AND THE AREA ON A KLEIN SURFACE
MONICA ROS¸IU and PETER KESSLER
We prove some properties of the Φ-metric and thee Φ-area associated with ane N-quadratic differential on a Klein surface.
AMS 2000 Subject Classification: 30F50.
Key words: Klein surface, N-meromorphic quadratic differential.
1. INTRODUCTION
We continue the study of theN-quadratic differentials on a Klein surface.
Our approach is from an alternative theory to the standard one in Alling and Greenleaf [1] and is based on the fact that according to a classical result due to F. Klein, the calculus on nonorientable manifolds can be canonically reduced to calculus on orientable manifolds. Such a model of calculus was developed by Bˆarz˘a [3] and is used here to develop a parallel theory to that in Strebel [6], for the local study of the Φ-metric and thee Φ-area associated with ane N-quadratic differential on a Klein surface X.
2. BASIC NOTIONS
A Klein surface is a pair (X, A) where X a surface and A a maximal dianalytic atlas on X not containing any analytic subatlas.
A symmetric Riemann surface is a pair (O2, k) consisting of a Riemann (orientable) surface O2 and an antianalytic involution k :O2 → O2 without fixed points.
LetHbe the group generated byk, with respect to the usual composition of functions.
Theorem2.1 (Klein). If (X, A) is a Klein surface, then there is a sym- metric Riemann surface (O2, k) such that X is dianalytically equivalent to O2/H. If Y is a Klein surface dianalytically equivalent to X, then in the
REV. ROUMAINE MATH. PURES APPL.,54(2009),3, 267–274
group of all analytic or antianalytic automorphisms of O2,there is an antian- alytic involution k0 :O2 → O2, conjugated with k, such that Y is isomorphic to O2/H0, where H0 is the group generated by k0. Conversely, if (O2, k) is a symmetric Riemann surface, then on the orbit space O2/H there exists a dianalytic atlas A such that (O2/H, A) is a Klein surface.
LetB1 be the maximal analytic atlas onO2andB2 the maximal analytic atlas on k(O2). Then k(O2) is O2 endowed with the second orientation. Ifq is the canonical covering projection ofO2 ontoO2/H, then the diagram
O2 −→k O2 q
y .q X
is commutative andqis a dianalytic mapping which mix the two structures on O2 to get the dianalytic structure ofX. The Klein surfaceX will be identified with the orbit space O2/H (see [4])
For convenience of notation, the points and the curves inO2andXwill be identified with their images in the Euclidean planes, through the corresponding charts.
IfUe is a parametric disk onX, thenq−1(Ue) =U∪k(U) is ak-symmetric set of O2. Thus, it is natural to consider restrictions on U ∪k(U) for the local study of the meromorphic quadratic differentials on O2. Since k is an involution without fixed points, one can suppose that U ∩k(U) = ∅. Let z be the local parameter on U and ze = {z, k(z)} the H-orbit of z. Thus, ze=k(z) =g q(z) =q(k(z)) ={z, k(z)}is the local parameter on Ue (see [3]).
LetQ2(O2) be the linear space of (meromorphic) quadratic differentials on O2 (see [6]) and Q2(O2) the linear space of (antimeromorphic) quadratic differentials on O2, i.e., the linear space of (meromorphic) quadratic differen- tials on O2 endowed with its second orientation.
Let Φ∈ Q2(O2) with local representation Φ∗/U ∪k(U) =
( ϕ(z)dz2 ifz∈U, ϕ(w)dwb 2 ifw∈k(U),
where z is the local parameter on U and w = k(z) the local parameter on k(U). According to the definition of a (meromorphic) quadratic differential on a Riemann surface (see [6]), ϕ= Φ/U and ϕb= Φ/k(U) are meromorphic functions on U and k(U), respectively.
Then the symmetrykwill induce the isomorphismK :Q2(O2)→Q2(O2) with
K(Φ)∗/U ∪k(U) = (
ϕ(k(z))dk(z)b 2 ifz∈U, ϕ(k(w))dk(w)2 ifw∈k(U),
or
K(Φ)∗/U ∪k(U) =
ϕ(k(z))b ∂k
∂z(z) 2
dz2 ifz∈U, ϕ(k(w))
∂k
∂w(w) 2
dw2 ifw∈k(U) because kis an antianalytic function.
LetQs(O2) =Q2(O2)⊕
k Q2(O2) =
Φ +K(Φ)|Φ∈Q2(O2) be the set of k-symmetric quadratic differentials on O2.
Definition 2.1. Let X be a Klein surface and A =
(Uei, hi, Vi) | i∈ I the induced atlas onX;Φ is an N-quadratic differential one X iff:
1. for every (U , h, Ve )∈A,Φ has the local representatione Φe∗/Ue =Φe∗1/Ue + Φe∗2/Ue, whereΦe1∈Q2(O2) and Φe2 ∈Q2(O2);
2. if (Ue1, h1, V1), (Ue2, h2, V2) ∈ A such that Ue1∩Ue2 6= ∅ and Φe∗/Ue1 = (Φe01)∗/Ue1 + (Φe001)∗/Ue1 ,Φe∗/Ue2 = (Φe02)∗/Ue2 +(Φe002)∗/Ue2 are the corresponding local representation ofΦ, then for every connected componente Ue ofUe1∩Ue2the local representations of differentialsΦe01,Φe02,Φe001,Φe002 onUesatisfy the conditions:
b1)
( (Φe01)∗/Ue = (eΦ02)∗/Ue (Φe001)∗/Ue = (Φe002)∗/Ue ifh2◦h−11 is an analytic function on h1(Ue),
b2)
( (Φe01)∗/Ue = (eΦ002)∗/Ue (Φe001)∗/Ue = (eΦ02)∗/Ue ifh2◦h−11 is an antianalitic function on h1(Ue).
Let Q2(X) be the linear space of N-quadratic differentials on X with respect to C.
Theorem2.2. There is an isomorphism qebetween Qs(O2) and Q2(X).
Proof. The mapping qe:Qs(O2)→ Q2(X) is defined by eq(Φ +K(Φ)) =Φe for every Φ∈Q2(O2) and is an isomorphism (see [4]).
Let Φ be an N-quadratic differential one X with Φe 6= 0. By the iso- morphism qewe can identify Φ with the correspondinge k-symmetric quadratic differential on O2. Let Φ be the the corresponding (meromorphic) quadratic differential onO2, so thatΦe∗/Ue = Φ∗/U+k(Φ)∗/U for every parametric disk Ue onX. Ifh∈Asuch thatze=h(Pe) =zis the local parameter corresponding
toUe, then
Φe∗/Ue =ϕ(z)dz2+ϕ(k(z))dk(z)b 2,
whereϕandϕbare meromorphic functions of the local parametersz, andk(z), respectively.
3. THE MAIN RESULTS
Let∆ be an open subset one Xandec: [0,1]→∆ a piecewise continuouslye differentiable curve in∆.e The curveechas exactly two liftings toq−1(∆) whiche are piecewise continuously differentiable. If ec(0) = ez0 = {z0, k(z0)} and c : [0,1]→∆ is the lifting ofecwith the initial pointz0, then k◦cis the lifting of ecwith the initial pointk(z0). Since there is a dianalytic isomorphism between O2/H and X, and an arbitrary rectifiable curve on O2 which does not go through any critical point of Φ can be subdivided into intervals each one of which lies in a parametric disk, we identify a curve ec in a parametric diskUe around a regular point Pe0 ∈X with the k-symmetric curveq−1(ec) =c∪k(c) in U∪k(U).
The Φ-lengths of the two curves c and k◦c, which means their lengths with respect to the Φ-metric, that is (see [6]),
ds(z) =p
|ϕ(z)| |dz|
are lΦ(c) = R
c
p|ϕ(z)| |dz| and lΦ(k(c)) =R
c
p|ϕ(k(z))| |dk(z)|, respectively,b where ϕ(z)dz2 and ϕ(k(z))dk(z)b 2 are the local representations of the (mero- morphic) quadratic differential Φ in ∆ and k(∆), respectively.
The critical points of an N-quadratic differentialΦ are its zeroes and itse poles. All other points of X are called regular points of Φ. A holomorphice point is either a regular point or a zero. The poles of the first order and the zeroes are called finite critical points, the poles of order greater or equal to two are called infinite critical points (see [6]).
Proposition3.1. In a neighborhood of every regular point P0 of Φone can introduce a local parameter w in terms of which the representation of Φ is dw2.The parameter is given by the integral w= Ψ(z) =R p
ϕ(z)dz and is uniquely determined up to a transformation w→ ±w+constant (see [6]).
Let ∆ be a parametric disk around a regular point P0. The curve c : [0,1] → ∆ is mapped by a branch Ψ0 of Ψ(z) = R p
ϕ(z)dz onto a curve c0 : [0,1]→ Ψ0(∆) = ∆0. The Euclidean length ofc0 does not depend on the selected branch of Ψ in ∆. Then the Φ-length ofcis denoted bylΦ(c) and can be computed by means of the differential dw=p
ϕ(z)dzin terms of the local
parameter was
lΦ(c) = Z
c0
|dw|= Z
c
p|ϕ(z)| |dz|.
Thus, the Φ-length ofcis the Euclidean length of c0;lΦ(c) is well defined even ifc passes through critical points ofϕbut become infinite ifc passes through an infinite critical point, which means a pole of order greater or equal to two.
Since the Φ-metric is not k-symmetric, one defines a k-symmetric Φ- length element on O2 as
des= 1
2(ds+ ds◦k).
Thus, for every z∈O2, des(ez) = 1
2
p|ϕ(z)| |dz|+p
|ϕ(k(z))| |dk(z)|
is invariant with respect to kand so the lengths ofcand k◦cwith respect to this metric are the same.
TheΦ-metric one X is defined by
dS(e ez) = des(z) = des(k(z)), so that, by definition,
lΦe(ec) = 1
2(lΦ(c) +lΦ(k(c))) is the Φ-length ofe ec (see [5]).
Let∆ be a domain one X. The Φ-distance of two pointse ze1 and ez2 in ∆e is defined to be
d(eze1,ze2) =inf
ec l
Φe(ec),
where ecranges over the piecewise smooth curves in ∆ joininge ze1 andze2. The Φ-distance of two pointse ze1 and ze2 depends of the domain which is selected.
Let ze0 = {z0, k(z0)} ↔ Pe0 be a point on X and ϕ(e z)de ez2 = ϕ(z)dz2+ ϕ(k(z))dk(z)b 2 the local representation of Φ in the parametric diske V(ze0, r), where r1 and r2 are the radii of convergence of the meromorphic functions ϕ and ϕbnear z0, respectivelyk(z0) whiler = min
i∈{1,2}ri .
The next theorem is an extension to Klein surfaces of a similar result on Riemann surfaces (see [6]).
Theorem3.1. The Φ-distance between a critical pointe ze0 and a regular neighboring point ez1 is finite if and only if ez0 is a finite critical point.
Proof. Letze0 be a point of Φ ande ϕ(e ez)dez2 =ϕ(z)dz2 +ϕ(k(z))dk(z)b 2= ∞
P
i=n
αi(z−z0)i
dz2+ ∞ P
i=n
βi [k(z)−k(z0)]i
dk(z)2 the representation of
the quadratic differential Φ in terms of the local parametere ez∈V(ze0, r), with q−1(ez) ={z, k(z)}, where αn6= 0, βn6= 0,n∈Z.
If an orientation ofO2 is given, then letz1 = (q/U)−1(ze1)∈V(z0, r).
Ifez0is a finite critical point, then n2 ≥ −12 >−1. Letcbe the straight arc joining z0 and z1 and c0 the straight arc joining w0 =k(z0) andw1 =k(z1).
Thus c : [0,1] → O2, c(t) = (1−t)z0 +tz1 and c0 : [0,1] → O2, c0(t) = (1−t)k(z0) +tk(z1) for everyt∈[0,1].
By the definition ofd(eze0,ze1) we have d(eze0,ez1) ≤ 12[lΦ(c) +lΦ(c0)]. Since
z→zlim0
p|αn+αn+1(z−z0) +· · · |=p
|αn|, there isM1 >0, such thatp
|ϕ(z)|
=|z−z0|n2 p
|αn+αn+1(z−z0) +· · · | ≤M1|z−z0|n2 for everyz∈V(z0, r).
Also lim
w→w0
p|βn+βn+1(w−w0) +· · · | = p
|βn|, so that there is M2 > 0 such that p
|ϕ(w))| ≤b M2|w−w0|n2 for everyw∈V(w0, r).
IfM = max(M1, M2) then d(eez0,ez1)≤ M
2 Z
c
|z−z0|n2 |dz|+ Z
c0
|w−w0|n2 |dw|
. But |z−z0|< rimplies
Z
c
|z−z0|n2 |dz|=
Z
c
|z0−z1|n2+1tn2dt≤ 2
n+ 2rn2+1tn2+1
1
0 = 2
n+ 2rn2+1<∞.
Similarly, R
c0|w−w0|n2 |dw|<∞, for|w−w0|< r. Thus, d(ez0,ez1)<∞.
It is obvious that theΦ-distance between a regular pointe ez0 and a point ze1 ∈V(ez0, r) is finite.
Ifze0 is a pole of order greater or equal to two, then n2 ≤ −1.
Let ec be a piecewise smooth curve in V(ze0, r) joining ez1 and ze0. Since
z→zlim0
p|αn+αn+1(z−z0) +· · · |=p
|αn|, there is a sufficiently small neigh- borhood of z0 such thatp
|αn+αn+1(z−z0) +· · · |=
∞
P
m=0
βm(z−z0)m, with β0=p
|αn|,βm ∈Cfor every m≥0. Thus, lΦ(c) =
Z
c
|z−z0|n2 p
|αn+αn+1(z−z0) +· · · | |dz|, where cis the lifting ofec toO2 with the initial pointz0.
One can assume that|z1−z0| ≤1. Then Z
c
|z−z0|n2 |dz| ≥ Z
c
|z−z0|−1|dz| ≥
Z 1 0
γ0(t)dt γ(t)−z0
=∞, so that lΦ(c) =∞ and also l
Φe(ec) =∞.
LetEe be a Lebesgue measurable subset of ∆. By the definition ofe E,e E and k(E) are Lebesgue measurable subsets of ∆ andk(∆).
The Φ-areas of the two subsets E and k(E) of O2, that are their areas with respect to the Φ-area element
da(z) =|ϕ(z)|dxdy, are
AΦ(E) = Z Z
E
|ϕ(z)|dxdy and
AΦ(k(E)) = Z Z
k(E)
|ϕ(w)|b dudv, with k(z) =wand w=u+ iv (see [6]).
Define ak-symmetric Φ-area element onO2 by dea= 1
2(da+ da◦k).
TheΦ-area element one X, namely,
dA(e ez) = dea(z) = dea(k(z))
is a dianalytic invariant on X, i.e., is independent of the parametrization (see [5]). By definition,
AΦe(E) =e Z Z
Ee
dAe is the Φ-area ofe Ee (see [5]).
The next theorem also generalizes to Klein surfaces a similar result on Riemann surfaces (see [6]).
Theorem3.2.An isolated singularity of Φe has a neighborhood with finite Φ-area if it is a first order pole.e
Proof. Let Pe0 be an isolated singularity of Φ such thate h(Pe0) = ez0, where h is the corresponding chart of the dianalytic atlas on X, and let ϕ(e ez)dze2 =ϕ(z)dz2+ ϕ(k(z))dk(z)b 2 =
∞
P
l=n
αl(z−z0)l
dz2+ ∞
P
l=n
βl[k(z)− k(z0)]l
dk(z)2 be the local representation of the quadratic differential Φ ine terms of the local parameter ze∈ V(ez0, r), with αl, βl ∈ C, for every l ≥ n, n∈Z,αn6= 0 and βn6= 0.
Then αl = 2πi1 R
c ϕ(z)
(z−z0)l+1dz and βl = 2πi1 R
c0
ϕ(w)b
(w−w0)l+1dw, with w = k(z), w0=k(z0),c, c0 : [0,2π]→C,c(θ) =z0+reiθ and c0(θ) =w0+r0eiθ.
Thus, 2π|αl|= R
c ϕ(z) (z−z0)l+1dz
≤R
c
|ϕ(z)|
|z−z0|l+1|dz|= rl+11
R
c|ϕ(z)| |dz| ⇒ 2π|αl|rl+1 ≤R2π
0
ϕ(z0+reiθ)
rdθ=RR
V
|ϕ(z)|dxdy, for a sufficiently smallr and l∈Z, whereV =V(z0, r). IfRR
V
|ϕ(z)|dxdy <∞ then αl= 0 forl≤ −2 and, similarly, βl= 0 forl≤ −2, which means that ez0 is a first order pole.
Conversely, assume thatez0 is a first order pole. Letϕ(e ez)dze2=ϕ(z)dz2+ ϕ(k(z))dk(z)b 2 =
∞
P
l=−1
αl(z−z0)l
dz2+ ∞
P
l=−1
βl[k(z)−k(z0)]l
dk(z)2be the local representation of the quadratic differentialΦ in terms of the local param-e eter ez∈ V(ze0, r), with αl, βl ∈Cfor every l≥ −1, α−1 6= 0, β−1 6= 0. Since
|ϕ(z)| ≤ |α−1|r−1 + const. ⇒ RR
V
|ϕ(z)|dxdy =R2π 0
ϕ(z0+reiθ)
rdθ ≤R2π 0
(|α−1|+r·const.)dθ < ∞. In the same way, RR
V0
|ϕ(w)|b dudv < ∞, where V0=V(w0, r), so that, A
Φe(Ve)<∞.
REFERENCES
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[3] I. Bˆarz˘a,Calculus on non orientable Riemann surfaces.Libertas Math.15(1995), 1–45.
[4] I. Bˆarz˘a,Fonctions et formes diff´erentielles de type m´eromorphe sur les surfaces de Rie- mann non orientable. II. Rev. Roumaine Math. Pures Appl.28(1983), 933–948.
[5] M. Ro¸siu, The metric and the area associated with a quadratic differential on a Klein surface. Rev. Roumaine Math. Pures Appl.44(1999), 645–651.
[6] K. Strebel,Quadratic Differentials. Ergeb. Math. Grenzgeb. (3). Springer-Verlag, Berlin, 1984.
Received 15 May 2008 University of Craiova
Faculty of Mathematics Str. Al.I. Cuza nr. 13
Craiova, Romania
