Vector fields on nonorientable surfaces

PII. S0161171203204038 http://ijmms.hindawi.com © Hindawi Publishing Corp.

VECTOR FIELDS ON NONORIENTABLE SURFACES

ILIE BARZA and DORIN GHISA

Received 4 April 2002

A one-to-one correspondence is established between the germs of functions and tangent vectors on a NOSXand the bi-germs of functions, respectively, elemen-tary fields of tangent vectors (EFTV) on the orientable double cover ofX. Some representation theorems for the algebra of germs of functions, the tangent space at an arbitrary point ofX, and the space of vector fields onXare proved by us-ing a symmetrisation process. An example related to the normal derivative on the border of the Möbius strip supports the nontriviality of the concepts introduced in this paper.

2000 Mathematics Subject Classification: 30F15, 30F50.

1. Introduction. It was Felix Klein who first had the idea that, in order to do more than topology on NOS, they should be endowed with dianalytic struc-tures. This was exploited much later by Schiffer and Spencer [13], who under-took, for the first time, the analysis of NOS. In the monograph [2], the category of Klein surfaces was introduced, and, in [3], Andreian Cazacu clarified com-pletely the concept of morphism of Klein surfaces. She proved that the interior transformations of Stoïlow [14], on which he had based his topological prin-ciples of analytic functions, are essential in the definition of the morphisms of Klein surfaces. The category of Riemann surfaces appears as a subcategory of the category of Klein surfaces. Moreover, since the last one contains border free, as well as bordered surfaces, it also contains the subcategory of bordered Riemann surfaces. For this reason, if the contrary is not explicitly mentioned, by a surface, Riemann surface, or Klein surface, we will understand in this paper a bordered or a bordered free surface, respectively Riemann surface or Klein surface.

In this paper, the methods used in [4,5,6,7,8] are extended to the study of vector fields on NOS. The extension required new concepts and techniques. The first three sections present the general frame of the problems we are dealing with and which are based on a result of Felix Klein (Theorem 2.1).

Insection 4, the concept of bi-germ of functions on a symmetric Riemann surface is defined.

Section 6deals with the study of vector fields on symmetric Riemann sur-faces and on NOS. A symmetrisation process, dual to the symmetrisation of bi-germs, is defined. The connection between the vector fields on a nonori-entable Klein surface and the vector fields on its orinonori-entable double covering is given (Theorem 6.2).

InSection 7, we give the detailed construction of the unit normal vector field to the border of the Möbius strip.

2. Prerequisites. LetXbe a surface. Ifp∈X, a chart atpis a couple(U , ϕ)

consisting of an open neighborhoodUofpand a homeomorphismϕ:U→V

whereVis a relatively open subset of the closed upper half-plane

C+_{:=z∈C|Im(z)≥0. (2.1)}

Adianalytic atlasonXis a family of charts᐀= {(Uα, ϕα)|α∈I}such that

X=α∈IUαand, for every two charts(Uα, ϕα)and(Uβ, ϕβ)∈᐀, the transfer

function ϕβ◦ϕ−α1 is either conformal or anticonformal on each connected

component ofϕα(Uα∩Uβ).

The couple (X,᐀) is called aKlein surfaceif᐀is amaximal dianalytic atlas onX.

We notice that a Klein surface can be an orientable as well as a nonorientable surface. However, the universal coveringXof any Klein surfaceX, being sim-ply connected, is necessarily orientable. Consequently, there is a conformal structure onXthat makes the canonical projection

p:X →X (2.2)

locally analytic or antianalytic in terms of each chart; we call such a map dian-alytic.

LetᏳbe the group of covering transformations ofXoverX. Some elements ofᏳmight be conformal and some anticonformal mappings ofX on itself. If

Xis a nonorientable Klein surface, thenᏳcontains necessarily anticonformal mappings. Suppose that this is the case and letᏳ1be the subgroup ofᏳformed with all its conformal elements. Then, the orbit space

ᏻ2:=ᏻ2(X):=X/Ᏻ1 (2.3)

is an orientable surface. Moreover,ᏻ2has a unique analytic structure, which makes the canonical projection

an analytic mapping. Ifg∈Ᏻ\Ᏻ1, thenᏳ=Ᏻ1∪gᏳ1andᏳ1∩gᏳ1= ∅. There-fore, we can define

h:ᏻ2 →ᏻ2 (2.5)

byh(x) :=gx, whereyis the fiber ofy∈Xwith respect toπ:X→ᏻ2, that is,

y= {gy|g∈Ᏻ1}.

Obviously,his anantianalytic involutionsinceg∈Ᏻ\Ᏻ1and g2∈Ᏻ1. It is fixed point freesinceh(x) =xwould implyg∈Ᏻ1, contrary to the hypothesis. Klein calledsymmetryan involution of a Riemann surface of the type pre-viously described. There are lots of other types of symmetries (see, e.g., [1, 2,10,11], and so on); however, in this paper, we will only use the concept of symmetry in the sense of Klein.

The following theorem has its origins in Klein’s works.

Theorem2.1. If (ᏻ2,h)is a symmetric Riemann surface and if h is the two-element group generated byh, then the covering projection

q:ᏻ2 →ᏻ2/h (2.6)

induces a structure of nonorientable Klein surface onᏻ2/h with respect to whichqis a morphism of Klein surfaces.

Conversely, if X is a nonorientable Klein surface, there exists a symmetric Riemann surface(ᏻ2,h)such thatXis dianalytically equivalent toᏻ2/h .

We callᏻ2:=ᏻ2(X)the orientable double cover ofX, and we notice that this concept is different from those defined by other types of symmetries.

3. Vector fields. LetXbe a Klein surface. ForP∈X, we denote byᏯ∞_{(P )the}

algebra of germs of complex functions defined on neighborhoods ofPwhose real and imaginary parts are functions of class Ꮿ∞. To every differentiable curveα:]−ε, ε[→X(ε >0 andα(0)=P ), we associate the operator

→_X

P ,α=→X P:Ꮿ∞(P ) →C (3.1)

defined by

→

XP(f ):=

d(f◦α)

dt (0), (3.2)

for everyf ∈Ꮿ∞_{(P ). The map} →_{X P is a linear operator and satisfies Leibniz}

rule for the derivation of the product of functions. This operator→XP is called

a tangent vector atPtoX. The vector spaceTPXof all tangent vectors atPto

If(U , ϕ)is a chart atPand ifϕ(Q)=(x, y)are the local coordinates inU , ϕ(P )=(0,0),thenx→ϕ−1_{(x,0)andy→ϕ}−1_{(0, y),the so-called coordinate} curves atP, determine a basis forTPX

Ꮾ=Ꮾ(U ,ϕ):=

_∂

∂x

P ,

_∂

∂y

P

(3.3)

defined by

∂ ∂x

P

(f ):=∂

f◦ϕ−1

∂x

ϕ(P ),

∂ ∂y

P

(f ):=∂

f◦ϕ−1

∂y

ϕ(P ),

(3.4)

where∂/∂xand∂/∂yare the usual partial derivatives in the complex planeC. (See, e.g., [9,11,12].)

Every→X P∈TPXhas the form

→

XP=X1(P )

∂ ∂x

P

+X2(P )

∂ ∂y

P

, (3.5)

whereX1_{(P )andX}2_{(P )are complex coefficients.} Thetangent bundleofXis the set

TX:=

P∈X

TPX. (3.6)

It is well known thatTXhas a canonical structure of differential manifold of real dimension 4.

IfAis a subset ofX, avector fieldonAis any function

→_{X:A →TX (3.7)}

such that→X (P ) :=→X P∈TPXfor everyP∈A.

In particular, for a chart(U , ϕ), we have the vector fields∂/∂xand∂/∂y

∂ ∂x,

∂

∂y :U →TX, (3.8)

P→(∂/∂x)(P ):=(∂/∂x)P, respectively,P→(∂/∂y)(P ):=(∂/∂y)P as defined

by (3.4).

If→X :A→TXis a vector field of classᏯ∞, then the restriction of→X toUhas the form

→

XU=X1 ∂ ∂x+X

2 ∂

∂y, (3.9)

We will study in more detail the vector fields onXandᏻ2:=ᏻ2(X).

4. Bi-germs of functions onᏻ2. In all what follows,Xwill be a nonorientable Klein surface,ᏻ2:=ᏻ2(X)will be the corresponding double cover ofX,hwill be the symmetry ofᏻ2as inTheorem 2.1, andXwill be identified with the orbit spaceᏻ2/h . The mapq:ᏻ2→ᏻ2/h will be the canonical projection.

Different parametric disks or half-disks, if centered on the borderᏮ(X), will beevenlycovered byq. Then, the inverse image byqof such a parametric disk (half-disk)Dconsists of a pairDandhDof symmetric disks (half-disks) onᏻ2

q−1 D=D∪hD, D∩hD= ∅. (4.1)

This brings us to the necessity of considering germs ofᏯ∞-functions on sym-metric subsets ofᏻ2as defined next.

ForP∈ᏻ2, we denote by{P;hP}the orbit ofP with respect toh and by

P=hPthe image byqofPandhPonX:=ᏻ2/h .

Sinceqis a dianalytic map, ifF:X→Cis a function of classᏯ∞_{or ifF is}

harmonic or dianalytic, then so isF◦qand vice versa.

A subsetΣofᏻ2will be calledsymmetricif it ish-invariant, that is,hΣ= Σ. Obviously, the restriction ofhto any symmetric setΣcontinues to be an invo-lution.

In the study of local properties of functions at pointsP∈X, we only need symmetric neighborhoods of{P;hP}. Such a neighborhood is, by definition, a subsetΣsuch that there is a parametric disk (half disk)Dcentered atP,D⊂Σ. Implicitly,hDis a parametric disk (half disk) centered athPandD∪hD⊆Σ.

IfᏰis the family of parametric disks (half-disks) centered atP, the family {D∪hD|D∈Ᏸ}is a fundamental system of symmetric neighborhoods of {P;hP}. IfD∩hD= ∅, then

D:= P|P∈D (4.2)

is a parametric disk (half-disk) onX, and it is evenly covered byq. LetᏯ∞(P )

be the algebra of germs of functions of classᏯ∞_{atP, and letF}

P∈Ꮿ∞(P ) .

We will denote byF any class representative of the germ F_P, and we will consider that the domain ofF is a disk or a half-disk as previously defined (i.e., evenly covered byq). The functionsfP:=F◦qDandfhP :=F◦qhD(qDand

qhDbeing the restrictions ofqtoDand, respectively, tohD) are functions of

classᏯ∞. Having in mind the conventional notations, we can write

Thus, the germF_Pinduces onXthe two-element set of germs{fP;fhP}. This

set is anonorderedpair of elements of the setᏯ∞(P )∪Ꮿ∞_{(hP ), with the pair}

having an element in each member of this union. With this preparation in mind, we can give the following definition.

Definition4.1. A bi-germ of classᏯ∞at{P;hP}is a function

χ:P;hP →Ꮿ∞_{(P )∪Ꮿ}∞_{(hP ) (4.4)}

such thatχ(Q)∈Ꮿ∞(Q)for everyQ∈ {P;hP}.

We denote byᏯ∞(P;hP )the set of bi-germs of classᏯ∞at{P;hP}.The alge-braic operations inᏯ∞_{(P;hP )are the natural operations induced by the algebra}

Ꮿ∞_{(Q), that is, for arbitraryχ, λ∈Ꮿ}∞_{(P;hP ),Q∈ {P;hP}, anda∈C,}

(χ+λ)(Q):=χ(Q)+λ(Q),

(χλ)(Q):=χ(Q)λ(Q),

(aχ)(Q) =aχ(Q).

(4.5)

Obviously,Ꮿ∞(P;hP )with these operations is a commutative algebra with unit. We call itthe algebra of bi-germs of classᏯ∞_{at {P;hP}.}

Remark 4.2. The algebra Ꮿ∞_{(P;hP ) is not the product algebra Ꮿ}∞_{(P )×}

Ꮿ∞_{(hP )since the last one consists of ordered pairs of germs.}

It is convenient to denote the bi-germχ∈Ꮿ∞_{(P;hP )by{χP;χhP}. The two}

germsχP ∈Ꮿ∞(P )andχhP ∈Ꮿ∞(hP )are different, and, generally, there is no

connection between them. With these notations, the algebraic operations in Ꮿ∞_{(P;hP )become}

χP;χhP

+λP;λhP

=χP+λP;χhP+λhP

,and so on. (4.6)

The pullback algebra isomorphism of Ꮿ∞(hP ) onto Ꮿ∞(P ) induced by h is usually denoted byh∗P

h∗_P :Ꮿ∞_{(hP ) →Ꮿ}∞_{(P ), h}∗

P(f ):=f◦h, (4.7)

(wheref◦his the germ atP off◦hhD,Dbeing the domain off). Obviously,

the inverse ofh∗_P ish∗_hP :Ꮿ∞_{(P )→Ꮿ}∞_{(hP ). It is also obvious thath}∗

P andh∗hP

The involutionh:{P ,hP} → {P ,hP}also induces a pullback map defined as follows and denoted, to avoid confusions, byhb

hb_:Ꮿ∞_{(P;hP ) →Ꮿ}∞_{(P;hP ), h}b_χ

P;χhP:=h∗P

χhP;h∗hP

χP. (4.8)

The algebraic properties ofh∗P imply thathbisan involutive algebra

isomor-phism.

We now considerF∈Ꮿ∞(P ). It induces the bi-germ

qb_{(F ):=q}∗

P(F );q∗hP(F )

∈Ꮿ∞_{(P;hP ). (4.9)}

In this way, we obtain the map

qb:Ꮿ∞ P →Ꮿ∞_{(P;hP ). (4.10)}

The main properties ofqb_{are given in the following proposition.}

Proposition4.3. (i)The mapqb_{is an injective algebra morphism;}

(ii) hb_◦qb_=qb_{, that is,q}b_ishb_{-left invariant;}

(iii) qb_(Ꮿ∞_{(P )) = {χ|h}b_(χ)=χ}.

Proof. The assertions (i) and (ii) are obvious. According to (ii),qb_(Ꮿ∞_{(P ))}

consists of bi-germs which arehb_{-invariant, that is,q}b_(Ꮿ∞_{(P )) ⊆ {χ|h}b_(χ)= χ}.

Reciprocally, letχ= {χP;χhP}be anhb-invariant bi-germ. We have the

fol-lowing equivalences:

hb(χ)=χ⇐⇒h∗P

χhP;h∗hP

χP=χP;χhP⇐⇒h∗P

χhP=χP,

h∗_hPχP=χhP⇐⇒h∗P

χhP=χP

(4.11)

(becauseh∗hP=(h∗P)−1).

IfDis a disk or (half-disk) centered atPsuch thatD∩hD= ∅, we consider the mapq−1

D, the inverse ofqD:D→D.

Let (q−D1)∗P be the pullback algebra isomorphism induced by q −1

D. With

h∗_P(χhP)=χP∈Ꮿ∞(P ), we get successively:

Ꮿ∞_{(P ) F:=q}−1

D

∗

P

χP=q−D1

∗

P

h∗P

χhP=q−D1)∗P◦h ∗

P

(χhP

=h◦q−D1

∗

P

χhP

=q−_hD1∗_PχhP

,

qb_{(F )=q}∗

P(F );q∗hP(F )

=χP;χhP

;

(4.12)

The main assertion of this section is that finding the germs ofᏯ∞-functions atPis equivalent to finding thehb_{-invariant elements ofᏯ}∞_{(P;hP ).}

We will study thehb_{-invariant bi-germs in the next section.}

5. Symmetrisation and antisymmetrisation. We consider an arbitrary bi-germχ∈Ꮿ∞_{(P;hP ). We define the bi-germ᏿χby}

᏿χ:=1₂χ+hb_χ=

₁

2

χP+h∗P

χhP

;1 2

χhP+h∗hP

χP

. (5.1)

It is obvious that the equalityhb_{(χ)=χis equivalent to᏿χ=χ, that is,h}b

and ᏿ have the same set of fixed points. Thus, the differenceχ−᏿χgives an “estimation” of the deviation ofχfromhb_{-invariance. This prompts us to}

define a new operatorᏭ:Ꮿ∞(P;hP )→Ꮿ∞_{(P;hP )by}

Ꮽχ:=χ−᏿χ=1

2

χ−hbχ. (5.2)

Clearly, the equalityᏭχ=χis equivalent tohb_{(χ)= −χ. In this way, we are}

led to the following definition.

Definition5.1. A bi-germχ∈Ꮿ∞_{(P;hP )is called symmetric orh-invariant}

(resp., antisymmetric orh-antiinvariant) if and only ifhb_{(χ)=χ(resp.,h}b_(χ)=

−χ).

Thus, the symmetric bi-germs are the fixed points of᏿and the antisymmet-ric ones are the fixed points ofᏭ. We denote byᏵthe identity ofᏯ∞(P;hP ).

The following theorem gives the most important properties of the operators ᏿andᏭ.

Theorem5.2. (A)(᏿,Ꮽ)is a pair of orthogonal projectors ofᏯ∞(P;hP ), that is, the following five assertions hold:

(i) ᏿,Ꮽ:Ꮿ∞(P;hP )→Ꮿ∞_{(P;hP )are linear operators;}

(ii) ᏿◦᏿=᏿andᏭ◦Ꮽ=Ꮽ;

(iii) ᏿◦Ꮽ=Ꮽ◦᏿=0=the null operator of Ꮿ∞_{(P;hP );}

(iv) ᏿+Ꮽ=Ᏽ=the identity of Ꮿ∞(P;hP ); (v) ᏿−Ꮽ=hb_{=an involution of Ꮿ}∞_{(P;hP ).}

(B)With the notationsµs:=᏿µandµa:=Ꮽµforµ∈Ꮿ∞(P;hP ), we have

(vi) (µλ)s=µsλs+µaλa;

(vii) (µλ)a=µsλa+µaλs, for everyµ, λ∈Ꮿ∞(P;hP ).

Proof. (i) The linearity ofᏵandhb_{, together with the equalities᏿=(1/2)} (Ᏽ+hb_{)andᏭ=(1/2)(Ᏽ−h}b_{), implies that᏿andᏭare linear.}

(ii)᏿◦᏿=(1/2)(Ᏽ+hb_{)◦(1/2)(Ᏽ+h}b_{)=(1/4)(Ᏽ+2h}b_+hb_◦hb_{)=(1/4)(2Ᏽ+}

2hb_{)=᏿. Analogously,Ꮽ◦Ꮽ=Ꮽ.}

(iii)᏿◦Ꮽ=(1/2)(Ᏽ+hb_{)◦(1/2)(Ᏽ−h}b_{)=(1/4)(Ᏽ−h}b_◦hb_{)=0 sinceh}b _is

(iv) and (v) are clear. Now we shall prove (vi). (vi) The assertions

µsλs+µaλa=

1 4

µ+hb_µλ+hb_λ+1

4

µ−hb_µλ−hb_λ

=1₄µλ+µhb_λ+hb_µλ+hb_µhb_λ

+1₄µλ−µhb_λ−hb_µλ+hb_µhb_λ

=1 4

2µλ+2hb_µhb_λ

=1 2

µλ+hb_(µλ)

=᏿(µλ)=(µλ)s.

(5.3)

In a similar way, we prove (vii).

Having in viewProposition 4.3(iii) and the properties of᏿andᏭ, the follow-ing notations are justified:

Ꮿ∞

s (P;hP ):=

χ∈Ꮿ∞_{(P;hP )|h}b_χ=χ,

Ꮿ∞

a(P;hP ):=

χ∈Ꮿ∞_{(P;hP )|h}b_{χ= −χ.} (5.4)

Clearly, Ꮿ∞s (P;hP )=᏿Ꮿ∞(P;hP ) and Ꮿ∞a(P;hP )=ᏭᏯ∞(P;hP ). These

equal-ities are consequences of Theorem 5.2and of the equivalences hb_χ=χ

᏿χ=χandhb_{χ= −χᏭχ=χ.}

The proofs of the following two corollaries ofTheorem 5.2are straightfor-ward.

Corollary 5.3. (i) If µ, λ ∈ Ꮿ∞s(P;hP ) or µ, λ ∈ Ꮿ∞a(P;hP ), then λµ ∈

Ꮿ∞

s(P;hP );

(ii)Ifλ∈Ꮿ∞

s(P;hP )andµ∈Ꮿ∞a(P;hP ), thenλµ∈Ꮿ∞a(P;hP ).

Corollary 5.4. The set Ꮿ∞s(P;hP ) is a subalgebra of Ꮿ∞(P;hP ), but

Ꮿ∞

a(P;hP )is not; however,Ꮿ∞a(P;hP )is a subspace of Ꮿ∞(P;hP ).

The statement (ii) in the next theorem gives a representation of the algebra Ꮿ∞_{(P ) intoᏯ}∞_{(P;hP ).}

Theorem5.5. (i)The vector spaceᏯ∞_{(P;hP )has the following direct sum}

decomposition:

Ꮿ∞_{(P;hP )=Ꮿ}∞

s(P;hP )⊕Ꮿ∞a(P;hP ). (5.5)

(ii) The natural mapqb_:Ꮿ∞_{(P ) →Ꮿ}∞

s(P;hP )is an algebra isomorphism.

Proof. (i) We have seen that Ꮿ∞s (P;hP ) =᏿Ꮿ∞(P;hP ) and Ꮿ∞a(P;hP ) =

ᏭᏯ∞_{(P;hP ). It is also clear thatᏯ}∞

s(P;hP )∩Ꮿ∞a(P;hP )= {0}, that is, the single

For everyµ∈Ꮿ∞_{(P;hP ), we have}

µ=᏿µ+Ꮽµ (5.6)

with᏿µ=µs∈Ꮿ∞s(P;hP )andᏭµ=µa∈Ꮿ∞a(P;hP ). Thus, (i) holds true.

(ii) This part of the proposition is a consequence ofProposition 4.3(i), (iii), and the definition ofᏯ∞

s(P;hP ).

As C. Constantinescu noticed, we can prove that the subspaceᏯ∞a(P;hP )of

the algebraᏯ∞(P;hP )is isomorphic with the vector space of germs of functions of odd type in the sense of G. de Rham (see [5, page 27]).

6. Vector fields onXandᏻ2

Definition6.1. The vector field→Z :ᏻ2→Tᏻ2is called symmetric ordh -invariant (resp., antisymmetric ordh-antiinvariant) if and only if(dh)(→Z ) =→Z

(resp.,(dh)(→Z ) = −→Z ).

We denote, as usual, byχ(X)andχ(ᏻ2)the sets of vector fields onXand, respectively, ᏻ2 endowed with their algebraic structures (vector spaces and modules over the algebra of complex functions). We introduce the following notations:

χs

ᏻ2

:=→Y ∈χᏻ2

|(dh)→Y =→Y ;

χa

ᏻ2

:=→Y ∈χᏻ2

|(dh)→Y = −→Y .

(6.1)

The main result of this paper is assertion (iii) of the next theorem.

Theorem6.2. (i)χs(ᏻ2)andχa(ᏻ2)are subspaces ofχ(ᏻ2); (ii) the following direct sum decomposition holds true:

χᏻ2

=χs

ᏻ2

⊕χa

ᏻ2

; (6.2)

(iii) χ(X)is canonically isomorphic withχs(ᏻ2).

We present the proof of this theorem in several steps.

Step1(elementary fields of tangent vectors (EFTV) onᏻ2). Let→X P∈TPX.

The differentialsq_∗,P=dPqandq∗,hP=dhPqare bijective mappings. They lead

to the tangent vectors→Y P:=(dPq)−1(→X P)∈TPᏻ2and→Y hP:=(dhPq)−1(→X P)∈ ThPᏻ2. Thus, to every tangent vector →X P ∈TPX corresponds a nonordered

Definition6.3. An elementary field of tangent vectors (EFTV) onᏻ2(at the pair of symmetric points{P;hP}) is a function

→_Z:{P;hP}→T

Pᏻ2∪ThPᏻ2 (6.3)

such that→Z (Q) :=→Z Q∈TQᏻ2for everyQ∈ {P;hP}.

We denote byET{P;hP}ᏻ2 the set of all EFTV at {P;hP}, and the element

→

Z∈ET{P;hP}ᏻ2will be denoted by{→Z P;→Z hP}.

The setET_{P;hP}ᏻ2will be endowed with its natural structure of vector space: If→Z = {→ZP;→Z hP},W→ = {W→ P;W→ hP} ∈ET{P;hP}ᏻ2anda∈C, then→Z +W→ and

a→Z are defined as: →

Z+W→ :=→Z P+W→ P;→Z hP+W→ hP

,

a→Z :=a→Z P; a→Z hP

.

(6.4)

It is obvious that the differential

dh=h∗:Tᏻ2 →Tᏻ2 (6.5)

is a fixed-point freeinvolution. If→Z Q∈TQᏻ2, then(dh)(→Z Q)∈ThQᏻ2. We see that the restriction ofdhtoET{P;hP}ᏻ2(still denoted bydh) continues to be an involution. We have

dPh−1=dhPh:ThPᏻ2 →TPᏻ2. (6.6)

If →Z = {→Z P;→Z hP} ∈ ET{P;hP}ᏻ2, then (dh)(→Z ) = {(dh)(→Z hP);(dh)(→Z P)} ∈

ET{P;hP}ᏻ2.

We now formulate the counterpart ofDefinition 5.1.

Definition6.4. The EFTV→Z is called symmetric ordh-invariant (resp., an-tisymmetric ordh-antiinvariant) if and only if(dh)(→Z ) =→Z (resp.,(dh)(→Z ) = −→Z ).

If→X P∈TPX, then(dq)−1(

→

XP)= →

Y = {→Y P;→Y hP}, where→Y P =(dPq)−1(→X P) and→Y hP=(dhPq)−1(→X P).

Proposition6.5. (i)For every→X _P∈T_PX, the EFTV→Y =(dq)−1₍→_X

P)isdh

-invariant.

(ii)Conversely, if→Z = {→Z P;→Z hP} ∈ET{P;hP}ᏻ2isdh-invariant, then

dPq→Z P=dhPq→Z hP:=→X P∈TPX (6.7)

and(dq)−1₍→_X

P)=

→_Z.

The following simple proposition gives the rule of derivation on nonori-entable Klein surfaces.

Proposition6.6. Let→X P∈TPXandf∈Ꮿ∞(P ) . If

→

Y = {→Y P;→Y hP} =(dq)−1 (→X _P)and{fP;fhP}:=qb(f ) = {q∗P(f ) ;qh∗P(f ) } ∈Ꮿ∞(P ,hP ), then

→

XP f

=→Y P

fP

=→Y hP

fhP

. (6.8)

Proof. SincefP=h∗P(fhP)and→Y isdh-invariant, we get

→

YP

fP

=→Y P

h∗P

fhP

=dPh

→

YP

fhP

=→Y hP

fhP

. (6.9)

On the other hand,→Y P(fP)=→Y P(q∗P(f )) =(dq)(→Y P)(f ) =→X P(f ).

Step2(symmetrisation and antisymmetrisation of EFTV). We consider→Y = {→Y P;→Y hP} ∈ET{P;hP}ᏻ2. Generally, there is no connection between→Y Pand→Y hP.

We remember that(dPh)−1=dhPhand that the restriction ofdhtoET{P;hP}ᏻ2 (also denoted bydh) is an involution ofET{P;hP}ᏻ2.

As in the case of bi-germs, we define the operators ᏿,Ꮽ : ET{P;hP}ᏻ2 →

ET_{P;hP}ᏻ2by

᏿→Y :=1

2

→

Y+(dh)→Y ,

Ꮽ→Y :=1

2

→

Y−(dh)→Y

(6.10)

for every→Y ∈ET{P;hP}ᏻ2.

Clearly,(dh)(→Y ) =→Y if and only if᏿→Y =→Y and(dh)(→Y ) = −→Y if and only if Ꮽ→Y =→Y . Based onDefinition 6.1, we call the operators᏿andᏭthe operators of symmetrisationandantisymmetrisation, respectively, of EFTV onᏻ2.

We denote byᏵthe identity ofET_{P;hP}ᏻ2. Formulas (6.10) become

᏿ =1

2[Ᏽ+dh],

Ꮽ:=1

2[Ᏽ−dh].

(6.11)

The next theorem is similar toTheorem 5.2and the proof will be omitted.

Theorem6.7. The operators᏿andᏭare a pair of orthogonal projectors of the spaceET{P;hP}ᏻ2, that is, the following five assertions hold:

(iii) ᏿◦Ꮽ=Ꮽ◦᏿=0=the null operator of ET{P;hP}ᏻ2; (iv) ᏿+Ꮽ=Ᏽ=the identity of ET{P;hP}ᏻ2;

(v) ᏿−Ꮽ=dh=an involution of ET_{P;hP}ᏻ2.

Step3(a representation theorem forT_PX). We introduce the following no-tations:

ETs;{P;hP}ᏻ2:=

→

Y ∈ET{P;hP}ᏻ2|(dh)

→

Y=→Y ;

ETa;{P;hP}ᏻ2:=

→

Y ∈ET{P;hP}ᏻ2|(dh)

→

Y= −→Y .

(6.12)

Now, we can formulate the following theorem.

Theorem 6.8. (i) The sets ETs;{P;hP}ᏻ2 and ETa;{P;hP}ᏻ2 are subspaces of

ET{P;hP}ᏻ2;

(ii) the following direct sum decomposition holds true:

ET{P;hP}ᏻ2=ETs;{P;hP}ᏻ2⊕ETa;{P;hP}ᏻ2; (6.13)

(iii) the vector spacesT_PXandETs;{P;hP}ᏻ2are canonically isomorphic.

Proof. (i) The operators᏿andᏭare linear operators and᏿(ET{P;hP}ᏻ2)=

ETs;{P;hP}ᏻ2andᏭ(ET{P;hP}ᏻ2)=ETa;{P;hP}ᏻ2. This proves (i).

(ii) Obviously,ETs;{P;hP}ᏻ2∩ETa;{P;hP}ᏻ2= {0}and→Y =᏿→Y +Ꮽ→Y for every

→

Y ∈ET{P;hP}ᏻ2.

(iii) We have seen that if→Z = {→Z P,→Z hP} ∈ETs;{P;hP}ᏻ2 then (dPq)(→Z P)= (dhPq)(→Z hP). We define

qb:ETs;{P;hP}ᏻ2 →TPX (6.14)

byqb(→Z ) :=(dPq)(→Z P)=(dhPq)(→Z hP). The mappingqbis the natural

isomor-phism mentioned in (iii).

Step 4(global symmetrisation and antisymmetrisation). The global sym-metrisation operator᏿and the global antisymmetrisation operatorᏭ(or the operators ofdh-invarianceanddh-antiinvariance, respectively) are defined in a way similar to the case of EFTVs, namely,᏿,Ꮽ:χ(ᏻ2)→χ(ᏻ2), where

᏿→X :=1₂→X +(dh)→X ,

Ꮽ→X :=1₂→X −(dh)→X ,

(6.15)

for every→X ∈χ(ᏻ2)or, equivalently, in terms ofᏵanddh,

᏿:=1₂[Ᏽ+dh],

Ꮽ:=1₂[Ᏽ−dh].

If{P;hP}is an arbitrary pair of symmetric points onᏻ2, we denote by᏾the operator of restriction to{P;hP}of the vector fields→X ∈χ(ᏻ2)

᏾→X :=→X |{P;hP}=→X P;→X hP

. (6.17)

Indeed, the restriction of→X to the two-element set{P;hP}may be identified with the image by →X of this set, namely, with {→X P;→X hP}. This means that

᏾(→X ) ∈ET{P;hP}ᏻ2, that is,

᏾:χᏻ2

→ET{P;hP}ᏻ2. (6.18)

It is obvious that the following two diagrams are commutative:

χᏻ2

᏿

᏾

χᏻ2

᏾

ET{P;hP}ᏻ2 ᏿ ET{P;hP}ᏻ2,

χᏻ2

Ꮽ

᏾

χᏻ2

᏾

ET{P;hP}ᏻ2 Ꮽ ET{P;hP}ᏻ2.

(6.19)

As a consequence of the fact that dh:Tᏻ2→Tᏻ2 is an involution, we can formulate the following theorem, which is the globalisation ofTheorem 6.7. The proof will be omitted.

Theorem 6.9. The pair of operators(᏿,Ꮽ)defined by (6.15) is a pair of orthogonal projectors of the vector spaceχ(ᏻ2), that is,

(i) ᏿,Ꮽ:χ(ᏻ2)→χ(ᏻ2)are linear operators; (ii) ᏿◦᏿=᏿andᏭ◦Ꮽ=Ꮽ;

(iii) ᏿◦Ꮽ=Ꮽ◦᏿=0=the null operator ofχ(ᏻ2); (iv) ᏿+Ꮽ=Ᏽ=the identity ofχ(ᏻ2);

(v) ᏿−Ꮽ=dh=an involution ofχ(ᏻ2).

Step5(final remarks). We can now complete the proof ofTheorem 6.2. (i) The global operators᏿andᏭare linear operators on the vector space

χ(ᏻ2). Moreover,᏿χ(ᏻ2)=χs(ᏻ2)andᏭχ(ᏻ2)=χa(ᏻ2). The same conclusion follows directly from (6.1). However, these equalities highlight the fact that the elements of χs(ᏻ2) and χa(ᏻ2)appear, respectively, as thedh-invariant anddh-antiinvariant components of the elements ofχ(ᏻ2).

(ii) Clearly,→X ∈χs(ᏻ2)∩χa(ᏻ2)if and only if→X =→0 =thezerovector field onᏻ2. Since→X =᏿→X +Ꮽ→X for every →X ∈χ(ᏻ2)and since᏿→X ∈χs(ᏻ2)and Ꮽ→X ∈χa(ᏻ2), (ii) holds true.

(iii) LetM→ ∈χ(X)be an arbitrary vector field onXand letP∈X. According toProposition 6.5(i),→Y :=(dq)−1_(M)→ _{is adh-invariant EFTV onᏻ}

2. Since the h -orbits onᏻ2are either identical or disjoint and sinceᏻ2=P∈X.{P;hP}, we

can define→X ∈χs(ᏻ2)by its restrictions

→

X|{P;hP}:=(dq)−1M→ P

to the set of h -orbits {P;hP}. It is an easy exercise to check that the cor-respondence→M ↔→X is an isomorphism between the vector spacesχ(X)and

χs(ᏻ2).

7. The normal derivative to the border of the Möbius strip. As an appli-cation of the topics that we have dealt with so far, we present, in detail, the normal derivative to the border of the Möbius strip, which is a vector field defined on that border.

We consider the annulusARdefined as

AR:=

z∈C|_R1 ≤ |z| ≤R

, (7.1)

whereR >1 is fixed. The maph:AR→ARdefined by

hz:=h(z):= −1_z=w=u+iv (7.2)

for everyz=x+iy∈ARis a fixed-point free antianalytic involution ofARand

the orbit space

MR:=AR/h , (7.3)

whereh = {Id;h}is the two element group generated byh, is a Möbius strip. Ifz∈AR, itsh-orbit is the two-element set

z= {z;hz} =hz. (7.4)

Ifz=ρeiθ_{, 1/R≤ρ≤R, 0≤θ < π, theh-symmetric point ofzisw= −1/z=} (1/ρ)ei(θ+π )_.

Whenzruns over the line segment with end-pointsReiθ_and(1/R)eiθ_from Reiθ_{towards(1/R)e}iθ_{(i.e.,ρdecreases fromRto 1/R), its symmetricwruns}

on the line segment with endpoints(1/R)ei(θ+π ) _andRei(θ+π ) _{from the first}

point toward the second.

The border ofMRconsists of the orbitszsuch that|z| =Ror 1/R, that is,

∂MR=

Reiθ_;1

Re

i(θ+π )_{|0≤θ < π. (7.5)}

Ifq:AR→MRis the canonical projection andz0=Reiθ0, thenq−1(z0)consists ofz0and(1/R)ei(θ0+π )=w0.

Theunitnormal vector that we use in the definition of the normal derivative to the border ofARis taken with respect to the Euclidian metric. This metric

The unit inner normal (to∂AR) vectors with respect to the Euclidian metric ds2_=dx2_+dy2_{at the pointsz}

0andw0=hz0are

→_Y

z0= _∂

∂n

z0

= −cosθ0

∂ ∂x

z0

−sinθ0

∂ ∂y z0 , →

Yw0= _∂

∂n

w0

= −cosθ0 ∂

∂u

w0

−sinθ0 ∂

∂v

w0 .

(7.6)

With our earlier notations,

→_{Y =}→_Y

z0;

→_Y

w0

∈ET{z0;w0}AR. (7.7)

In all that follows,ARwill be endowed with the metricgsgiven by

gs(z)=dσ2=1

4

1+ 1 |z|2

2

[dx⊗dx+dy⊗dy] (7.8)

andMRwith the projectiongs of this metric byq(see [8]).

Thus,his anisometric involutionof(AR, gs)and

q:AR, gs

→MR,gs

(7.9)

is alocal isometry.

Ifz∈AR, we define the scalar product and the norm induced bygsonTzAR

as follows.

If →U =U1(∂/∂x)z+U2(∂/∂y)z,V→ =V1(∂/∂x)z+V2(∂/∂y)z∈TzAR, their

scalar product→U , →V = →U , →V gsis given by

→

U ,→V =gs(z)→U , →V =1

4

1+ 1 |z|2

2

U1V1+U2V2. (7.10)

The norm→U = →U gsof

→_{U, given by (7.10), is}

→

U=→U , →U 1/2=1₂

1+_|z1_|2

U12+U22. (7.11)

Thegs-length of the vectors→Y z0and

→_Y

w0is given by

→

Yz0=

1 2

1+_R12

cos2_θ

0+sin2θ0=

R2₊₁ 2R2 ;

→

Yw0=

1 2

1+R2cos2_θ

0+sin2θ0=

R2₊₁ 2 .

Thus,→Yz0<1<

→

Yw0. As

→

Y = {→Y z0;

→

Yw0}∉ETs;{z0;w0}AR,it cannot be used

to define the unit normal vector to∂MR!

The relations (7.12) suggest, instead,

→

N0:=

2R2

R2₊₁ →

Yz0;

2

R2₊₁ →

Yw0 := →

Nz0;

→

Nw0

∈ET{z0;w0}AR. (7.13)

Clearly,→N z0 =

→

Nw0 =1.

If z=x+iy ∈AR and w=u+iv = −1/z=hz, then the action of the

differentialdzh=h∗,zon the vectors(∂/∂x)z, (∂/∂y)zof the basis ofTzARis

given by

dzh

_∂

∂x

z

=ux(z)

_∂

∂u

w+ vx(z)

_∂ ∂v w ; dzh

∂ ∂y

z

=uy(z)

_∂

∂u

w+v

y(z) _∂ ∂v w, (7.14)

where{(∂/∂u)w;(∂/∂v)w}is the basis ofTwARandux(z)=(∂u/∂x)(z), and

so on. The Jacobian matrixJh;zofhat the pointzis

Jh;z=

u_x(z) v_x(z) uy(z) vy(z)

=     

x2_−y2 |z|4

2xy

|z|4 2xy

|z|4

−x2_−y2 |z|4

   

. (7.15)

If we takez=h−1_{w=hw, the analogues of formulas of (7.14) and (7.15) are}

dwh

_∂

∂u

w

=x_u(w) _∂

∂x

z+ y_u(w)

_∂ ∂y z ; dwh

∂ ∂u

w

=xv(w)

_∂

∂x

z+ yv(w)

_∂ ∂y z , (7.16) respectively,

Jh;w=

xu(w) yu(w) x_v(w) y_v(w) =     

u2_−u2 |w|4

2uv

|w|4 2uv

|w|4

−u2_−v2 |w|4

   

. (7.17)

Let→X = {→X z,X→ w} ∈ET{z;w}AR

→

Xz=Az

_∂

∂x

z+Bz

_∂

∂y

z, Az, Bz∈C;

→_X

w=Aw

_∂ ∂u w+ Bw _∂ ∂v w

, Aw, Bw∈C.

Clearly(dh)(→X ) = {(dwh)(→X w);(dzh)(→X z)}∈ET{z;w}AR, and the two elements

of(dh)(→X ) are obtained from (7.14) and (7.16) and are given by

dwh→Xw=Awxu(w)+Bwxv(w)

∂ ∂x

z+

Awyu(w)+Bwyv(w)

∂ ∂y

z ,

dzh

→

Xw

=Azux(z)+Bzuy(z)

∂ ∂u

w+

Azvx(z)+Bzvy(z)

∂ ∂v

w .

(7.19)

Taking into account (7.18) and (7.19), we can formulate the following proposi-tion.

Proposition7.1. The EFTVs→X isdh-invariant, that is,→X ∈ETs;{z;w}ARif

and only if

Awxu(w)+Bwxv(w)=Az;

Awyu(w)+Bwyv(w)=Bz.

(7.20)

We now consider the particular casez=z0=Reiθ0,w=w0=(1/R)ei(θ0+π ) and→X =→N 0=the EFTV given by (7.13).

The coefficientsAz,Bz,Aw, andBwbecome, in this case,

Az=−

2R2_cosθ 0

R2₊₁ ; Bz=

−2R2_sinθ 0

R2₊₁

Aw=−2 cos θ0

R2₊₁ ; Bw=

−2 sinθ0

R2₊₁ .

(7.21)

The corresponding Jacobian matrix is

Jh;w=

xu(w) yu(w) xv(w) yv(w)

=

R2_{cos 2θ}

0 R2sin 2θ0

R2_{sin 2θ}

0 −R2cos 2θ0

. (7.22)

We check the validity of relations (7.20) corresponding to the following case:

Awxu(w)+Bwxv(w)=−

2 cosθ0

R2₊₁ R 2_{cos 2θ}

0+− 2 sinθ0

R2₊₁ R 2_{sin 2θ}

0

=−2R2

cosθ0cos 2θ0+sinθ0sin 2θ0

R2₊₁

=−2R2cosθ0

R2₊₁ =Az.

(7.23)

The (inner) gs-unit normal vector to the border of MR at the point z0 is therefore the image of→N0bydq

! →

N0:=dq→N 0. (7.24)

We see now how→N! 0acts on germsfz∈Ꮿ∞(z) . Here, the answer is given by Proposition 6.6.

{fz, fw} =qb(fz)is a symmetric bi-germ and

! →

N0 fz

=→N z0

fz

=→N w0

fw

, (7.25)

where→N z0 and

→

Nw0 are given by (7.13). Thus,

! →

N0 fz

= −_R22R₊2₁

cosθ0

∂fz ∂x

z0

+sinθ0

∂fz ∂y

z0

. (7.26)

A similar result has been obtained in [6,7] where the derivative, with respect to the exterior normal, was taken.

As a final remark, we should notice that, since any nonorientable surfaceSis “created” by a symmetric Riemann surfaceR, all the objects onSshould be de-pendent on “similar” objects onR. Although the EFTVs might appear somehow artificial, their existence is dictated by the previously mentioned philosophy.

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Ilie Barza: Department of Engineering Sciences, Physics and Mathematics, Karlstad University, S-651 88-Karlstad, Sweden

E-mail address:[email protected]

Dorin Ghisa: Department of Mathematics, Glendon College, York University, 2275-Bayview Avenue, Toronto, Canada M4N 3M6