PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 63 (77), 1998, 47–54

SOME REMARKS ON THE NONORIENTABLE SURFACES

I. Bârză, D. Ghişa, S. Ianuş

Communicated by Miroljub Jevtić

Abstract. It is a classical result of F. Klein that for any nonorientable (regular enough) surface X there is an orientable surface O_{2} and an involution without fixed point of O_{2} such that X is isomorphic to the quotient space of O_{2} with respect to the group generated by the respective involution.

In this note a reinforcement of the Klein’s result is presented and the effect on the vector bundle of covariant tensors of second order on X produced by that involution is studied.

The projection p : O_{2} ⟶ X induces an isomorphism between the vector space of covariant tensors of order two on X and the space of covariant symmetric tensors of order two on O_{2} which are invariant with respect to the given involution.

By a nonorientable surface (NOS) X we understand in this paper a differentiable 2-dimensional connected manifold which does not admit a orientation. It is convenient to endow such a manifold with a so-called dianalytic structure, i.e., a structure defined by an atlas A such that every transition function is a conformal mapping or a mapping whose complex conjugate is conformal (called anticonformal mapping). It is evident that A does not admit an analtic subatlas A_{1}, since then (X, A_{1}) would be an orientable (Riemann) surface. Still Felix Klein has foreseen the role of these surfaces in the complex analysis. Despite the contradiction in terms, Teichmüller called them nonorientable Riemann surfaces. The name nonorientable Klein surface seems to become predominant (see [7]).

As in the orientable case, we can construct the universal covering

π : X̂ ⟶ X

where X̂ is a simply connected, hence orientable surface. This covering is a Galois covering, i.e., the automorphisms group G of X̂ which conserve the fibers and acts transitively on each fiber. It contains necessarily conformal, but also anticonformal mappings.

AMS Subject Classification (1991): Primary 30F50

If we denote by G_{1} the subgroup of G formed with the conformal transformations, then we can easily note that for A ∈ G \ G_{1} we have G \ G_{1} = AG_{1}, i.e., G is the disjoint union

G = G_{1} ∪ AG_{1}.

Moreover, G_{1} is itself transitive and therefore

O_{2} := X̂/G_{1}

is a Riemann surface whose universal covering is of the form π_{1} : X̂ ⟶ O_{2}. For a fixed A ∈ G \ G_{1} we consider the mapping

h_{A} := h : O_{2} ⟶ O_{2}

defined by h(û) = Âu, where û is the fiber of an u ∈ X̂ by the action of G_{1}. It is evident that h is correctly defined and it represents a fixed point free antianalytic involution of O_{2}.

There is a canonical identification of X with O_{2}/⟨h⟩, where ⟨h⟩ is the two elements group generated by h.

Reciprocally, if there is a fixed point free antianalytic involution h of the Riemann surface O_{2}, then the quotient space O_{2}/⟨h⟩ is a NOS endowed with a dianalytic structure.

On every NOS of class C^{1} we can define the Riemannian metric

ds = λ|dz + μdz̄|,

where λ is a positive function [6]. If μ(z) ≡ 0, the corresponding parameter and metric are called isothermal. It is easily seen that an isothermal metric on a NOS X induces a dianalytic structure on X.

Indeed, if the parameters z and z_{1} correspond to two overlapping local charts, then in their common domain,

ds = λ|dz| = λ_{1}|dz_{1}|,

and therefore for the transition function z → z_{1} we have:

|dz_{1}| = \frac{λ}{λ_{1}} |dz|.

But dz_{1} = \frac{∂z_{1}}{∂z} dz + \frac{∂z_{1}}{∂z̄} dz̄ and, by consequence, one of the two partial derivatives

\frac{∂z_{1}}{∂z} or \frac{∂z_{1}}{∂z̄} must be identically zero. It results that z → z_{1} is a conformal or an anticonformal mapping.

Reciprocally, if the NOS X is endowed with a dianalytic structure, i.e., for

every transition function z → z_{1} we have dz_{1} = \frac{∂z_{1}}{∂z} dz or dz_{1} = \frac{∂z_{1}}{∂z̄} dz̄, then there is a positive function ν such that |dz_{1}| = ν|dz|. By consequence,

ds = λ_{1}|dz_{1} + μ_{1}dz̄_{1}| = λ_{1}|dz_{1}| |1 + μ_{1} \frac{dz̄_{1}}{dz_{1}} | = λ_{1}ν|dz| |1 + μ\frac{dz̄}{dz}|,

which is possible if and only if μ = μ_{1} = 0, i.e., if and only if ds is isothermal.

The following theorem represents a reinforcement of a classical result due to Klein.

Theorem 1. If X is a NOS endowed with a dianalytic structure and if f : W ⟶ X is a covering of X with W orientable, then W admits a structure of Riemann surface with respect to which the projection f is dianalytic. For every symmetry k of W (if it exist any), there is a dianalytic atlas of the quotient space W/⟨k⟩ such that the canonical projection W ⟶ W/⟨k⟩ is a dianalytic function. Moreover, W/⟨k⟩ is a nonorientable cover of X. If the dianalytic structure of X has been induced by the isothermal metric ds = λ|dz|, then the analytic structure of W

is induced by the isothermal metric dσ = λ_{1}|dw|, where λ_{1}(w) = λ(z(w))|\frac{∂z}{∂w} + \frac{∂z}{∂w̄} |.

Proof. Let A be a dianalytic atlas on X. For every q ∈ W let us chose a neighborhood U_{q} of q such that f|_{U_{q}} is a one to one mapping and f(U_{q}) is included in the domain of a local parameter h_{p} on X, where p = f(q). It is easily seen that the atlas

B = {U_{q}, h_{p} ∘ (f|_{U_{q}}) | q ∈ W}

defines a dianalytic structure on W and f is a dianalytic function with respect to this structure.

Indeed, given the fact that for two charts (V_{1}, g_{1}), (V_{2}, g_{2}) in B with g_{i} = h_{i} ∘ (f|V_{i}), i = 1, 2, we have g_{2} ∘ g_{1}^{−1} = h_{2} ∘ h_{1}^{−1} in g_{1}(V_{1} ∩ V_{2}) the dianalyticity of the atlas B and of the function f is granted. We can enlarge B, if necessary, including for every chart, the chart defined by the complex conjugate parameter. Given the fact that W has been supposed orientable, there is a partial atlas of B previously enlarged which contains only charts compatible with a given orientation of W. This atlas defines an analytic structure on W.

Let π_{1} : W ⟶ W/⟨k⟩ be the canonical projection. For every p ∈ W/⟨k⟩ we chose q ∈ W such that p = π_{1}(q) and a neighborhood U_{q} of q with the property that π_{1}|U_{q} is a one to one mapping and U_{q} is contained in the domain of a local parameter h_{q} on W. A local chart at p is (π_{1}(U_{q}), h_{q} ∘ (π_{1}|_{U_{q}})^{−1}). It is easily seen that the set of these charts forms a dianalytic atlas on W/⟨k⟩ and that π_{1} is a dianalytic function with respect to this atlas. Taking into account that the partial order in the set of the coverings endows this set with a lattice structure and the fact that ⟨h⟩ contains only two elements, it results that we have the following chain of coverings

Let (V, g) be a local chart on W. Then V = U_{q_{0}} for a certain q_{0} ∈ W and g = h_{p_{0}} ∘ (f|_{V}), for a local parameter h_{p_{0}} on X, with p_{0} = f(q_{0}). If z = h_{p_{0}}(p), p ∈ f(V) and w = g(q), q ∈ V, then w → z is a conformal mapping or

anticonformal mapping. We define λ_{1}(w) = λ(z(w))|\frac{∂z}{∂w} | if w → z is a conformal

mapping and λ_{1}(w) = λ(z(w))|\frac{∂z}{∂w̄} | if w → z is an anticonformal mapping. In both cases λ_{1}(w)|dw| = λ(z)|dz|. It can be easily checked that λ_{1}(w)|dw| is symmetric and it induces the previously mentioned conformal structure of W.

In [1] and [2] a theory of integration on NOS endowed with dianalytic structures has been developed, in which the orientable surface O_{2} plays an essential role.

In the following we will define the coverings of certain geometric objects on X by O_{2} and we will study the effect on them produced by an antianalytic involution h of O_{2} .

LetS^{2}(T^{∗}X) = ⋃_{x∈X} S^{2}(T_{x}^{∗}X) where T_{x}^{∗}X is the cotangent space at the point x and S^{2}(T_{x}^{∗}X) is the space of covariant tensors of second order at x. The triplet

E = (S^{2}(T^{∗}X), π, X)

where π is the canonical projection, is a vector bundle.

Analogously we define the vector bundle Ẽ = (S^{2}(T^{∗}O_{2}), π̃, O_{2}).

Let us denote by S(Ẽ) the space of differentiable sections of class C^{∞} in the vector bundle Ẽ. A section θ ∈ S(Ẽ) is a covariant symmetric tensor field of order two on O_{2}. We say that θ is h-invariant (respectively h-antiinvariant) if h^{∗}θ = θ (respectively h^{∗}θ = −θ), where h^{∗} is the transformation induced by h on S(Ẽ).

Let us denote

S_{s}(Ẽ) := {θ ∈ S(Ẽ)|h^{∗}θ = θ} and S_{a}(Ẽ) := {θ ∈ S(Ẽ)|h^{∗}θ = −θ}.

Obviously, S(Ẽ) is a C^{∞}(O_{2}, ℝ)-module.

Let us define the symmetrisation and antisymmetrisation operators S and A on S(Ẽ) respectively by:

Sθ := \frac{1}{2}(θ + h^{∗}θ); Aθ := \frac{1}{2}(θ − h^{∗}θ).

It can be easily checked that S and A are orthogonal projectors on S(Ẽ), i.e., they fulfill the following equalities:

S ∘ S = S; A ∘ A = A; S ∘ A = A ∘ S = 0; S + A = Id_{S(Ẽ)} and S − A = h^{∗}.

(h^{∗} is an involution).

Let us notice that S_{s}(Ẽ) and S_{a}(Ẽ) are invariant spaces with respect to the operators S and A. Moreover, we have the following result:

Theorem 2. a) S_{s}(Ẽ) and S_{a}(Ẽ) are vector subspaces of S(Ẽ);

b) S(Ẽ) = S_{s}(Ẽ) ⨁ S_{a}(Ẽ);

c) The projection p : O_{2} ⟶ X induces an isomorphism between the vector space of covariant tensors of order two on X, on the space S_{s}(Ẽ).

Proof. The points a) and b) are obvious. To prove c), let us denote

α : S(E) ⟶ S_{s}(Ẽ)

an application defined punctually by α(θ) = θ̃, where θ̃ is chosen such that for every u ∈ O_{2},

(1)  θ̃_{u}(Ã, B̃) = θ_{x}(p_{∗}Ã, p_{∗}B̃),

where x = p(u) and p is the dual application of p^{∗}, p being a local diffeomorphism. Then

p_{∗} : T_{u}O_{2} ⟶ T_{x}(X)

is an isomorphism, therefore α is correctly defined and it is a one-to-one application.

Let us show now that θ̃ is h-invariant, i.e., it belongs to S_{s}(Ẽ). We have p ∘ h = p and this relation implies

(2)  p_{∗} ∘ h_{∗} = p_{∗}

where p_{∗} is the differential.

Let p^{∗} be the dual of p_{∗}. Then, for every u ∈ O_{2} and arbitrary Ã, B̃ in T_{u}O_{2}, we have

(h^{∗}θ̃)_{u}(Ã, B̃) = θ̃_{hu}(h_{∗}Ã, h_{∗}B̃) = θ̃_{u}(h_{∗}Ã, h_{∗}B̃)
= θ_{x}(p_{∗}h_{∗}Ã, p_{∗}h_{∗}B̃) = θ_{x}(p_{∗}Ã, p_{∗}B̃),

where x = p(u) = p(hu). Hence, indeed h^{∗}θ̃ = θ̃.

It remains to prove that α is onto. Let θ̃ ∈ S_{s}(Ẽ) be arbitrary. We chose θ in S(E) such that for every couple A, B ∈ T_{x}X,

(3)  θ_{x}(A, B) = θ̃_{u}(Ã, B̃),

where x = p(u) and where Ã, B̃ ∈ T_{u}O_{2} are uniquely defined by the isomorphism p_{∗} such that p_{∗}Ã = A and p_{∗}B̃ = B.

Let us show that θ_{x} is correctly defined, i.e., it does not depend on the choice of u for which x = p(u). Indeed, θ being h-invariant, we have

(4)  θ̃_{u}(Ã, B̃) = θ̃_{hu}(h_{∗}Ã, h_{∗}B̃).

On the other hand, taking into account (2), we have

(5)  p_{∗}h_{∗}Ã = p_{∗}Ã = A and p_{∗}h_{∗}B̃ = p_{∗}B̃ = B.

The relations (3), (4) and (5) imply

(6)  θ_{x}(A, B) = θ̃_{hu}(h_{∗}Ã, h_{∗}B̃),

which shows that θ_{x} is indeed correctly defined. The ℝ-linearity of α implies that it is an isomorphism.

We recall that a Riemannian metric g on X is a symmetric, doubly covariant, positively defined tensor field at every point of X. The set of Riemannian metrics on X is denoted by M(X). Every metric g ∈ M(X) is a section of the vector bundle E such that at every point x ∈ X,

g(x) ∈ J_{x}(X) ⊂ S^{2}(T^{∗}X),

where J_{x}(X) is the convex cone of bilinear symmetric and positively defined forms on T_{x}(X).

The previous theorem implies that there is an isomorphism between the cone of h-invariant Riemannian metrics on O_{2} and the cone M(X). From the same theorem it results that every Riemannian metric g̃ on O_{2} has a canonical decomposition

g̃ = g̃_{s} + g̃_{a}

where g̃_{s} is a h-invariant Riemannian metric and g̃_{a} ∈ S_{a}(Ẽ).

The involution h is an isometry with respect to the component g̃_{s} of g and the projection p is a local isometry between the Riemannian manifolds (O_{2}, g̃_{s}) and (X, g), where g is the element of M(X) corresponding to g_{s} by the isomorphism mentioned in the previous theorem.

Example 1. Let M = {z = x + iy | y > 0, −1/2 < x ≤ 1/2, |z| > 1} ∪ {e^{iϕ} | π/3 ≤ ϕ ≤ π/2}. For every τ ∈ M, let us denote by Σ_{τ} the lattice {m+nτ | m, n ∈ ℤ} identified with the group of translations of the complex plane ℂ, z ⟶ z+m+nτ.

Let us denote by G the group generated by the mappings U, V : ℂ ⟶ ℂ defined by

U(z) = z + τ; V (z) = z̄ + \frac{1}{2}.

Let G_{1} be the subgroup of G formed with the conformal mappings. It can be easily checked that G consists of all translations with elements from Σ_{τ} and by consequence the orbit space ℂ/G_{1} := T (τ) is a torus.

It is known (see [4]) that every torus is conformally equivalent to a torus T (τ) and that T (τ) is symmetric if and only if Re (τ) = 0 (see [3] and [7]).

If T (τ) is symmetric then it represents the orientable double cover of the Klein bottle ℂ/G.

The Euclidian metric g = |dz|^{2} in the complex plane ℂ being invariant with respect to both U and V, is invariant with respect to G_{1}, so it induces a Riemannian metric on the torus ℂ/G_{1}. Such a metric can be defined by g̃(ż) = |dz|^{2}, where ż is the orbit of z with respect to G_{1}:

ż = z + Σ_{τ} = {z + m + nτ | m, n ∈ ℤ}.

It is obvious that h : T (τ) ⟶ T (τ) defined by

h(ż) := z̄ + \frac{1}{2} + Σ_{τ}

is an antianalytic involution. The h-invariant component of g̃ is given by

g̃_{s}(ż) = \frac{1}{2} [g̃(ż) + g̃(h(ż))] = \frac{1}{2} [|dz|^{2} + |d(z̄ + m + nτ)|^{2}]

= \frac{1}{2}(|dz|^{2} + |dz̄|^{2}) = |dz|^{2} = g̃(ż).

By consequence g̃ is a symmetric metric. We should have expected this result since the Riemannian metric induced by g on the Klein bottle ℂ/G should be the same, given its obvious invariance with respect to G.

Example 2. Let R > 1 and let A_{R} be the annulus {z ∈ ℂ | 1/R < |z| < R}. Let h : A_{R} ⟶ A_{R} defined by h(z) = −1/z̄. (A_{R}, h) is a symmetric Riemann surface and the quotient space A_{R}/⟨h⟩ is a Möbius strip. The Euclidian metric

g̃ = ds^{2} = dx^{2} + dy^{2}

is not h-invariant. Its h-invariant component is

g̃_{s} = dσ^{2} = \frac{1}{4}(1 + \frac{1}{x^{2} + y^{2}} )^{2}(dx^{2} + dy^{2}).

Indeed, with the notations z = x + iy and w = u + iv = −1/z̄ we have:

dσ = \frac{1}{2} (ds + ds ∘ h) that is dσ(z) = \frac{1}{2} (|dz| + |dw|) = \frac{1}{2}(1 + \frac{1}{|z|^{2}} )|dz|,

equality which means exactly the previous formula for g̃_{s}.

Example 3. One denotes, as usual, by ℂ̂ the extended complex plane, identified with the Riemann sphere. Let h : ℂ̂ ⟶ ℂ̂ defined by h(z) = −1/z̄ (with the conventions 1/0 = ∞ and 1/∞ = 0). The pair (ℂ̂, h) is a symmetric Riemann surface and the quotient space ℙ^{2} := ℂ̂/⟨h⟩ is the real projective plane.

The natural (Riemannian) metric on ℂ̂ is the spheric metric g̃ = da^{2} defined by

da(z) = \frac{|dz|}{1 + |z|^{2}} .

The h-invariant component of g̃ is g̃_{s} = dσ^{2}, where

dσ(z) = \frac{1}{2} (\frac{|dz|}{1 + |z|^{2}} + \frac{|dh(z)|}{1 + |h(z)|^{2}} ) = da(z).

This last equality means that the spheric metric on ℂ̂ is h-invariant and therefore

g̃ = g̃_{s} = \frac{dx^{2} + dy^{2}}{(1 + x^{2} + y^{2})^{2}} .

Acknowledgements. We are grateful to the referee for the valuable suggestions which had as effect an improved final version of this paper. Particularly we put into evidence a more obvious connection between Example 1 and the topics of the paper and we added a new example.

References

[1] I. Bârză, Integration on Non Orientable Riemann Surfaces. Editors: S.Dimiev and K. Sekigawa, Almost Complex Structures, World Scientific, Singapore–New Jersey–London–Hong Kong, 1995.

[2] , Calculus on Non Orientable Riemann Surfaces, Libertas Mathematica 15 (1995), 1–45.

[3] , The dianalytic morphisms of the Klein bottles, in: I. Laine, S. Rickman and T. Sorvali (Eds), Complex Analysis, Juensuu 1987, Lecture Notes in Mathematics 1351, Springer-Verlag, Berlin–Heidelberg–New York–London–Paris–Tokyo, 1987.

[4] R.C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1972.

[5] J. Lehner, Discontinuous Groups and Automorphic Functions, Amer. Math. Soc., Rhode Island, 1964.

[6] O. Lehto, Univalent functions and Teichmüller Spaces, Springer-Verlag, 1987.

[7] M. Seppälä, T. Sorvali, The Geometry of Riemann Surfaces and Teichmüller Spaces, North-Holland Mathematics Studies 169, North Holand, Amsterdam, 1992

[8] K.J. Sibner, Symmetric Fuchsian Groups, Amer. J. Math. 90 (1968), 1237–1259.

Ilie Bârză Department of Mathematics Luleå University of Technology S-971 87 Luleå Sweden Dorin Ghişa Department of Mathematics Glendon College York University Toronto Canada M4N3M6.

Stere Ianus P.O. Box 10-88 Of. Postal 10 R-72200 Bucharest Romania

(Received 31 01 1997) (Revised 21 03 1998)