z(t) and v(t)=v+w(t),
where
a(b2−1) b(1−a2)
(15) z(t)= βt and w(t)= βt,
a2b2−1 a2b2−1
with
b2−a2
(16) α= β,
a2b2−1
showing that Y(u,v) is invariant by helicoidal motions. Observe that when a=±b
we have α=0, i.e., X is a rotational surface in S3. (cid:3)
Remark2. Itisimportanttonotethattheconstantaandbin(12)wereconsidered
in (1,+∞) in order to obtain non-zero constant curvatures with its well defined
torsions, and then to apply the Bianchi-Spivak construction. This is not a strong
restriction since the curvature function κ(t) = t2−1 assumes all values in R\{0}
t
when t ∈ (1,+∞). However, by taking a = 1 and b > 1 in (12), a long but
straightforward computation gives an unit normal vector field
1
N(u,v)= (n ,n ,n ,n ),
(cid:112) 1 2 3 4
2(1+b2)
HELICOIDAL FLAT SURFACES IN THE 3-SPHERE 7
where
(cid:16) v(cid:17)
n (u,v) = −bsin u+ +cos(u+bv),
1 b
(cid:16) v(cid:17)
n (u,v) = bcos u+ +sin(u+bv),
2 b
(cid:16) v(cid:17)
n (u,v) = bsin u− −cos(a−bv),
3 b
(cid:16) v(cid:17)
n (u,v) = −bcos u− −sin(u−bv).
4 b
Therefore,oneshowsthatthisparametrizationisalsobyasymptoticlineswherethe
anglefunctionisgivenbyω(u,v)= 1−b2v−π. Moreover, thisisaparametrization
b 2
of a Hopf cylinder, since the unit normal vector field N makes a constant angle
with the Hopf vector field (see section 4).
We will use the parametrization Y(u,v) given in (14), compose with the stereo-
graphic projection in R3, to visualize some examples with the corresponding con-
stants a and b.
Figure 1. a=2 and b=3.
√
Figure 2. a= 2 and b=3.
8 F.MANFIOANDJ.P.DOSSANTOS
√ √
Figure 3. a= 3 and b= 2.
4. Constant angle surfaces
InthissectionwewillcompleteourclassificationofhelicoidalflatsurfacesinS3,
byestablishingoursecondmaintheorem,thatcanbeseenasaconverseofTheorem
1. It is well known that the Hopf map h : S3 → S2 is a Riemannian submersion
and the standard orthogonal basis of S3
E (z,w)=i(z,w), E (z,w)=i(−w,z), E (z,w)=(−w,z)
1 2 3
hasthepropertythatE isverticalandE , E arehorizontal. ThevectorfieldE ,
1 2 3 1
usually called the Hopf vector field, is an unit Killing vector field.
ConstantanglesurfaceinS3 arethosesurfaceswhoseitsunitnormalvectorfield
makes a constant angle with the Hopf vector field E . The next result states that
1
flatness of a helicoidal surface in S3 turns out to be equivalent to constant angle
surface.
Proposition 3. AhelicoidalsurfaceinS3, locallyparametrizedby (4)andwiththe
profile curve γ parametrized by (6), is a flat surface if and only if it is a constant
angle surface.
Proof. Let us consider the Hopf vector field
E (x ,x ,x ,x )=(−x ,x ,−x ,x ),
1 1 2 3 4 2 1 4 3
and let us denote by ν the angle between E and the normal vector field N along
1
the surface given in (5). Along the parametrization (4), we can write the vector
field E as
1
E (X(t,s))=φ (t)(−x ,x ,0,x ).
1 α,β 2 1 3
Then, since φ (t)∈O(4), we have
α,β
x x(cid:48)
(cid:104)N,E (cid:105)(t,s)=(cid:104)N,E (cid:105)(s)=(β−α) 3 3 .
1 1 (cid:112)
β2x2+α2(x(cid:48))2
3 3
By considering the parametrization (6) for the profile curve γ, the angle ν = ν(s)
between N and E is given by
1
ϕ(cid:48)sinϕcosϕ
(17) cosν(s)=(β−α) .
(cid:113)
β2sin2ϕ+α2(ϕ(cid:48))2cos2ϕ
HELICOIDAL FLAT SURFACES IN THE 3-SPHERE 9
By taking the derivative in (17), we have
d (β−α)(cid:0)β2ϕ(cid:48)(cid:48)sin3ϕcosϕ−β2(ϕ(cid:48))2sin4ϕ+α2(ϕ(cid:48))2cos4ϕ(cid:1)
(cosν(s))= ,
ds (cid:0)β2sin2ϕ+α2(ϕ(cid:48))2cos2ϕ(cid:1)23
and the conclusion follows from the Proposition 1. (cid:3)
Given a number (cid:15) > 0, let us recall that the Berger sphere S3 is defined as the
(cid:15)
sphere S3 endowed with the metric
(18) (cid:104)X,Y(cid:105) =(cid:104)X,Y(cid:105)+((cid:15)2−1)(cid:104)X,E (cid:105)(cid:104)Y,E (cid:105),
(cid:15) 1 1
where (cid:104),(cid:105) denotes de canonical metric of S3. We define constant angle surface in
S3 in the same way that in the case of S3. Constant angle surfaces in the Berger
(cid:15)
spheres were characterized by Montaldo and Onnis [17]. More precisely, if M is
a constant angle surface in the Berger sphere, with constant angle ν, then there
exists a local parametrization F(u,v) given by
(19) F(u,v)=A(v)b(u),
where
(cid:0)√ √ √ √ (cid:1)
(20) b(u)= c cos(α u), c sin(α u), c cos(α u), c sin(α u)
1 1 1 1 2 2 2 2
√ √
is a geodesic curve in the torus S1( c )×S1( c ), with
1 2
1 (cid:15)cosν 2B 2B
c = ∓ √ , α = c , α = c , B =1+((cid:15)2−1)cos2ν,
1,2 2 2 B 1 (cid:15) 2 2 (cid:15) 1
and
(21) A(v)=A(ξ,ξ ,ξ ,ξ )(v)
1 2 3
is a 1-parameter family of 4×4 orthogonal matrices given by
A(v)=A(ξ)·A˜(v),
where
1 0 0 0
0 1 0 0
A(ξ)=
0 0 sinξ cosξ
0 0 −cosξ sinξ
and
cosξ cosξ −cosξ sinξ sinξ cosξ −sinξ sinξ
1 2 1 2 1 2 1 3
A˜(v)= cosξ1sinξ2 −cosξ1cosξ2 sinξ1sinξ3 sinξ1cosξ3 ,
−sinξ1cosξ3 sinξ1sinξ3 cosξ1cosξ2 cosξ1sinξ2
sinξ sinξ −sinξ cosξ −cosξ sinξ cosξ cosξ
1 3 1 3 1 2 1 2
ξ is a constant and the functions ξ (v), 1≤i≤3, satisfy
i
(22) cos2(ξ (v))ξ(cid:48)(v)−sin2(ξ (v))ξ(cid:48)(v)=0.
1 2 1 3
In the next result we obtain another relation between the function ξ , given in
i
(21), and the angle function ν.
Proposition 4. The functions ξ (v), given in (21), satisfy the following relation:
i
(23) (ξ(cid:48)(v))2+(ξ(cid:48)(v))2cos2(ξ (v))+(ξ(cid:48)(v))sin2(ξ (v))=sin2ν,
1 2 1 3 1
where ν is the angle function of the surface M.
10 F.MANFIOANDJ.P.DOSSANTOS
Proof. With respect to the parametrization F(u,v), given in (19), we have
F =A(cid:48)(v)·b(u)=A(ξ)·A˜(cid:48)(v)·b(u).
v
We have (cid:104)F ,F (cid:105)=sin2ν (cf. [17]). On the other hand, if we denote by c , c , c ,
v v 1 2 3
c the columns of A˜, we have
4
(cid:104)F ,F (cid:105)| =g (cid:104)c(cid:48),c(cid:48)(cid:105)+g (cid:104)c(cid:48),c(cid:48)(cid:105).
v v u=0 11 1 1 33 3 3
As (cid:104)c(cid:48),c(cid:48)(cid:105)=(cid:104)c(cid:48),c(cid:48)(cid:105), (cid:104)c(cid:48),c(cid:48)(cid:105)=0 and g +g =1, a straightforward computation
1 1 3 3 1 3 11 33
gives
sin2ν = (cid:104)F ,F (cid:105)=(g +g )(cid:104)c(cid:48),c(cid:48)(cid:105)
v v 11 33 1 1
= (ξ(cid:48)(v))2+(ξ(cid:48)(v))2cos2(ξ (v))+(ξ(cid:48)(v))sin2(ξ (v)),
1 2 1 3 1
and we conclude the proof. (cid:3)
Theorem 2. Let M be a helicoidal flat surface in S3, locally parametrized by (4),
and whose profile curve γ is given by (6). Then M admits a new local parametriza-
tion such that the fundamental forms are given as in (9) and ω is a linear function.
Proof. ConsidertheunitnormalvectorfieldN associatedtothelocalparametriza-
tion X of M given in (4). From Proposition 3, the angle between N and the Hopf
vector field E is constant. Hence, it follows from [17] (Theorem 3.1) that M can
1
be locally parametrized as in (19). By taking (cid:15) = 1 in (18), we can reparametrize
the curve b given in (20) in such a way that the new curve is a base curve γ . In
a
fact,bytaking(cid:15)=1,weobtainB =1,andsoα =2c andα =2c . Thisimplies
√ 1 2 2 √1
that (cid:107)b(cid:48)(u)(cid:107)=2 c c , because c +c =1. Thus, by writing s=2 c c , the new
1 2 1 2 1 2
parametrization of b is given by
1 (cid:16) s s (cid:17)
b(s)= √ acos ,asin ,cos(as),sin(as) ,
1+a2 a a
(cid:112)
where a= c /c . On the other hand, we have
1 2
A(v)·b(s)=A(ξ)X(v,s),
where X(v,s) can be written as
1
X(v,s)= √ (x ,x ,x ,x ),
1 2 3 4
1+a2
with
(cid:16)s (cid:17)
x = acosξ cos +ξ +sinξ cos(as+ξ ),
1 1 a 2 1 3
(cid:16)s (cid:17)
x = acosξ sin +ξ +sinξ sin(as+ξ ),
(24) 2 1 a(cid:16)s 2 (cid:17) 1 3
x = −asinξ cos −ξ +cosξ cos(as−ξ ),
3 1 a 3 1 2
(cid:16)s (cid:17)
x = −asinξ sin −ξ +cosξ sin(as−ξ ).
4 1 a 3 1 2
On the other hand, the product φ (t)·X(v,s) can be written as
α,β
1
φ (t)·X(v,s)= √ (z ,z ,z ,z ),
α,β 1 2 3 4
1+a2
where
(cid:16)s (cid:17)
z = acosξ cos +ξ +αt +sinξ cos(as+ξ +αt),
1 1 a 2 1 3
(cid:16)s (cid:17)
z = acosξ sin +ξ +αt +sinξ sin(as+ξ +αt),
(25) 2 1 a(cid:16)s 2 (cid:17) 1 3
z = −asinξ cos −ξ +βt +cosξ cos(as−ξ +βt),
3 1 a 3 1 2
(cid:16)s (cid:17)
z = −asinξ sin −ξ +βt +cosξ sin(as−ξ +βt).
4 1 a 3 1 2
