SURFACES OF CONSTANT CURVATURE IN 3-DIMENSIONAL SPACE FORMS

SURFACES OF CONSTANT CURVATURE IN 3-DIMENSIONAL SPACE FORMS

Jos´e A. G´alvez

1 Introduction.

The study of surfaces immersed in a 3-dimensional ambient space plays a central role in the theory of submanifolds. In addition, Riemannian manifolds with constant sectional curvature can be considered as the most simple examples. Thus, one can think of surfaces with constant Gauss curvature in the Euclidean space

R

³, hyperbolic space

H

³or 3-sphere

S

³as very natural objects of study.

Through these notes we will study some classical results on complete surfaces with constant Gauss curvature in 3-dimensional space forms, using a modern approach.

The notes are organized as follows.

In Section 2 we establish some notation and recall some necessary preliminary concepts. In Section 3 we center our attention on the complete surfaces with positive constant Gauss curvature in

S

^3,

R

^{3 or}

H

³. We shall prove the Liebmann theorem and show that the only complete examples must be totally umbilical round spheres. In particular, there is no complete surface in

S

³with constant Gauss curvatureK(I)∈ (0,1).

In Section 4 we prove the existence of a conformal representation for flat surfaces in the hyperbolic 3-space. We shall associate to each surface a pair of holomorphic 1-forms and see that the flat surface can be recovered in terms of this holomorphic pair. We prove that the two hyperbolic Gauss maps of a flat surface are holomorphic with respect to the conformal structure given by the second fundamental form, and classify the complete flat surfaces.

Section 5 is devoted to the Hartman-Nirenberg theorem for flat surfaces in the

The author is partially supported by MCYT-FEDER, Grant No MTM2007-65249 and Grupo de Exce- lencia P06-FQM-01642 Junta de Andaluc´ıa.

Euclidean 3-space. We shall show that a complete flat surface in

R

³must be a right cylinder on a planar curve which is defined for all values of its arc length parameter.

In Section 6 we shall study the Bianchi-Spivak representation of a flat surface in

S

³. We start showing the existence of global Tschebyscheff coordinates in a complete flat surface. We prove that the asymptotic curves of a flat surface have torsion±1 when the curvature of the curves does not vanish. Moreover, these asymptotic curves are characterized in a simple way which generalizes the torsion±1concept. We recall the quaternionic model of

S

³and show that any complete flat surface Σin

S

^{3 can}

be recovered by multiplication of the two asymptotic curves passing across a point p∈Σ. We also prove the converse of this result in terms of curves with generalized torsion±1. Finally, we note that every asymptotic curve of a flat torus is closed. Thus, flat tori can be classified.

In Section 7 we prove the Hilbert theorem. Thus, we show that there is no complete surface with constant negative Gauss curvatureK(I)in

R

^3or

S

³. The same happens for complete surfaces in

H

^3whenK(I)<−1. There exist complete surfaces in

H

³

with constant Gauss curvatureK(I)∈[−1,0[.

2 Some preliminary concepts.

Let us denote by

M

^3(c)to the simply-connected Riemannian 3-space with constant sectional curvaturec=−1,0,1. That is,

M

^3(c)denotes to the hyperbolic space

H

³

ifc=−1, to the Euclidean space

R

^{3whenc= 0}or to the sphere

S

^{3ifc= 1.}

Now, consider an oriented Riemannian surfaceΣwith induced metricI=h,iand f : Σ−→

M

^3(c)an isometric immersion. LetNbe a unit normal alongΣwhich is compatible with the orientation, andIIits associated second fundamental form. That is,

II(X, Y) =h−∇XN, Yi, X, Y ∈X(Σ),

whereX(Σ) denotes to the set of smooth vector fields inΣ, and∇ the Levi-Civita connection of the ambient space.

Ifk1, k2are the principal curvatures of the immersion, namely, the eigenvalues of

the shape operatorS, given bySX =−∇_XN, then we shall denote by K=k₁k₂, H =k₁+k₂

2

to the extrinsic curvature and to the mean curvature of the immersion, respectively.

These two quantities are extrinsic and depend on howSis immersed into

M

^3(c).

On the other hand, we denote byK(I)to the curvature of the metricI, namely, the Gauss curvature, which is an intrinsic quantity.

For the previous immersionf : Σ −→

M

^3(c)the following relations must be satisfied

Gauss equation: K(I) =K+c,

Mainardi-Codazzi equation: ∇XSY − ∇YSX−S[X, Y] = 0, X, Y ∈X(Σ), where∇is the Levi-Civita connection of the metricI.

In fact, Bonnet’s theorem asserts that given a simply-connected Riemannian sur- face Σe with induced metric I and a self-adjoint endomorphism S, the Gauss and Mainardi-Codazzi equations are sufficient for the existence of an immersionfe:Σe−→

M

³(c), whose induced metric isIand whose shape operator isS. Moreover, that im- mersion is unique up to rigid motions.

It is a remarkable fact that the Mainardi-Codazzi equation does not depend onc.

Hence, it agrees for the ambient spaces

H

^3,

R

^3and

S

^3.

Moreover, this equation appears in different contexts, which motivates the study of the equation from an abstract point of view, since every result obtained from the abstract setting will be applied in different situations. Thus, we introduce the concept ofCodazzi pair:

Codazzi pairs.

We start by defining the framework where the extrinsic curvature and mean curvature can be defined.

Definition 1. A fundamental pair on an oriented surfaceΣis a pair of real quadratic forms(I, II)onΣ, whereIis a Riemannian metric.

Associated with a fundamental pair(I, II)we define the shape operatorSof the pair as

II(X, Y) =I(S(X), Y) (1)

for any vector fieldsX, Y onΣ.

Conversely, from (1) it becomes clear that the quadratic formII is totally deter- mined byIandS. In other words, to give a fundamental pair onΣis equivalent to give a Riemannian metric onΣand a self-adjoint endomorphismS.

We define themean curvature, theextrinsic curvatureand theprincipal curvatures of(I, II)as half the trace, the determinant and the eigenvalues of the endomorphism S, respectively.

In particular, given local parameters(x, y)onΣsuch that

I=E dx²+ 2F dxdy+G dy², II=e dx²+ 2f dxdy+g dy²,

the mean curvature and the extrinsic curvature of the pair are given, respectively, by H =H(I, II) =Eg+Ge−2F f

2(EG−F²) , K=K(I, II) = eg−f² EG−F². Moreover, the principal curvatures of the pair areH±√

H²−K.

We shall say that the pair(I, II)isumbilicalatp∈ΣifIIis proportional toIat p, or equivalently:

• if both principal curvatures coincide atp, or

• ifSis proportional to the identity map on the tangent plane atp, or

• ifH²−K= 0atp.

We define theHopf differentialof the fundamental pair(I, II)as the (2,0)-part of II for the Riemannian metricI. That is, if we considerΣas a Riemann surface with respect to the metricIand take a local conformal parameterz, then

I= 2λ|dz|²

II =Q dz²+ 2λ H|dz|²+Q dz². (2)

The quadratic formQ dz², which does not depend on the chosen parameter, is known as the Hopf differential of the pair(I, II). We note that(I, II)is umbilical atp∈Σ if, and only if,Q(p) = 0.

All the above definitions can be understood as natural extensions of the corre- sponding ones for isometric immersions of a Riemann surface in a 3-dimensional am- bient space, whereIplays the role of the induced metric andIIthe role of its second fundamental form.

A specially interesting case for our study happens when the fundamental pair sat- isfies, in an abstract way, the Mainardi-Codazzi equation for surfaces in

M

^3(c),

Definition 2. We say that a fundamental pair(I, II), with shape operatorS, is a Codazzipair if

∇XSY − ∇YSX−S[X, Y] = 0, X, Y ∈X(Σ), (3) where∇stands for the Levi-Civita connection associated with the Riemannian metric I.

A first important remark is the following: given a fundamental pair(I, II)and a conformal parameterzforIthen, from (2), a straightforward computation gives that (I, II)is a Codazzi pair if and only if

Qz=λHz. (4)

In particular, this means the Hopf differentialQdz²is holomorphic if and only if the mean curvature is constant.

We observe that the existence of a holomorphic quadratic form for a class of sur- faces is a very useful tool, which can be used in order to obtain geometric properties about the behavior of those surfaces. As an example, we get

Theorem 1(Hopf). Let(I, II)be a Codazzi pair of constant mean curvatureHon a topological sphereΣ. Then(I, II)is totally umbilical.

When this result is used for immersions into

M

^3(c)asserts that a topological sphere of constant mean curvature in

M

^3(c)must be necessarily a round sphere [17].

Proof of Theorem 1. If the pair(I, II)has constant mean curvature thenQ_z ≡0.

The only holomorphic quadratic form on a sphere vanishes identically. ThusQdz²≡ 0and(I, II)is totally umbilical.

2 We shall apply the same technique in the next Section in order to prove the Lieb- mann theorem.

Finally, it should be observed that U. Abresch and H. Rosenberg have extended the Hopf theorem for surfaces immersed in 3-dimensional homogeneous manifolds with a 4-dimensional isometry group. They obtain a quadratic holomorphic differen- tial which allows them to characterize the spheres of constant mean curvature as the revolution examples [1, 2]. For the 3-dimensional homogeneous ambient spaceSol₃, a recent result has been obtained by B. Daniel and P. Mira [10].

3 Surfaces with positive constant Gauss curvature.

Let us start with a surfaceΣendowed with a complete Riemannian metricIof con- stant Gauss curvatureK(I) =k >0. Thus, ifΣis simply-connected thenΣis iso- metric to the standard sphere

S

^{2(r)of radiusr= 1/}

√

k, from the Cartan-Hadamard theorem.

The following theorem asserts how a complete Riemannian surface with positive constant Gauss curvature can be immersed in

H

^3,

R

^3or

S

^3.

Theorem 2 (Liebmann). Let Σbe a surface and f : Σ −→

M

^{3(c) a complete}

immersion with positive constant Gauss curvature. Thenf(Σ)is a totally umbilical round sphere.

Observe that, from the Gauss equation, a surface with constant Gauss curvature must also have constant extrinsic curvature. In

R

³ both curvatures agree, and they differ by a constant in

H

^3and

S

³. Moreover, if the Gauss curvature is positive then the extrinsic curvature of the surface is also positive in

H

^3and

R

³. However, if the Gauss curvatureK(I)is positive in

S

³then the extrinsic curvature is only positive if K(I)>1.

Bearing in mind these comments, the Liebmann theorem for

M

^{3(c)(withK(I)>}

1in

S

³) will be a consequence of the following result [23].

Theorem 3. Let (I, II) be a Codazzi pair on a surfaceΣ with positive constant extrinsic curvature. Then the (2,0)-part ofIwith respect to the Riemannian metricII is a holomorphic quadratic form.

In particular, ifΣis a topological sphere then the pair(I, II)is totally umbilical.

Proof.SinceK=k1k2>0, we can orientateΣsuch thatk1andk2are positive. In particular, with this orientationIIis a Riemannian metric.

Thus, if we take a local conformal parameterzfor the Riemannian metricII, then we can write

I=P dz²+ 2λ|dz|²+P dz², II = 2ρ|dz|².

Now, let us denote byΓ^k_ij to the Christoffel symbols of the Levi-Civita connection ofIwith respect to the parameterz. That is,

∇∂

∂z

∂

∂z = Γ¹₁₁∂

∂z+ Γ²₁₁∂

∂z, ∇∂

∂z

∂

∂z = Γ¹₁₂∂

∂z+ Γ¹₁₂∂

∂z. If we denote byD=λ²− |P|², a direct computation gives

Dz

2D = Γ¹₁₁+ Γ¹₁₂. The Codazzi equation for the parameterzgives

ρ_z

ρ = Γ¹₁₁−Γ¹₁₂.

AndK=ρ²/Dis constant, one has2ρ_z/ρ=D_z/D. Or equivalently,Γ¹₁₂ = 0.

Therefore,

P_z= ∂

∂zI ∂

∂z, ∂

∂z

= 0,

as we wanted to show.

Since a holomorphic 2-form in a sphereΣmust vanish identically thenP dz²≡0 inΣ. Hence,(I, II)must be totally umbilical.

2 In order to finish the proof of the Liebmann theorem, it is necessary to study the case of an immersionf : Σ−→

S

³with constant Gauss curvatureK(I) =k∈(0,1].

Whenk ∈ (0,1) then the extrinsic curvature is negative. Thus, the principal curvatures satisfyk₁k₂<0, andk₁6=k₂onΣ. But this fact contradicts the Poincar´e index theorem, since the universal cover ofΣis a sphere.

The casek= 1depends more strongly on the Gauss equation. For that, we argue as follows: Let p be a non-umbilical point on an immersed sphereΣ of constant Gauss curvature1in

S

³. Then, let us take doubly orthogonal parameters(u, v)in a neighborhood ofp, that is,

I=Edu²+Gdv², II =k2Gdv².

Here, one principal curvature must vanish since, from the Gauss equation, the extrinsic curvature also vanishes. Andk2is different from 0.

The Codazzi equation yieldsE_v= 0and(k₂²G)_u= 0.

Thus, takingeuwithdeu=p

E(u)du, if necessary, we can assumeE ≡1. And analogously, from the equation(k²₂G)_u= 0we have

I=du²+ 1

k₂²dv², II= 1 k2

dv². Therefore, the Gauss curvature takes the form

1 =−|k2| 1

|k2|

uu

. (5)

SinceΣis compact, eitherΣis totally umbilical or there exists a non-umbilical point wherek2has a maximum or minimum. But the latter case is not possible since every solution of the real equationh⁰⁰(t) +h(t) = 0has no positive local minimum, and 1/|k2|would have a positive local minimum, which contradicts (5).

Therefore,Σis a totally umbilical sphere, which finishes Liebmann theorem.

2

The techniques of Codazzi pairs in Theorem 3 have been used in the ambient spaces

H

^2×

R

^and

S

^2×

R

in order to obtain a Liebmann type theorem [3] (see also [8]).

There are some interesting open problems related with the existence of surfaces with positive Gauss curvature:

IfΣis a surface with positive Gauss curvature in

R

³ and boundary lying on a planeP then it is easy to see that∂Σis a convex Jordan curve. Moreover, given a convex Jordan curveΓlying on a planeP, there exists a constantk >0such that for allk⁰ ∈(0, k]there is a surfaceΣwith Gauss curvaturek⁰whose boundary isΓ(see for instance [15]).

An open problem is to find the best value of k in terms of the curveΓ. It is known that some restrictions must be satisfied. For instance,kandΓmust satisfy the following inequality

k≤ π AΓ

,

whereA_Γis the planar area enclosed byΓ[11]. Moreover, the equality in the inequal- ity above is only possible whenΓis a circle andΣis a hemisphere.

On the other hand, given a convex planar Jordan curveΓit is not known if fixed a constantk >0there exist exactly two non-isometric surfaces with Gauss curvaturek and boundaryΓ.

A specially interesting problem is to find the closed surfaces in

R

³with constant Gauss curvature and with a small number of singularities, that is, with a finite number of isolated singularities. As we have proved, if there is no singularity then the surface must be a round sphere. It is not difficult to prove that there is no closed surface with only one singularity, by using Alexandrov reflection principle. And revolution surfaces are the only surfaces with two singularities. However, the existence of these surfaces for a finite number of singularities greater than two is not known.

4 Flat surfaces in the hyperbolic 3-space.

In the Lorentz-Minkowski model,

L

⁴, the hyperbolic 3-space is described by one connected component of a two-sheeted hyperboloid. More precisely,

H

³⁼

n

(x0, x1, x2, x3)∈

R

^{4: −x2}0+x²₁+x²₂+x²₃=−1, x0>0o (6) endowed with the induced metric of

L

⁴given by the quadratic form−x²₀+x²₁+x²₂+ x²₃.

Let

N

³+=n

(x₀, x₁, x₂, x₃)∈

R

^{4: −x2}0+x²₁+x²₂+x²₃= 0, x₀>0o

be the positive null cone in

L

⁴. And consider the following relation in

N

³⁺

x, y∈

N

³+are identified if and only if there existsλ >0 : x=λy.

With this identification

N

³+/

R

⁺ is a topological sphere. In fact

N

³+/

R

^{+ inherits a}

natural conformal structure and can be regarded as the ideal boundary

S

²∞of

H

^3.

We shall regard

L

⁴ as the space of2×2 Hermitian matrices, Herm(2), in the standard way, by identifying(x0, x1, x2, x3)∈

L

⁴with the matrix

x0+x3 x1+ix2

x1−ix2 x0−x3

. (7)

Under this identification the inner product of

L

⁴can be recovered from the equal- ityhm, mi =−det(m)for allm ∈Herm(2). Then

H

³ is the set ofm∈ Herm(2) withdet(m) = 1. The action of

SL

^(2,

C

⁾on these Hermitian matrices defined by

g·m=gmg^∗,

whereg ∈

SL

^(2,

C

^{),g∗ = t}g, preserves the inner product, leaves

H

³invariant and is transitive. The kernel of this action is {±I2} ⊆

SL

^(2,

C

^)and

PSL

^(2,

C

^{) =}

SL

^(2,

C

^)/{±I2}can be regarded as the identity component of the special Lorentzian group

SO

^(1,3).

The space

N

³+is seen as the space of positive semi-definite2×2Hermitian matri- ces of determinant0and its elements can be written asw^tw, where^tw= (w1, w2)is

a non zero vector in

C

²uniquely defined up to multiplication by a unimodular com- plex number. The mapw^tw−→[(w₁, w₂)]∈

CP

¹becomes the quotient map of

N

³+

on

S

²∞and identifies

S

²∞to

CP

¹. Thus, the natural action of

SL

^(2,

C

^)on

S

²∞is the action of

SL

^(2,

C

^)on

CP

¹by M¨obius transformations.

Now, letΣbe a surface andψ: Σ−→

H

³an immersion with Gauss mapη. Then, the image of the mapψ+ηlies in

N

³+, sincehψ, ψi=−1,hψ, ηi= 0andhη, ηi= 1.

Therefore, we can define the (positive) hyperbolic Gauss map of the immersion as G⁺= [ψ+η] : Σ−→

S

²∞≡

N

³+/

R

^+.

Analogously, the negative hyperbolic Gauss map can be defined asG⁻= [ψ−η] : Σ −→

S

²∞ ≡

N

³+/

R

^+. It is important to observe that in general both hyperbolic Gauss maps have nothing to do, that is, they are not related.

The geometric interpretation of the previous definitions is the following (See Fig- ure 1). Letp∈ Σandγ(t)the oriented geodesic in

H

^{3such thatγ(0) = ψ(p)and}

γ⁰(0) = η(p). ThenG⁺(p)(analog. G⁻(p)) can be seen as the intersection ofγ(t) and the ideal boundary

S

²∞whent→ ∞(analog. whent→ −∞).

Figure 1:G⁺(p)andG⁻(p)in the Poincar´e model of

H

^3.

Let us start with a flat surfaceΣin

H

³, that is, with intrinsic curvatureK(I) = 0.

From the Gauss equation the product of its principal curvaturesk1k2 = 1. Thus, we can choose a unit normal inΣsuch thatk1 andk2are positive, or equivalently, the second fundamental form is a Riemannian metric. We shall always fix that normal throughout this section.

Proposition 1. LetΣbe a surface andψ : Σ −→

H

³ a flat immersion with unit normalη. If we considerΣas a Riemann surface with the conformal structure induced by II, thenψ+η, ψ −η : Σ −→

N

³+ are conformal maps . In particular, the hyperbolic Gauss mapsG⁺, G⁻ : Σ−→

S

²∞are conformal forII.

Proof. Letp∈Σand{e1, e2}an orthonormal basis atpsuch that−dη(e1) =k1e1

and−dη(e2) =k₂e₂. Then atp,

II(e₁, e₁) =k₁ hd(ψ+η)(e₁), d(ψ+η)(e₁)i= (1−k₁)² II(e1, e2) = 0 hd(ψ+η)(e1), d(ψ+η)(e2)i= 0 II(e2, e2) =k2 hd(ψ+η)(e2), d(ψ+η)(e2)i= (1−k2)² Therefore,ψ+ηis conformal atpif and only if

(1−k1)² k1

= (1−k2)² k2

.

Or equivalently,(k₁k₂−1)(k₁−k₂) = 0; which is satisfied becausek₁k₂= 1inΣ.

Analogously, it can be proved thatψ−ηis conformal.

2 Now, let us start with a simply connected surface Σand a flat immersion ψ : Σ−→

H

³with unit normalη. Then, from Proposition 1,G⁺: Σ−→

S

²∞≡

CP

^1is

conformal and, so, there exist holomorphic functionsA, Bsuch thatG⁺= [ψ+η] = [(A, B)]. Then,

ψ+η=R1

A B

A, B

=R1

AA AB

AB BB

,

for some positive regular functionR1.

Analogously, sinceG⁻ : Σ −→

S

²∞ ≡

CP

¹is conformal, there exist holomor- phic functionsC, Dsuch thatG⁻= [ψ−η] = [(C, D)]. Then,

ψ−η=R₂ C

D

C, D

=R₂

CC CD

CD DD

,

for some positive regular functionR2. By taking

g=

A C

B D

we have

ψ+η=g

R₁ 0 0 0

g^∗, ψ−η=g

0 0 0 R₂

g^∗. (8) So,

ψ=1 2g

R1 0 0 R2

g^∗, η= 1 2g

R1 0 0 −R2

g^∗. (9) Since,hψ, ψi=−1we have

1 = 1 4det

g

R1 0 0 R2

g^∗

= 1

4R1R2|det(g)|². (10) As|det(g)| 6= 0anddet(g)is holomorphic, there exists a well defined holomor- phic functionh1 inΣsuch thath1 =p

det(g). Thus, by substituting the previous A, B, C, DbyA/h₁, B/h₁, C/h₁, D/h₁, respectively, one hasdetg= 1. Moreover, R1R2= 4from (10), that is,R2= 4/R1.

With this g⁻¹g_z=

D −C

−B A

A⁰ C⁰ B⁰ D⁰

=:

a11 a12

a21 −a11

, (11)

and so

(ψ+η)z=g

a11 a12

a₂₁ −a11

R1 0 0 0

+

(R1)z 0

0 0

g^∗. (12) By usingh(ψ+η)z, ψ−ηi= 0one has from (8) and (12)

a11+(R1)z

R1

= 0. (13)

sinceR2= 4/R1.

Thus,h₂= log(R₁)is a harmonic function. That is, there exists a non vanishing holomorphic functionh3inΣsuch thatR1=e^h2 = 2|h3|².

Therefore, replacing againA, B, C, DbyAh₃, Bh₃, C/h₃, D/h₃, one has from (9)

ψ=gg^∗, η=g

1 0 0 −1

g^∗. (14)

Moreover, from (13), the new holomorphic functiona₁₁ given by (11) vanishes identically. That is, we can write

g⁻¹gz=

0 a₁₂ a₂₁ 0

.

With all of this, we have found a holomorphic mapg: Σ−→

SL

^(2,

C

^{)given by}

g=

A C

B D

(15) such that the immersionψand its unit normalηcan be recovered by (14). Moreover, the hyperbolic Gauss maps are given by

G⁺= A

B ∈

S

²∞, G⁻= C

D ∈

S

²∞ (16)

andgsatisfies

g⁻¹dg=

0 ω θ 0

, whereω, θare holomorphic 1-forms inΣ.

Let us suppose there exists another holomorphic mapg0: Σ−→

SL

^(2,

C

^)such

that

ψ=g₀g₀^∗, η=g₀

1 0 0 −1

g₀^∗.

Then, from (14), gg^∗ = g₀g₀^∗ and so g⁻¹g₀(g⁻¹g₀)^∗ = I₂, whereI₂ denotes the identity matrix. Thus,

g⁻¹g0=

p q

−q p

, (17)

for certain complex functionsp, qsuch that|p|²+|q|²= 1.

Sinceg, g0are holomorphic thenp, qmust be constant. In addition, as g

1 0 0 −1

g^∗=g₀

1 0 0 −1

g^∗₀

one has thatqmust vanish identically. Hence, from (17), g0=g

e^iϕ 0 0 e^−iϕ

,

for a real constantϕ.

In particular,g₀also satisfies (16) and g⁻¹₀ dg0=

0 e^−2iϕω e^2iϕθ 0

. (18)

Thus, we obtain the following representation of flat immersions in terms of holo- morphic data [12].

Theorem 4(Conformal representation).

(i) LetΣbe a simply connected surface andψ: Σ−→

H

³a flat immersion with unit normal η. Consider Σas a Riemann surface with the conformal structure induced by its second fundamental form. Then, there exists a holomorphic mapg : Σ −→

SL

^(2,

C

⁾and two holomorphic 1-formsω, θwith|ω|>|θ|such that ψ=gg^∗, η=g

1 0 0 −1

g^∗, (19)

and

g⁻¹dg=

0 ω θ 0

. (20)

Besides, its hyperbolic Gauss maps are given as in (15) and (16).

Ifg0 : Σ −→

SL

^(2,

C

⁾is another holomorphic curve in the conditions above then

g0=g

e^iϕ 0 0 e^−iϕ

, for a real constantϕ.

Moreover, its induced metric and its second fundamental form are given by I=ωθ+ (|ω|²+|θ|²) +ωθ, II =|ω|²− |θ|². (21) (ii) Conversely, letΣbe a Riemann surface andg: Σ−→

SL

^(2,

C

⁾a holomorphic map such that

g⁻¹dg=

0 ω θ 0

,

for certain 1-formsω, θwith|ω|>|θ|. Then,ψ=gg^∗is a flat immersion in

H

^3with

unit normal vector fieldηgiven by (19). Moreover, its induced metric and its second fundamental form are given by (21).

Proof. In order to prove (i), we need to compute the induced metric and second fundamental form of the immersion.

Thus, if we take a local conformal parameterzand we writeω=ω₁dz, θ=θ₁dz, then from (19) and (20)

ψz=g

0 ω₁ θ₁ 0

g^∗, ηz=g

0 ω₁ θ₁ 0

1 0 0 −1

g^∗.

Hence, one obtains

hψz, ψzi=ω1θ1, hψz, ψzi=1

2(|ω1|²+|θ1|²), hψz,−ηzi= 0, hψz,−ηzi= 1

2(|ω1|²− |θ1|²).

Thus (21) is proved, and sinceIIis positive definite|ω|>|θ|.

The converse is a straightforward computation.

2

The pair (ω, θ) given by the theorem above is usually called theWeierstrass data associated with the flat immersion. As it was proved in (18), they are unique up to multiplication by a unit complex numberpin the following way (p ω, p θ).

In particular,ωθ,|ω|and|θ|are uniquely defined.

Remark 1. Since |ω| > 0one obtains thatG⁻ is a local diffeomorphism because (G⁻)⁰(z) 6= 0at every point. This was first observed by L. Bianchi [5]. In fact, he proved that the mapG⁻ →G⁺is holomorphic and gave a representation of flat immersions using Euclidean congruence of spheres.

Using the conformal representation above, we give a new proof of the classifica- tion of the complete flat surfaces in

H

³, obtained by Volkov and Vladimirova [31].

Theorem 5. LetΣbe a surface andψ: Σ−→

H

³a complete flat immersion. Then ψ(Σ)is either a horosphere or the set of points at a fixed distance from a geodesic.

Proof. Passing to the universal cover ofΣ, if necessary, we can assume thatΣis simply connected. Hence, with the conformal structure determined by the second

fundamental form,Σmust be biholomorphic either to the unit disk

D

or to the complex plane

C

^.

If(ω, θ)is the Weierstrass data associated with the immersion, then from (21) and since|ω|>|θ|

I≤4|ω|².

Thus,4|ω|²is a conformal complete flat metric inΣ. Thus,Σwith this metric must be isometric to the Euclidean plane. In particular,Σis conformal to

C

^.

On the other hand, the modulus of the holomorphic functionθ/ωis less than one inΣ≡

C

^{. Hence,θ/ω}is a complex constantc0, with|c0|<1.

By using|ω| 6= 0, we can take a local conformal parameterzsuch thatω =dz.

Thus, from (20), we must find a holomorphic immersion g=

A C

B D

∈

SL

^(2,

C

⁾

satisfying

g⁻¹gz=

0 1 c₀ 0

. Or equivalently,

A⁰ C⁰ B⁰ D⁰

=

c0C A c0D B

.

Thereby,A, B, C, Dare solutions of the differential equationX⁰⁰=c0X.

Ifc0 = 0the solutions of the previous ODE are X(z) = a+bz for suitable constantsa, b∈

C

. In such a case

g=m0

1 z 0 1

,

wherem0is a constant matrix in

SL

^(2,

C

^).

Therefore, the flat immersion is, up to the isometrym₀ψm^∗₀, given by ψ(z) =

1 z 0 1

1 0 z 1

=

1 +|z|² z

z 1

.

Since the last coordinate of the matrix is constant one has from (7),ψ0+ψ3= 1. That is,ψ(Σ)lies on a horosphere, andψ(Σ) is the whole horosphere by completeness.

(See Figure 2.)

Figure 2: A horosphere in the half-space model of

H

^3.

Ifc0 6= 0the solutions of the previous ODE areX(z) = ae^√c0z+be^−√c0zfor suitable constantsa, b∈

C

. In such a case

g=m₀ p4 c₀

4 e^{−√c0z √}₄⁻¹

4c0e^−√c0z p4 c0

4 e^√c0z √4¹

4c₀e^√c0z

! ,

wherem₀is a constant matrix in

SL

^(2,

C

^).

Then, up to an isometry, the flat immersion is given by

ψ(z) = 1 2|c₁|

(|c1|²+ 1)e^−c1z−c1z (|c1|²−1)e^−c1z+c1z (|c1|²−1)e^c1z−c1z (|c1|²+ 1)e^c1z+c1z

wherec1=√ c0.

The surface, given by the points whose distance to the geodesic e^s 0

0 e^−s

is a constantR >0, can be parametrized by

F(s, t) =

coshR e^s sinhR e^it sinhR e^−it coshR e^−s

.

Thus,ψ(z)lies on the surfaceF(s, t)forR = arcsinh_1−|c

1|² 2|c1|

as we wanted to show. (See Figure 3.)

Figure 3: A surface equidistant from a geodesic in the half-space model of

H

^3.

2

The conformal representation is a useful tool and can be used for investigating the geometric behavior of flat immersions. For instance, it was used for proving that a properly embedded annular end must be asymptotic to a revolution surface (see [12]

and [6]).

Since the complete flat immersions in

H

³are well known, it is interesting to pose the study of flat surfaces with singularities. These “surfaces” are being intensively studied by different authors (see [6], [13], [24], [25], [26], [28]). To this respect, and bearing in mind that a flat surface is locally convex in

H

³, it seems of special interest to research the complete surfacesΣin

H

³such that they are the boundary of a convex body with a small number of singularities. This problem is still open (see [13], [6]).

5 Flat surfaces in the Euclidean space.

We start with a flat immersionψ : Σ−→

R

³. From the Gauss equation the product of the principal curvatures of the immersion vanishes identically. We consider two subsets of Σgiven by the interior of the umbilical points, Σ_U, and the set of non umbilical pointsΣN.

It is clear that the closure ofΣU ∪ΣN is the whole surface. Thus, we study both

subsets independently in order to understandψ(Σ).

The setΣ_Uis made of umbilical points with vanishing principal curvatures. Hence, every connected component ofψ(ΣU)lies on a plane.

Thus, we will focus our attention on the study ofΣN.

Lemma 1. LetΥ⊆ΣN be a line of curvature corresponding to the vanishing prin- cipal curvature, thenψ(Υ)is a straight segment in

R

³with no umbilical point.

Moreover, ifψis a complete immersion andΥis a line of curvature associated to the principal curvature0with a non umbilical pointp0 ∈Υ, thenψ(Υ)is a straight line of

R

³with no umbilical point.

Proof.SinceΥ⊆Σ_N there exist local coordinates(u, v)such that I=Edu²+Gdv², II =k₂Gdv².

Here, the coordinate curves are the lines of curvature of the immersion.

The Codazzi equation givesE_v = 0and(k²₂G)_u = 0. Thus, up to a change of parameters as we did in Section 3, we can assume

I=du²+ 1

k²₂dv², II= 1 k₂dv².

The vectorψ_uuhas no normal part sincek₁≡0. Thus,ψ_uu = Γ¹₁₁ψ_u+ Γ²₁₁ψ_v, and from the expression ofIone hasΓ¹₁₁= 0 = Γ²₁₁.

Therefore the curvesα(u) = ψ(u, v0)which are parametrized by the length arc satisfyα⁰⁰(u) = 0. That is,α(u)is a segment of line in

R

^3.

The Gauss equation yields

1 k2

uu

= 0. (22)

That is, for the segmentα(u)one has that the second principal curvaturek2satisfies the equation (22) for the arc length parameter of the curveu. Thus, if there exists a first umbilical point whenu→u₀thenk₂must tend to0, which contradicts (22).

Therefore,α(u)has no umbilical point. In particular, ifψis a complete immersion α(u)must be a straight line.

2

It is important to observe the following fact from the lemma above. Ifψ: Σ−→

R

³is a complete flat immersion andp1, p2are non umbilical points with respective straight linesr₁, r₂associated with the vanishing principal curvature thenr₁=r₂or they do not intersect. Otherwise, there would exist a unique pointp∈r1∩r2satisfying that two different directions (associated withr₁andr₂) are principal directions with vanishing principal curvatures. Thenpwould be umbilical, which contradicts thatr1

has no umbilical point.

With all of this, we can classify the complete flat surfaces in

R

^3[16].

Theorem 6(Hartman-Nirenberg). Letψ : Σ−→

R

³be a complete flat immersion.

Thenψ(Σ)is a right cylinder on a planar curve which is defined for all values of its arc length parameter.

Proof. Passing to the universal cover ofΣ, if necessary, we can assume thatΣis simply connected. So,Σis a simply connected flat surface and therebyΣis isometric to the Euclidean plane

R

². Hence, we shall work with

R

^{2instead ofΣ.}

Ifψis totally umbilical then the image is a plane. Thus, we shall assume there is a non umbilical point.

Letp0 ∈

R

²be a non umbilical point. By using Lemma 1 there exists a line of curvatureΥ₀ ⊆

R

^{2such thatψ(Υ0)}is a straight line. Sinceψ(Υ₀)is a geodesic of

R

^3thenΥ0is a geodesic of the surface andΥ0is a straight line in

R

^2.

Up to an isometry in

R

²we can assume thatΥ₀is a vertical line. Then, let us show that the image of every vertical liner⊆

R

²is a straight line in

R

^3.

Letr⊆

R

²be a vertical line different fromΥ0containing a non umbilical point p1. LetΥ1⊆

R

²be its associated line of curvature such thatψ(Υ1)is a straight line.

Reasoning as aboveΥ₁must be a straight line in

R

². In addition,Υ₁must be vertical because otherwise would intersectΥ0which contradicts the previous remark. Hence,

r= Υ₁and its image is a straight line.

Now, letr⊆

R

²be a vertical line contained in the interior of the set of umbilical pointsΣU ⊆

R

^{2. Sincer}is a geodesic in

R

^2thenψ(r)is a geodesic inψ(ΣU). And sinceψ(Σ_U)is a piece of a plane thenψ(r)is a straight line.

Ifr ⊆

R

²is a vertical line which is not in the previous conditions thenris the limit of vertical linesrnin the conditions above. And sinceψ(rn)is a straight line for anynthenψ(r)must also be a straight line.

With all of this, if we consider usual coordinates(x, y)in

R

²the immersionψis parametrized as

ψ(x, y) =α(x) +yβ(x).

Here,ψ(x0, y)is a straight line with unit vectorβ(x0).

Thus,

1 =hψ_x, ψ_xi=hα⁰(x), α⁰(x))i+ 2yhα⁰(x), β⁰(x)i+y²hβ⁰(x), β⁰(x)i, 0 =hψx, ψyi=hα⁰(x), β(x)i+yhβ⁰(x), β(x)i,

1 =hψy, ψyi=hβ(x), β(x)i.

In particularhβ⁰(x), β⁰(x)i = 0andβ(x)is a constant unit vectorβ₀. That is, the image of vertical lines in

R

²are parallel straight lines in

R

^3.

Moreoverhα⁰(x), β0i= 0and by integratinghα(x)−α(x0), β0i= 0. This means α(x)is contained in a plane perpendicular to the vectorβ₀.

2

6 Flat surfaces in the 3-sphere.

This section is devoted to expose the basics of the theory of flat surfaces in the unit3- sphere

S

³. We shall begin by describing the fundamental equations of flat surfaces in terms of asymptotic parameters. Then we shall describe

S

³as well as the usual Hopf fibration in terms of quaternions. By means of this model for

S

³, we shall explain the classical Bianchi method via which flat surfaces in

S

³are constructed by multiplying two intersecting asymptotic curves. We shall also describe a refinement of this method due to Kitagawa [18], which has been the fundamental tool for studying flat surfaces

in

S

³ from a global viewpoint. Afterwards, we shall expose the most significative global results regarding complete flat surfaces and flat tori in

S

³. Finally, we shall discuss some of the most important open problems of the theory. The basic references for most of what follows are [14, 18, 30, 33].

6.1 Asymptotic parameters

Generally, the fundamental equations of a flat surface in

S

³are better understood by means of parameters whose coordinate curves are asymptotic curves on the surface.

First, let us observe that, as the surface is flat, its intrinsic Gauss curvature vanishes identically. Consequently, by the Gauss equation, the extrinsic curvature of the surface isK = −1. In this situation, asKis a negative constant, there existTschebyscheff coordinatesaround every point. This simply means that we can choose local coordi- nates(u, v)such that: (a) theu-curves and thev-curves are asymptotic curves of the surface, and (b) these curves are parametrized by arclength.

Theorem 7. Let (I, II)be a Codazzi pair in a surface Σwith negative constant extrinsic curvatureK. Then there exist local coordinates(u, v)and a smooth function ω(u, v)∈(0, π)such that

I=du²+ 2 cosω dudv+dv², II = 2√

−Ksinω dudv.

(23) Moreover,

1. IfΣis simply connected there exist global functionsu, v : Σ−→

R

^{such that}

(u, v)are local coordinates in the conditions above in a neighborhood of every point.

2. IfΣis simply connected andIis complete then the previous map(u, v) : Σ−→

R

²is a global diffeomorphism.

Proof.Let us consider local coordinates (global coordinates ifΣis simply connected) such that

I=E du²+ 2F dudv+G dv², II = 2f dudv, f >0.

Let us denote by Γ^k_ij to the Christoffel symbols of the Levi-Civita connection of the induced metric with respect to the parameters(u, v). That is,

∇∂

∂u

∂

∂u = Γ¹₁₁∂

∂u + Γ²₁₁∂

∂v, ∇∂

∂u

∂

∂v = Γ¹₁₂∂

∂u + Γ²₁₂∂

∂v,

∇∂

∂v

∂

∂v = Γ¹₂₂∂

∂u+ Γ²₂₂∂

∂v.

DenoteD=EG−F², then a straightforward computation gives Du

2D = Γ¹₁₁+ Γ²₁₂, Dv

2D = Γ²₂₂+ Γ¹₁₂. Besides, from the Codazzi equation for(I, II)one has

fu

f = Γ¹₁₁−Γ²₁₂, fv

f = Γ²₂₂−Γ¹₁₂.

Now, we use thatK=−f²/Dis a negative constant. Thus, we have 0 =−f_u

f +D_u

2D = 2Γ²₁₂, 0 =−f_v f +D_v

2D = 2Γ¹₁₂. Hence,∇∂

∂u

∂

∂v = 0and soEv= 0 =Gu. By replacingu, vby the new local coordinates

ue= Z

pE(u)du, ev= Z

pG(v)dv

one has

I=du²+ 2F dudv+dv², II = 2f dudv.

Since the extrinsic curvature is a negative constantK=−f²/(1−F²), we have F²+ (f /√

−K)²= 1. And so there exists a smooth functionωsuch that I=du²+ 2 cosω dudv+dv², II = 2 sinω dudv,

with0< ω < π, since1−F²>0.

WhenΣ is simply connected then we have proved that(u, v) : Σ −→

R

^{2 is}

a local diffeomorphism. Thus, if we consider the new Riemannian metric III = du²+ 2 cosω dudv+dv², thenI+III = 2(du²+dv²). Hence du²+dv² is a

complete flat metric whenIis complete. So,(u, v) : (Σ, du²+dv²)−→

R

^{2is a local}

isometry and must be a global diffeomorphism.

2

As a consequence we have

Theorem 8. LetΣbe a surface andψ : Σ −→

M

^3(c)an immersion with negative constant extrinsic curvatureKin a space form. Then the asymptotic curves ofψhave constant torsionτ, withτ²=−Kat points where the curvature of the curve does not vanish. Moreover, two asymptotic curves through a point have torsions of opposite signs if they have non vanishing curvature at that point.

Proof.Using Theorem 7 there exist local coordinates(u, v)such that its induced met- ric and second fundamental form can be written as in (23). Thus, with this parametriza- tion, the coordinate curves are the asymptotic curves of the immersion.

Let us take the asymptotic curveα(u) = ψ(u, v0). Thenαis parametrized by the arc length andh∇α⁰(u)α⁰(u), N(u, v0)i= 0, whereNis the normal vector of the immersion and∇is the Levi-Civita connection in

M

^3(c).

Hence, if we assume the curvature of α(u)does not vanishes atu = u0 then N(u0, v0) is the binormal vector atu = u0. And the normal vector of α(u) is N(u, v0)∧α⁰(u)at the points where the curvature ofα(u)does not vanish.

If we writeN(u, v0)∧α⁰(u) =a(u)_∂u^∂ +b(u)_∂v^∂ , for suitable functionsa, b, then the torsionτ(u)ofα(u)can be computed as

τ(u) = −h∇_α⁰_(u)N(u, v0), N(u, v0)∧α⁰(u)i

= −h∇α⁰(u)N(u, v0), a(u)∂

∂u+b(u)∂

∂vi

= b(u)hN(u, v₀),∇∂

∂u

∂

∂vi=b(u)√

−Ksinω(u, v₀).

Here,b(u)can be calculated by considering the inner product with_∂u^∂ and _∂v^∂ in the

equalityN(u, v₀)∧α⁰(u) =a(u)_∂u^∂ +b(u)_∂v^∂ 0 =hN(u, v0)∧α⁰(u), ∂

∂ui = a(u) +b(u) cosω sinω=hN(u, v0)∧α⁰(u), ∂

∂vi = a(u) cosω+b(u).

That is,b(u) = 1/sinω andτ(u) = √

−Kas we wanted to show. Analogously, it can be proved that that the torsion ofβ(v) =α(u₀, v)is−√

−K, whereN(u₀, v)is the binormal vector of the curves.

2

For a flat immersion in

S

³the functionω, called theangle function, has two basic properties: Firstly, asIis regular then0 < ω < π, that isωis bounded. Secondly, the Gauss equation of the surface translates intoω_uv = 0. In other words, the angle ωverifies the homogeneous wave equation, and thus it can be locally decomposed as ω(u, v) =ω₁(u)+ω₂(v), whereω₁andω₂are smooth real functions. Let us point out here that as the flat surfaces in

S

³are described by the homogeneous wave equation, which is hyperbolic, it turns out that flat surfaces in

S

³ will not be real analytic in general.

As these Tschebyscheff coordinates (T-coordinates from now on) are essential for the local study of flat surfaces, it is important to understand when are theyglobally available on a surface, in order to develop a global theory. We have proved that any simply-connected complete flat surface in

S

³has globally definedT-coordinates.

If we drop the trivial topology assumption, this is no longer true. It turns out that simply connected flat surfaces in

S

³ do not possess in general globally definedT- coordinates, but instead, they admit a globally definedTschebyscheff immersion. In other words, we can take two mapsu, v from the surface into

R

that verify all the properties ofT-coordinates, except for the fact that the map(u, v)into

R

^{2may not be}

injective. The existence of thisT-immersion is enough in many cases to deal globally with non-complete flat surfaces in

S

³. It was proved in [14] thatT-coordinates are globally available on any (not necessarily complete) simply connected real-analytic flat surface in

S

^3.

6.2 The quaternionic model for S

³

The best way to describe explicitly flat surfaces in

S

³is to regard the3-sphere as the set of unit quaternions. Let us explain this model for

S

^3briefly.

We begin by identifying

R

⁴ with the quaternions in the standard way, that is, (x₁, x₂, x₃, x₄)is viewed as the quaternionx₁+ix₂+jx₃+kx₄. Recall that in the product of quaternions

i j=k j k=i k i=j

j i=−k k j=−i i k=−j

i i=−1 j j=−1 k k=−1.

In that way the unit3-sphere

S

^3⊂

R

⁴is regarded as the space of unit quaternions, i.e. quaternionsxwith unit norm,kxk= 1. We also point out that

S

^2≡

S

^{3∩{x1= 0}}

can be seen as the space of purely imaginary unit quaternions.

The advantage of this model is that, by using the usual product of quaternions, the space

S

³receives in a natural way a Lie group structure. Indeed, ifx, y ∈

S

^{3, then}

x y ∈

S

³. Moreover, the left and right translationsx7→x aandx7→a xturn out to be isometries for the standard Riemannian metric of

S

³. In other words, the metric of

S

³ is bi-invariant with respect to this Lie group structure. Thus we have for any x, y, a∈

S

^{3thathx, yi=ha x, a yi=}hx a, y ai.

Apart from multiplication, there is another operation with quaternions that will be useful to us: theconjugationx:=x₁+ix₂+jx₃+kx₄7→¯x=x₁−ix2−jx3−kx4. It turns out that conjugation is geometrically an orientation reversing isometry in

S

^3.

Moreover we havex¯=x⁻¹wheneverx∈

S

³, and in additionx¯ =−xifx∈

S

^{2 ⊂}

S

³. Let us remark thathx, xi=xxandx y= ¯yx.¯

We end up the description of the geometry of

S

³via quaternions with the intro- duction of the usualHopf fibration. Let us defineAd(x)y:=x yx, where¯ x, y∈

S

^3.

Then the Hopf fibration h :

S

^{3 −→}

S

² is given by h(x) = Ad(x)i = xix.¯ It follows immediately that the fibers h⁻¹(p) are great circles of

S

³. In fact, if

p=ip2+jp3+kp4is a point in

S

^{2thenh−1(p)}is the great circle given by h⁻¹(p) =

x1+ix2+jx3+kx4∈

S

^{3: −p}_p^{4x1+p3x2−(1 +p3)x3= 0,}

3x₁+p₄x₂−(1 +p₂)x₄= 0

, ifp26=−1

h⁻¹(−i) ={jx3+kx4∈

S

^{3: x3, x4∈}

R

^}.

The Hopf fibration will be crucial for describing flat surfaces in

S

³. Let us also remark that one can defineskew Hopf fibrations by means ofhξ(x) := Ad(x)ξ :

S

^3−→

S

^{2, whereξ∈}

S

²is fixed but arbitrary. The fibers will still be great circles of

S

^3.

6.3 First examples of flat surfaces in S

³

The simplest way to obtain flat surfaces in

S

³is by means of the Hopf fibration. This is due to the following remark by H.B. Lawson (see [30, 27]):

Proposition 2. Ifcis a regular curve in

S

^{2, thenh−1(c)}is a flat surface in

S

^3.

Proof. First, we observe that from the expression above ofh⁻¹(p)for p ∈

S

^{2, it}

is a direct computation thath⁻¹(c) ⊆

S

³ is the product of two circles contained in perpendicular planes whencis a circle. Hence, up to an isometry,h⁻¹(c)is

S

^1(r)×

S

¹⁽

√1−r²)with0< r <1. That is,h⁻¹(c)is a flat surface. These tori are usually called Clifford tori.

On the other hand, letcbe a regular curve in

S

^{2. For anyq ∈ h−1(c)we can}

consider a circleec ⊆

S

²with a contact of order two ath(q)with the curvec. Then, h⁻¹(c)andh⁻¹(ec)have a contact of order two atq. Thus, since the Clifford torus h⁻¹(ec)is flat and the Gauss curvature only depends on derivatives up to order two from the Gauss equation, we have thatqis a flat point inh⁻¹(c).

2

As the fibers of the Hopf fibration are geodesics of

S

³, it turns out thath⁻¹(c)has, in general, the topology of a cylinder. This is the reason why these surfacesh⁻¹(c) are calledHopf cylinders.