Convex isometric folding

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CONVEX ISOMETRIC FOLDING

E. M. ELKHOLY

(Received 27 September 1993 and in revised form 5 June 1996)

Abstract.We introduce a new type of isometric folding called “convex isometric folding.” We prove that the infimum of the ratio VolN/Volϕ(N)over all convex isometric foldings ϕ:N→N, whereNis a compact 2-manifold (orientable or not), is 1/4.

Keywords and phrases. Isometric and convex foldings, regular and universal covering spaces, manifolds, group of covering transformations, fundamental regions.

2000 Mathematics Subject Classification. Primary 51H10, 57N10.

1. Introduction. A mapϕ:M→N, whereMandNareC∞_{Riemannian manifolds} of dimensionsmandn, respectively, is said to be an isometric folding ofMintoNif and only if for any piecewise geodesic pathγ:J→M, the induced pathϕ◦γ:J→Nis a piecewise geodesic and of the same length. The definition is given by Robertson [4]. The set of all isometric foldingsϕ:M→Nis denoted by᏶(M,N).

Letp :M→N be a regular locally isometric covering and let G be the group of covering transformations ofp. An isometric foldingφ∈᏶(M)is said to bep-invariant if and only if for allg∈Gand allx∈X,p(ϕ(x))=p(ϕ(g,x)). See Robertson and Elkholy [5]. The set ofp-invariant isometric foldings is denoted by᏶i(M,p).

Definition1.1. Letϕ∈᏶(M,N), whereMandNareC∞_{Riemannian manifolds} of dimensionsmandn, respectively. We say thatϕis a convex isometric folding if and only ifϕ(M)can be embedded as a convex set inRn_.

We denote the set of all convex isometric foldings ofMintoNbyC(M,N), and if C(M,N)≠, then it forms a subsemigroup of᏶(M,N).

Definition1.2. We say thatϕ∈᏶i(M,p)is ap-invariant convex isometric folding

if and only ifϕ(M)can be embedded as a convex set inRm_.

We denote the set ofp-invariant convex isometric foldings of M byCi(M,p). If

Ci(M,p)≠ , then for any covering map, p :M →N, Ci(M,p) is a subsemigroup

ofC(M).

To solve our main problem we need the following:

(1) Robertson and Elkholy [5] proved that ifNis ann-smooth Riemannian manifold, p:M→Nis its universal covering, andGis the group of covering transformations of p, then᏶(N)is isomorphic as a semigroup to᏶i(M,p)/G.

corresponding foldingψ∈᏶i(M,p)maps each fiber ofpto a single point.

(3) Elkholy and Al-Ahmady [3] proved that under the same conditions of (2), ifNis a compact 2-manifold, then

VolN Volϕ(N)=

VolF

Volψ(F), (1.1)

whereF is a fundamental region ofGinM.

2. Convex isometric folding and covering spaces. The next theorem establishes the relation between the set of convex isometric folding of a manifold,C(N), and the set ofp-invariant convex isometric folding of its universal covering space,Ci(M,p).

Theorem2.1. LetNbe a manifold andp:M→Nits universal covering. LetGbe the group of covering transformations ofp. IfC(N)≠, thenC(N)is isometric as a semigroup toCi(M,p)/G.

Proof. Let C(N)≠ϕ. Then by using (1), there exists an isomorphism f from ᏶i(M,p)/Ginto᏶(N). SinceCi(M,p)is a subsemigroup of᏶i(M,p),Ci(M,p)/Gis a

subsemigroup of᏶i(M,p)/G.

Leth=f|(Ci(M,P)/G). SinceCi(M,p)/Gis a semigroup,his a homeomorphism

and also it is one-one. To show thath is an onto map, we suppose thatϕ∈C(N). Hence,ϕ∈᏶(N) and, consequently, there existsψ∈᏶i(M,p)/G. Sinceϕ∈C(N),

ϕ∗is trivial and hence for allx∈M,ψ(G,x)=ψ(x), and thereforeψ∈Ci(M,p)/G.

Theorem2.2. LetNbe a compact orientable 2-manifold and consider the universal covering space(R2_{,P)ofN. Letϕ∈C(N)andψ∈C}

i(R2,p). Then for allx,y∈R2,

d(ψ(x),ψ(y))≤∆, where∆is the radius of a fundamental region for the covering space.

Proof. Elkholy [1] proved the truth of the theorem forN=S2_{. So, we have to} prove it for the connected sum ofn-tori. First, letN=T be a torus homomorphic to the quotient space obtained by identifying opposite sides of a square of length “a” as shown in Figure 1(a)

T

a a

b b

(a) (b)

g1·x g2·x g3·x

g4·x α1 α2 α3

α4 _y

Suppose thatϕ:T→T is a convex isometric folding. Thenϕ∗(π1(T ))is trivial. By Theorem 2.1, there exists a convex isometric foldingψ:R2_→R2_{such that for allx,} y∈R2_{and for allg∈G,pψ(x)=pψ(g,x). Equivalently, for all(P,Q)∈R}2_and for allg∈Z×Z, there exists a uniqueh∈Z×Zsuch thath◦ψ(P,Q)=ψg(P,Q), i.e.,

ψP,Q+2∆m,2∆n_{=ψP+2∆m,Q+2∆n, wherem,n,m,n∈Z.} (2.1)

Consider any fundamental regionF of the covering space(R2_{,p)ofT, i.e., a closed} square of length “a” with sides identified as shown in Figure 1(b). Sinceϕ∗is trivial, by (2), for allx∈R2_{,ψ(G,x)=ψ(x). Now, letxandybe distinct points ofR}2_such thatx=g·yfor allg∈Gand letd(x,y)=α1. Then there exists a pointx∗_=g·x such that

dy,x∗_=min(α

i), αi=dy,gi,x, i=1,...,4. (2.2)

Thus, there are always four equivalent pointsgi·x, i=1,...,4 which form the

ver-tices of a square of length “a” and such that d(gi·x,y)≤2∆. From Figure 1(b), it

is clear that maxd(x∗_{,y)≤∆and sinceψis an isometric folding, by Robertson [4],} d(ψ(x),ψ(y))≤d(x,y), i.e.,

dψ(x),ψy=dψgi·x,ψy≤dgi·x,y=dx,y≤∆, (2.3)

and this proves the theorem forN=T.

Now, consider the connected sum of two tori, obtained as a quotient space of an octagon with sides identified as shown in Figure 2(a). The group of covering transfor-mationsG is isometric toZ×Z×Z×Z. Using the same previous technique, we can

(a) T#T

(b) g1·x g2·x g3·x

g4·x

y a1 a2 a3

a4

Figure2.

Theorem2.3. LetNbe a compact nonorientable 2-manifold and consider the uni-versal covering space (M,p) of N. Let φ ∈ C(N)and ψ ∈Ci(M,p). Then for all

x,y∈M,dψ(x),ψ(y)≤∆, where∆is the radius of a fundamental region for the covering space.

Proof. By Elkholy [2], the theorem is true forN=p2_andM=S2_{. Now, consider} the connected sum of two projective planes, the Klein bottleK, homeomorphic to the quotient space obtained by identifying the opposite sides of a square as shown in Figure 3(a).

b b

a

a

(a) K

(b)

g1·x

g2·x g3·x g4·x

y

Figure3.

Suppose thatϕ:K→K is a convex isometric folding. Then there exists a convex isometric foldingψ:R2_→R2_{such that for allx∈R}2_{andg∈G,p(ψ(x))=p(ψ(g·} x)). Equivalently, for all(P,Q)∈R2 _{and for all g∈Z×Z2, there exists a unique} h∈Z×Z2such thath◦ψ(P,Q)=ψ(g(P,Q)), i.e.,

ψP,Q+2∆m_,2∆n

=ψP+2∆m,2∆n+(−)m_{Q, wherem,n,m,n∈Z.} (2.4)

Any fundamental regionF of the covering space(R2_{,p)ofKis a closed square of} diameter 2∆with the boundary identified as shown in Figure 3(b). Sinceϕ∗is trivial, for allx∈R2_{,ψ(G·x)=ψ(x).}

Now, letxandy be distinct points ofR2_{such thaty≠g·xfor allg∈G, and let} d(x,y)=α1. Thus, there exists a pointx∗_{=g·xsuch that}

dy,x∗_=min(α

i), αi=dy,gi·x, i=1,...,4. (2.5)

Thus, there are always four equivalent pointsgi·xwhich form the vertices of a

par-allelogram such that the shortest diameter is of length less than 2∆.

Now, the pointyis either inside or on the boundary of a triangle of verticesg1·x=

x,g2·x,g3·x. Lety_{be a point equidistant from the vertices of this triangle, i.e.,}

From Figure 3(b), it is clear thatd(y_{,x) <∆and, hence,d(x}∗_{,y) <∆. Therefore,}

dψ(x),ψy=dψgi·x,ψy≤dg·xi,y=dx∗,y<∆ (2.7)

and the result follows.

Now, letNbe the connected sum of three projective planes obtained as the quotient space of a hexagon with the sides identified in pairs as indicated in Figure 4(a). In this case,(R2_{,p)is the universal cover ofNandGZ×Z×Z2. Using the same method as} that used above, we can always have equivalent pointsgi·x,i=1,...,4 which form

the vertices of a parallelogram whose shortest diameter is of length less than 2∆. From Figure 4(b), we can see that maxd(y,x∗_{) <∆and the theorem is proved.}

In general and by using the same technique, the theorem is also true for the con-nected sum ofn-projective planes.

P#P#P

(a)

g1·x

g2·x g3·x g4·x

y

(b) Figure4.

3. Volume and convex folding. The following theorem succeeds in estimating the maximum volume we may have if we convexly folded a compact 2-manifold into itself.

Theorem3.1. The infimum of the ratio

eN=_Volϕ(N)VolN , (3.1)

whereNis a compact 2-manifold over all convex isometric foldingsϕ∈C(N)of degree zero, is4.

Proof. Robertson [4] has shown that ifNis a compact 2-manifold, andϕ:N→N is a convex isometric folding, any convex isometric folding is an isometric folding, then degϕis±1 or 0. We consider only the case for which degϕis zero otherwise ϕ(N)cannot be embedded as a convex subset ofR2_{unlessNis. In this case, the set of} singularities ofϕdecomposesNinto an even number of strata, sayk, each of which is homeomorphic toϕ(N)and, hence,

that is,eNshould be an even number. To calculate the exact value ofeN, consider first

an orientable 2-compact manifoldN. By using (1.1)

eN=_Volϕ(F)VolF (3.3)

and this means thateN can be calculated by calculating the volume ofF and of its

imageϕ(F), butF is a closed square of diameter 2∆andϕ(F)is a closed subset of F such that the distanced(x,x_{)between any two pointsx,x∈ϕ(F)is at most∆.} The supremum of 2-dimensional volume of such set isφ(∆/2)2_{and, hence, 2< e}

N.

ButeN is an even number. Hence,eN=4.

Now, let N be a nonorientable 2-compact manifold, i.e., a connected sum of n-projective planes. Elkholy [2] proved the theorem forn=1.

The fundamental region in this case is a square or a rectangle of diameter 2∆ ac-cording to whethernis even or odd. Ifnis an even number, then

VolF=2∆2 _(3.4)

and the result follows. Now, letnbe an odd number. ThenFis a rectangle of lengths ((n+1)/2)a,((n−1)/2)aand hence

VolF=4∆2_{sinθcosθ=4∆}2 a(n+1)/2 a(n2_+1)/2

a(n−1)/2 a(n2_+1)/2=

n2₋₁

n2₊₁2∆2. (3.5) Therefore,eN>2 for alln >1. SinceeN is an even number,eN=4.

References

[1] E. M. Elkholy,Isometric and topological folding of manifolds, Ph.D. thesis, Southampton University, England, 1982.

[2] , Maximum isometric folding of the real projective plane, J. Natur. Sci. Math. 25 (1985), no. 1, 25–30. MR 87d:53081. Zbl 589.53052.

[3] E. M. Elkholy and A. E. Al-Ahmady,Maximum volume of isometric foldings of compact2 -Riemannian manifolds, Proc. Conf. Oper. Res. and Math. Methods Bull. Fac. Sc. Alex. 26 A(1977), no. 1, 54–66.

[4] S. A. Robertson,Isometric folding of Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect. A79(1977), no. 3–4, 275–284. MR 58 7486. Zbl 418.53016.

[5] S. A. Robertson and E. M. Elkholy,Invariant and equivariant isometric folding, Delta J. Sci. 8(1984), no. 2, 374–385.