EXCERPTS OF ANALYSIS ON KLEIN SURFACES

ILIE B ˆARZ ˘A and DORIN GHIS¸A

We present some results reflecting the dichotomy between symmetric Riemann surfaces and Klein surfaces in dealing with objects proper to analysis on these surfaces.

AMS 2000 Subject Classification: 30F50.

Key words: Klein surface, symmetric Riemann surface, dianalytic atlas, orientable double covering surface.

1. INTRODUCTION

The category of Klein surfaces crystallized itself historically as a result of pioneering works of Felix Klein, as well as Teichm¨uller [19], Schiffer and Spencer [18], Alling and Greenleaf [1] and Cabiria Andreian Cazacu [3]. It includes as subcategories those of Riemann surfaces, and as these last ones they can be bordered, or border free.

Unlike Riemann surfaces, which are known to be orientable, a Klein surface might lack orientation, a fact involving some problems when dealing with analysis concepts, like vector fields, differential forms, etc. However, since any surface is locally orientable, as long as the analysis objects involved have a local character, one should be able to circumvolve difficulties. This fact doesn’t mean that there is no need for special language and techniques different of those proper to Riemann surface theory. Since differential geometry deals equally with orientable and non orientable manifolds, we were tempted to adopt the techniques and use the jargon of that field (see [6], [7], [10] and [13]). We found great inspiration in the de Rham work (see [17]) related to differential forms. However, when dealing with boundary value problems (see [8], [9], [14] and [15]), the language of analysis on Riemann surfaces has been more appropriate.

We present here some excerpts of analysis on Klein surfaces, which sum- marize our eforts related to this topic, as well as the main results we used.

For the modern concept of Klein surface, we suggest as reference [1, Chap- ter 1] and for that of morphism of Klein surfaces we suggest [3].

REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 395–406

Our construction of Klein surfaces is entirely based on the concept of symmetric Riemann surface in the sense of Klein. It applies equally to bor- dered, or border free Riemann surfaces and we will specify the type of surface only if necessary.

Definition1. A symmetric Riemann surface is a pair (O₂;k) formed with a Riemann surfaceO₂ and an antianalytic fixed point free involutionkof O₂.

We denote byhkithe two element group generated by k.

The result below has the origins in Klein’s work. It played a crucial role in the analysis on Klein surfaces we have undertaken in [4]–[7].

Theorem 1. Let (O₂;k) be a symmetric Riemann surface. Then the canonical projection p : O₂ → O₂/hki induces on O₂/hki a structure of non orientable Klein surface. Conversely, for every non orientable Klein surface X, there is a symmetric Riemann surface (O₂;k) such that X is isomorphic to O₂/hki.

There are a lot of proofs of this theorem. We are particularly fond of the one using isothermal coordinates to generate Riemann or Klein surfaces and which originates as well in Klein works (see [16]). The proof of the second part of the theorem is based on the remark that if π:Xb →Xis the universal covering of X, then Xb is simply connected, hence orientable. The covering is a Galois covering, i.e., the automorphisms group G of Xb associated to π conserves the fibers and acts transitively on each one of them. It contains necessarily conformal and anticonformal mappings. If we denote by G₁ the subgroup ofGformed with conformal transformations andA∈G\G1, thenG appears as the disjoint unionG=G₁∪AG₁. Moreover,G₁ itself is transitive and, therefore, O₂ := X/Gb 1 is a Riemann surface whose universal covering is of the form π₁ : Xb → O₂. The mapping k_A = k : O₂ → O₂ defined by k(cl[u]) := cl[Au], where cl[u] is the fiber of u ∈ X, is the fixed point freeb antianalytic involution we were looking for (see [5]).

For the first part, it is known that any surface X of class C¹ carries isothermal parameters h, i.e., parameters such that, where defined, h₁◦h⁻¹₂ is conformal or anticonformal ([2], p. 125). The Riemannian metric ds(z) = λ(z)|dz+µd¯z|they define is such thatµ(z)≡0. Then, ifzand z₁ correspond to two overlapping local charts, we have ds=λ|dz|=λ1|dz₁|in their common domain. Therefore, for the transition function z7→z1 we have|dz₁|= _λ^λ

1|dz|.

But dz1 = ^∂z_∂z¹dz+ ^∂z_∂¯z¹d¯z. Consequently, one of the two partial derivatives

∂z1

∂z or ^∂z_∂¯z¹ must be identically zero. It results that z 7→ z1 is a conformal or anticonformal mapping.

We succeeded to prove a reinforcement of this classical result (see [13]), namely

Theorem 2. If X is a Klein surface and if f : W → X is a covering of X with W orientable then W admits a conformal structure with respect to whichf is a morphism of Klein surfaces. For every symmetryk(in the sense of Klein)ofW, the surfaceW/hkiis a Klein surface and the canonical projection π :W → W/hki is a morphism of Klein surfaces. If the dianalytic structure of Xhas been induced by the metric ds=λ|dz|, then the analytic structure of W is induced by the metric dσ=λ₁|dw|, where λ₁(w) =λ(z(w))|_∂w^∂z +_∂^∂z_w¯|.

We will use the shorthand NOKS for anon orientable Klein surface.

2. FUNCTIONS ON NOKSs

The following definition is necessary in order to study the relationship between the functions on an NOKS and those on its orientable double cover.

Definition2. LetXbe an NOKS and (O₂;k) its orientable double cover.

A set ∆⊆ O₂ will be called symmetric iff ∆ =k∆. A function f : ∆ →Cb will be called symmetric, or k-invariant ifff(z) =f(k(z)) for everyz∈∆ and antisymmetric, ork-anti-invariant iff f(z) =−f(k(z)) for every z∈∆.

For applications F : O₂ → O₂ we will have a special interest in those commuting with k, i.e., such thatF◦k=k◦F.

Theorem 3. If (O₂;k) is the orientable double cover of the NOKS X, then any dianalytic mapping Fe :X → X can be lifted to a unique analytic mapping F : O₂ → O₂ commuting with k and such that p ◦F = Fe◦ p.

Conversely, if F :O₂ → O₂ is a continuous mapping commuting with k, then there is a unique continuous mapping Fe : X → X such that p◦F = Fe◦p.

If F is analytic, then Fe is dianalytic. Moreover, if we denote by F^◦n the nth iterate of F, then p◦F^◦n=Fe^◦n◦p.

For theproof see [11].

The real projective planeP² can be made into an NOKS having as ori- entable double cover (bC,k), wherek(z) =−1/¯z. We have proved the following results (see [11]).

Theorem 4. The dyanalytic transformations of P² are all of the form Fe(cl[z]) = cl[F(z)] where cl[z] = p(z) = p(k(z)) is the equivalence class of z with respect to the two element group hki, and

F(z) = e^iθ a0z²ⁿ⁺¹+a1z²ⁿ+· · ·+a2n+1

−¯a2n+1z²ⁿ⁺¹+ ¯a2nz²ⁿ− · · ·+ ¯a0

, where |a₀|+|a_2n+1|>0 and θ∈R.

The dianalytic automorphisms ofP²are all of the formG(cl[z]) = cl[G(z)]e or G(cl[z]) = cl[G(¯e z)], where

G(z) = e^iθ az+b

−¯bz+ ¯a, with |a|+|b|>0.

We denote by P²∗ the NOKS P²\ {cl[0]} which has as orientable double cover (C^∗,k), where C^∗=C\ {0}.

Theorem 5. The dianalytic transformations of P²∗ are of the form Fe(cl[z]) = cl[F(z)] = cl[χ(z)] where χ(z) = F(z) or χ(z) = F(¯z) with F(z) = e^iθz²ⁿ⁺¹, n∈Z, or

F(z) = 1 zexp

"_∞ X

n=1

anzⁿ+

∞

X

n=1

(−1)ⁿa¯n

zⁿ

# , and the series

∞

P

n=1

anzⁿ converges in C and not all an are zero.

It is known that every Klein bottle is an NOKS having as orientable double cover a torus. See [4] for details.

If X is a Klein bottle, then there exists a real number b >0 such that X is isomorphic with the orbit space C/H, H being the group of mappings generated by S, V :C→C,Sz= ¯z+¹₂ and V z=z+ ib.

Theorem 6. The mapf :C/H →C/H is an automorphism if and only if f has one of the following forms:

(1)f(cl[z]) = cl[αz+a]; (2)f(cl[z]) = cl[αz¯+a];

(3)f(cl[z]) = cl[αz+a+^bi₂]; (4) f(cl[z]) = cl[αz¯+a+^bi₂], where α=±1 and a∈[0,¹₂[.

We have denoted by cl[z] the orbit ofzwith respect to the groupH. For the proofsee [4].

3. BOUNDARY VALUE PROBLEMS ON AN NOKS

We presented in [8] the rationale behind the study of boundary value problems on an NOKS. It comes from applications in physics, chemistry and biology and it is a natural continuation of topological studies done in this field since 1929. By 1978 a systematic topological analysis led to the remarkable conclusion of a double topological character of distortions in liquid crystals differentiated for “energetic reasons”. This conclusion gave important weight to the question of solving boundary value problems for the surfaces of those ribbon-crystals, which may be orientable, as well as non orientable. Hence,

the necessity to study the non orientable surfaces not only from a topolo- gical point of view, but also from an analytical one, a fact which required a dianalytic structure on them, i.e., it required them to be NOKSs. Following the tradition of generating an NOKS X from a symmetric Riemann surface O₂ = O₂(X), we proceeded to solving first boundary value problems on O₂ and then transporting the solutions to X.

LetXbe an NOKS and letDbe a relatively compact region ofXhaving a nonempty boundary ∂D. In the case of a bordered NOKS we might have D=X and then∂D is the border of X.

Although the concept of analytic function on an NOKS has no mea- ning, every NOKS can carry harmonic functions. The Dirichlet problem for harmonic functions on Xrelative to the region D and the boundary function feon ∂D consists in finding a continuous function eu on D = D∪∂D which coincides with feon ∂D and is harmonic inD.

Let us denoteG=p⁻¹(D); then ∂G=p⁻¹(∂D). A real valued function f can be defined on ∂G by f = fe◦p. Since p◦k = p, we have f ◦k = f. We reduce the initial problem to that of solving a boundary value problem on O₂ relative toG and the boundary functionf. By using Perron’s method we conclude that this problem has a unique solution uwhenever ∂G has only regular points or, in particular, no component of∂Greduces to a single point.

Moreover, it is easy to prove

Theorem 7. If u is the solution of the Dirichlet problem with the boun- dary function f, then f =f ◦k on∂G implies u=u◦k in G. The function ue defined on D by u(cl[ze ]) :=u(z) =u(k(z)) is the solution to the Dirichlet problem for harmonic functions onXrelative to the regionDand the boundary function fe.

To be able to formulate the Neumann problem for an NOKS one would need to know what the normal derivative on the border of such a surface might look like. We have found that the proper way to provide an answer to this question was to express the normal derivative in terms of a generating metric of an NOKS.

We had to deal first with such a generating metric onO₂ and succeeded to solve the Neumann and Dirichlet problems for the M¨obius strip regarded as a bordered NOKS.

We also dealt in [14] with polyharmonic equations on the M¨obius strip by first solving the corresponding equations on the annulus.

Let us define A_R = {z ∈ C | _R¹ < |z| < R}, on which the symmetric component

dσ(z) = 1 2

1 + 1

|z|²

|dz|

of the Euclidean metric |dz| is defined. A model for the M¨obius strip X is obtained by identifying the points of A_R which are k-symmetric. The metric dσ induces a metric dσ^∗ on X which defines a dianalytic structure on it, with respect to which the canonical projection p:A_R→ X is a morphism of Klein surfaces.

Two important questions arise from this fact. First, is there ak-invariant Green function? Second, can a process of symmetrization similar to that of the Euclidean metric on AR be carried out for any metric on O₂?

In [8] we gave an affirmative answer to the first question by explicitly constructing k-invariant Green functions. This result allowed us to undertake the study of potentials on an NOKS (see [14]).

Following the tradition, we call hyperbolic a domainG⊆ O₂ iff there are non constant positive superharmonic functions on G. If Gis hyperbolic then p(G) is a hyperbolic domain on X = O₂/hki. Let G ⊆ O₂ be an arbitrary hyperbolic symmetric domain and let a∈G. Letga be the Green function of G with pole a. Then the function g_hai defined byg_hai(b) =g_a(b) +g_k(a)(b) is k invariant and verifies the identityg_hai(b) = g_hbi(a) for every a and b in G.

Moreover, there is a unique harmonic (up to cl[a]) functioneu:p(G)→Rsuch that ghai=eu◦p. The functionuehas in cl[a] a logarithmic singularity. This is the Green function on Xrelative to p(G) with pole cl[a].

Theorem 8.If O₂ is the orientable double cover of the NOKS X, then there is a Borel set A ⊆ O₂ such that A∩k(A) = ∅, O₂ =A∪k(A)and the canonical projection p:O₂ →X is injective on bothA and k(A).

For any k-invariant Borel measure µ on O₂, the set function eµ defined on the Borel sets of X by µ(B) =e µ(p⁻¹(B)∩A) = µ(p⁻¹(B)∩k(A)) is a Borel measure on X. Moreover, the measures µ and eµ generate potentials p^µ and p^µe, where p^µ(a) = R

ga(b)dµ(b) and p^µe(cl[a]) =R

g_cl[a]dµe such that for any fe∈C_o²(X) we have

Z

fdµ=− 1 2π

Z

p^µd∗df and Z

fedµe=− 1 2π

Z

p^µed∗df .e

In [12] we presented a comprehensive theory of measure and integration on an NOKS based on the positive answer to the second question.

4. MEASURE AND INTEGRATION ON AN NOKS

In this section a relationship between an NOKS and its orientable double cover is established, in terms of measures they support, as well as of the integrals with respect to those measures.

Theorem 9. Let µe be a Borel measure on an NOKS X. Then there is a unique k-invariant Borel measure µ onO₂ =O₂(X) such that for every Borel set B ⊆ O₂ on which the canonical projection p is injective we have µ(B) =µ(p(B)).e

In what follows the concept of covering partition of O₂ will appear as a working tool. It is represented by the couple (A,k(A)) from Theorem 8. Let us denote byB(O₂) andB(X) theσ-algebra of Borel sets onO₂,respectively on X, and by B_s(O2) the symmetric sets ofB(O2).

Theorem 10. (1) B_s(O₂) ={Z∪k(Z) |Z ∈B(O₂)}.

(2) If Z₁, Z₂ ∈B_s(O₂) and (A, k(A)) is a covering partition of O₂, then the assertions below are equivalent:

(j) Z₁=Z₂;

(jj)Z1∩A=Z2∩A;

(jjj)Z₁∩k(A) =Z₂∩k(A).

Let us denote by Mes(B(O₂)) and Mes(B(X)) the vector spaces of finite signed Borel measures onO₂ respectively onX.Ifµ∈Mes(B(O₂)) thenk^∗(µ) defined by (k^∗(µ))(Z) =µ(k(Z)) for everyZ ∈B(O₂) is another signed Borel measure on O₂. One can easily check that k^∗ : Mes(B(O₂))→ Mes(B(O₂)) is a linear involution.

We say thatµ∈Mes(B(O₂)) isk-invariant ork-antiinvariant iffµ(kZ) = µ(Z),respectively µ(kZ) =−µ(Z) for everyZ ∈B(O₂).

Let us use the notation

Mes_s(B(O₂)) :={µ∈Mes(B(O₂))|k^∗(µ) =µ}, Mesa(B(O₂)) :={µ∈Mes(B(O₂))|k^∗(µ) =−µ}.

It is obvious that Mess(B(O₂)) and Mesa(B(O₂)) are linear subspaces of Mes(B(O₂)) and they have in common only the null measure.

Theorem 11. (1) Mes(B(O₂)) = Mes_s(B(O₂))⊕Mes_a(B(O₂)).

(2)p^∗(Mes(B(X)) = Mess(B(O₂)).

(3)p^∗ : Mes(B(X))→Mes_s(B(O₂))is an isomorphism of vector spaces.

According to this theorem, every finite signed Borel measureµonO₂ can be decomposed into the sum of a k-invariant signed Borel measure µ_s and a k-antiinvariant signed Borel measureµa.The componentµs “projects” onto a signed Borel measure onX.The componentµaprojects onto the null measure.

5. SYMMETRIZATION AND ANTI-SYMMETRIZATION OF FUNCTIONS ON O₂

We now introduce the following notation/definitions:

• F(O₂) :={f :O₂ →Cb |f = function},

• F_s(O₂) :={f :O₂ →Cb |f = symmetric function},

• F_a(O₂) :={f :O₂ →Cb |f = antisymmetric function}, and {z∈ O₂|f(z) =∞}is afinite set for every f.

Thesymmetrization operatorSand theanti-symmetrization ope- rator A withS,A:F(O₂)→ F(O₂), are defined by

Sf := 1

2(f+f◦k) :=f_s, Af := 1

2(f −f ◦k) :=f_a. The main properties of these two operators are summed up in

Theorem 12. (1) The operatorsS and A are linear.

(2)The images of these operators areS(F(O₂)) =F_s(O₂)andA(F(O₂))

=F_a(O₂).

(3) The behavior of these operators with respect to the multiplication in the algebra F(O₂) is given by

(f g)s=fsgs+faga, (f g)a=fsga+fags

for every f, g∈ F(O₂).

(4) F_s(O₂) is a subalgebra of F(O₂) and F_a(O₂) is a vector subspace of F(O₂).

(5) S ◦ S = S, A ◦ A = A, S ◦ A=A ◦ S =O = the null-operator of F(O₂) andS+A=I = the identity operator ofF(O₂).

(6)Thedirect sum decomposition F(O₂) =F_s(O₂)⊕ F_a(O₂) holds.

Let us denote byM(O₂) the field of meromorphic functionsf :O₂→Cb on O₂, and by M(O₂) the field of antimeromorphic functions on O₂, i.e., functions which are antianalytic up to poles.

Let us denote by F(X) the algebra of the complex functions on the NOKS X and by p^∗ :F(X) → F_s(O₂) the pull-back mapping induced by p.

It is obvious that p^∗ is analgebra-isomorphism.

We point out next some properties of the pull-back. Define k^∗:F(O₂)→ F(O₂), f 7→k^∗f :=f ◦k.

Here, F(O₂) is the algebra of functions f : O₂ → Cb for which the set of points of O₂ where f takes the value ∞ is finite. It is obvious that k^∗ is an involution and that its restriction

k^∗ :M(O₂)→ M(O₂)

is a field isomorphism whose inverse is k^∗ : M(O₂) → M(O₂). It is also obvious that the restriction

k^∗:M(O₂)∪ M(O₂)→ M(O₂)∪ M(O₂)

of k^∗ :F(O₂)→ F(O₂) is an involution, too, permutingM(O₂) withM(O₂).

From the very definition of the symmetrization operatorS we get (since S ◦k^∗ =S) the relations

S(M(O₂)) =S(M(O₂))⊂ M(O₂)⊕ M(O₂).

This prompts us to state

Definition 3. A function fe : X → Cb is called dimeromorphic or a function of meromorphic type ifp^∗(fe)∈ S(M(O₂)).

Remark. The functions of meromorphic type were introduced and studied in [5]. There the Cauchy’s Theorem of Residues is given for functions of mero- morphic type on compact non-orientable Klein surfaces.

6. DIFFERENTIAL FORMS OF FIRST DEGREE ON X We recall that for the symmetric (open) subset ∆ = k∆⊆ O₂ a diffe- rential form ω of first degree on ∆ is a sectionof the cotangent fiber bundle of O₂, i.e., ω: ∆→T^∗(O₂), ∆3P 7→ω(P)∈T_P^∗(O₂). We denote by D¹(∆) the set of differential forms of first degree on ∆, endowed with the structure of module over the algebra of functions on this set.

The differential forms of first degree in open subsets of the NOKSXare defined similarly. Locally, these differential forms look like A(z)dz+B(z)d¯z.

The symmetrykinduces the pull-back mapk^∗ :D¹(∆)→ D¹(∆) which is well-known from the differential geometry. (See [6] for details.) For simpli- city, we shall always consider that ∆ =O₂.Because k is an involution,k^∗ is an isomorphism and its inverse also is k^∗, i.e., k^∗ also is an involution.

In analogy with Definition 2 we state

Definition 4. The differential form ω ∈ D¹(O₂) is called symmetric (resp. antisymmetric) ifk^∗ω =ω (resp. k^∗ω=−ω).

Notation: (i) D¹_s(O₂) :=

ω∈ D¹(O₂)| k^∗ω=ω ; (ii)D¹_a(O₂) :=

ω∈ D¹(O₂)| k^∗ω=−ω .

It is obvious that D¹_s(O₂) is a module over the algebraF_s(O₂).

One can prove (see [6]) that the module of the differential formsD¹(X) of first degree on the surface X, is canonically isomorphic to the previous module, via the pull-back map p^∗ induced by the projectionp.

Now, we define the symetrization/antisymmetrization operators S and A on the module D¹(O₂). Here, they are S and A :D¹(O₂) → D¹(O₂) that are defined as

Sω:= 1

2(ω+k^∗ω) :=ω_s, Aω:= 1

2(ω−k^∗ω) :=ω_a.

The main properties of these operators are given in

Theorem 13. (1) The operatorsS and A are linear.

(2)The images of these operators areSD¹(O₂) =D¹_s(O₂)andAD¹(O₂) = D_a¹(O₂).

(3)Sω=ω ⇔ω ∈ D¹_s(O₂) and Aω=ω ⇔ω∈ D_a¹(O₂) for ω ∈ D¹(O₂).

(4) S ◦ S = S; A ◦ A = A; S ◦ A=A ◦ S =O = the null-operator of D¹(O₂) and S+A=I =the identity operator of D¹(O₂).

(5)Thedirect sum decompositionD¹(O₂) =D_s¹(O₂)⊕ D¹_a(O₂)holds.

(6) The F(X) module D¹(X) is canonically isomorphic via p^∗ with the F_s(O₂)-moduleD¹_s(O₂).

Remark.Professor C. Constantinescu (ETH-Z¨urich) noticed that the fac- tor D_a¹(O₂) of the previous direct sum decomposition is isomorphic with the space of differential forms of odd type in the sense of G. de Rham, on the surface X.

7. DIFFERENTIAL FORMS OF SECOND DEGREE ON X For a symmetric subset ∆ of O₂ = O₂(X) we consider the differen- tial form

Ω : ∆→Λ²(T^∗O₂) := [

z∈∆

Λ²(T_z^∗O₂)

of second degree on ∆, where Λ²(T_z^∗O₂) is the second exterior power of the cotangent (complexified) space ofO₂at the pointz. For everyz∈∆ its image Ω(z)∈Λ²(T_z^∗O₂) looks like A(z)dz∧d¯z.

IfD is a parametric disk onO₂ such thatD∩kD=∅ and if z=x+ iy is a local parameter in D then w = u+ iv =kz is a local parameter in kD and therestriction of Ω to D∪kDis

Ω|_D^S_kD(Q) = (

A(z)dz∧d¯z forQ=z∈D, A⁰(w)dw∧d ¯w forQ=w∈kD.

By an easy computation we get the restriction (k^∗Ω)|_D∪kD(Q) toD∪kD of the pull-back of Ω by the symmetry kas

(k^∗Ω)|_D∪kD(Q) =











A⁰(kz)D(u, v)

D(x, y)dz∧d¯z forQ=z∈D, A(kw)D(x, y)

D(u, v)dw∧d ¯w forQ=w∈kD, where ^D(u,v)_D(x,y) is the Jacobian of the mapz7→kz=w.

Symmetricandantisymmetricdifferential forms of second degree are defined in a similar way as in the case of differential forms of first degree.

We denote byD²(O₂),D²_s(O₂) andD²_a(O₂) the sets of differential forms of second degree onO₂, the set of symmetric differential forms of second degree and the set of antisymmetric differential forms of second degree, respectively.

Thus,

(D²_s(O₂) :={Ω∈ D²(O₂)| k^∗Ω = Ω}, D²_a(O₂) :={Ω∈ D²(O₂)| k^∗Ω =−Ω}.

The symmetric/antisymmetric differential forms of second degree are the fixed points of the symmetrization/antisymmetrization operatorsS and A: D²(O₂)→ D²(O₂) defined as

SΩ := 1

2(Ω +k^∗Ω) := Ω_s, AΩ := 1

2(Ω−k^∗Ω) := Ω_a

Theorem 11 has an analogous formulation for the setting of differential forms of second degree. In particular, we mention the direct sum decomposi- tion D²(O₂) = D_s²(O₂)⊕ D²_a(O₂) and that the module D²_s(O₂) is canonically isomorphic to the module of differential forms of second degree on the sur- face X.

Constantinescu’s remark remains valid for this last case, as well as for the case of functions, i.e., differential forms of degree zero:

The vector space of differential forms of odd type and second degree (degree zero) on X is isomorphic to the component of the anti-symmetric elements of the vector space of differential forms of second degree (respectively functions) on O2.

The analysis on non orientable Klein surfaces represents the natural ex- tension of that on Riemann surfaces and, in particular, of classical complex analysis. We have found recently that Blaschke products-like functions can be defined on the real projective plane and the study of these functions project a new light on the global mapping properties of meromorphic functions. Some results of these studies are already in print and many new questions are waiting for an answer.

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Received 10 January 2009 Karlstad University

Department of Mathematics SE-6651 88 Karlstad, Sweden

[email protected] and Glendon College Department of Mathematics

Toronto, Canada [email protected]