Discrete curvature and the Gauss-Bonnet theorem

0.17 MB·English

by  Joakim Arnlind

DISCRETE CURVATURE AND THE
GAUSS–BONNET THEOREM
0
JOAKIMARNLIND,JENSHOPPE,ANDGERHARDHUISKEN
1
0
2 Abstract. Formatrixanaloguesofembeddedsurfaceswedefinediscretecur-
vaturesandEulercharacteristics,andanon-commutativeGauss–Bonnetthe-
n
orem is shownto follow. We derivesimpleexpressions forthe discrete Gauss
a
curvature in terms of matrices representing the embedding coordinates, and
J
providealargeclassofexplicitexamplesillustratingthenewnotions.
3
1

]
h
1. Introduction
p
- A particularwayof discretizing surfaces by replacing functions by matrices has for
h
a long time been used in physics to obtain a quantum theory of surfaces (mem-
t
a
branes) moving in Minkowski space, sweeping out 3-manifolds of vanishing mean
m
curvature [Hop82]. The discretization, sometimes called “Matrix Regularization”,
[ is of independent mathematical interest and contains many interesting structures.

1 One of the main features of the correspondence between functions on the surfaces
v andmatricesis that the Poissonbracketoftwofunctions becomes the commutator
3 of two matrices. This allows for an easy construction of discrete analogues of any
2
expressioninvolving products and Poissonbrackets of functions. In a recent paper
2
[AHH10], the geometry of surfaces embedded in Riemannian manifolds has been
2
. expressed in terms of Poisson brackets of the embedding coordinates. Using these
1
formulas, one can define discretizations of the Gaussian curvature and the Euler
0
characteristic, and it is immediate to prove a discrete Gauss-Bonnet theorem (see
0
1 Theorem 3.10).
: Apart from being interesting in their own right, these discrete concepts might
v
also help to solve questions related to the regularization in the above mentioned
i
X
MembraneTheory. Forinstance,solvingtheequationsofmotioninMembraneThe-
r oryyields matrices correspondingto a discrete surface. As solutionscorresponding
a
to surfaces of arbitrary topology exist, one would like to be able to determine the
geometry from the matrices in some way. In Theorem 3.11 we provide formulas
for computing the discrete curvature and the discrete Euler characteristic given
the matrix analoguesof the embedding coordinates (which are the solutions to the
equationsofmotioninMembraneTheory). Thus,inthelimitoflargematricesone
may determine the Euler characteristic, and hence the topology, of the surface.
2. Surface geometry and Poisson brackets
Let us recall some of the results obtained in [AHH10]. Consider a surface Σ em-
bedded in a Riemannian manifold M, of dimension m=2+p, via the coordinates
x1(u1,u2),...,xm(u1,u2), where u1,u2 are local coordinates on Σ. Furthermore,
letn1(u1,u2),...,nm(u1,u2)forA=1,...,pdenotethecomponentsofporthonor-
A A
mal vectors N normal to the surface at each point. Indices i,j,k,l will run from
A
1
2 JOAKIMARNLIND,JENSHOPPE,ANDGERHARDHUISKEN
1 to m and indices a,b will run from 1 to 2. The metric of M is denoted by g¯ ,
ij
the Christoffel symbols by Γ¯i and the covariant derivative by ¯. Regarded as a
jk ∇
subspace of TM, the tangent space TΣ is spanned by the vectors e =(∂ xi)∂ .
a a i
Letting ρ(u1,u2)beanarbitrarynon-vanishingdensityonΣ,onedefinesaPois-
son bracket on C∞(Σ) by setting
1
(2.1) f,h = εab ∂ f ∂ h ,
a b
{ } ρ
(cid:0) (cid:1)(cid:0) (cid:1)
whereεab isantisymmetricwithε12 =1and∂ = ∂ . Withthisbracketwedefine
a ∂ua
the tensors
(2.2) ij = xi,xj
P { }
(2.3) ij = 1εab ∂ xi ¯ N j = xi,nj + xi,xk Γ¯j nl ,
SA ρ a ∇b A { A} { } kl A
(cid:0) (cid:1)(cid:0) (cid:1)
and one can also consider them as maps TM TM by lowering the second index
→
with the ambient metric g¯, i.e.
(2.4) (X)= ikg¯ Xj∂
P P kj i
(2.5) (X)= ikg¯ Xj∂ .
SA SA kj i
With these definitions, one finds that
2
(2.6) Tr 2 i j = det(h )
SA ≡ SA j SA i −ρ2 A,ab
(cid:0) (cid:1) (cid:0) (cid:1)
g
(2.7) Tr 2 i j = 2 ,
P ≡PjPi − ρ2
where g =det g¯(e ,e ) is the determinant of the induced metric on Σ, and h
a b A,ab
is the second fu(cid:0)ndament(cid:1)al form corresponding to the normal vector NA.
In the main part of this article we will make use of the following result:
Theorem 2.1 ([AHH10]). Let K denote the Gaussian curvature of Σ. Then
1 ρ2 p
(2.8) K = g¯ R¯(e ,e )e ,e Tr 2,
g 1 2 2 1 − 2g SA
(cid:0) (cid:1) AX=1
where R¯ is the curvature tensor of M.
3. Matrix regularizations and discrete curvatures
Inthefollowing,weshallassumethatΣisacompactclosedorientablesurface. Let
us first define what is meant by a “matrix regularization”,and then show some of
its properties.
Definition 3.1. Let N ,N ,... be a strictly increasing sequence of positive inte-
1 2
gers,let Tα forα=1,2,...belinearmapsfromC∞(Σ,R)tohermitianN N
α α
matrices{and}let ~(N) be a real-valued strictly positive decreasing function×such
that lim N~(N)< . Furthermore, let ω be a symplectic form on Σ and let
N→∞
∞
, denotethePoissonbracketinducedbyω. If Tα hasthefollowingproperties
{· ·} { }
DISCRETE CURVATURE AND THE GAUSS–BONNET THEOREM 3
for all f,h C∞(Σ)
∈
(3.1) lim Tα(f) < ,
α→∞|| || ∞
(3.2) lim Tα(fh) Tα(f)Tα(h) =0,
α→∞|| − ||
1
(3.3) lim [Tα(f),Tα(h)] Tα f,h =0,
α→∞(cid:12)(cid:12)(cid:12)(cid:12)i~α − (cid:0){ }(cid:1)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
(3.4) lim (cid:12)2(cid:12)π~ TrTα(f)= fω, (cid:12)(cid:12)
α
α→∞ Z
Σ
where denotestheoperatornormand~ =~(N ),thenwecallthepair(Tα,~)
α α
||·||
a matrix regularization of (Σ,ω).
Givenlocalcoordinatesu1,u2 onΣ, wewrite ω =ρ(u1,u2)du1 du2,anditiseasy
∧
to see that the induced Poisson bracket becomes
1
f,h = εab ∂ f ∂ h .
a b
{ } ρ
(cid:0) (cid:1)(cid:0) (cid:1)
Definition 3.2. If fˆ,fˆ,... is a sequence of matrices such that fˆ has dimension
1 2 α
N and if it holds that
α
(3.5) lim fˆ Tα(f) =0,
α
α→∞(cid:12)(cid:12) − (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
then we say that the sequence con(cid:12)v(cid:12)erges to the(cid:12)f(cid:12)unction f.
Definition 3.3. A matrix regularization (Tα,~) is called unital if
(3.6) lim 1 Tα(1) =0.
α→∞|| Nα − ||
Remark 3.4. Although unital matrix regularizations seem natural, and all our ex-
amples fall into this category,it is easy to constructexamples ofnon-unital matrix
regularizations. Namely, let (Tα,~) be a matrix regularization and consider the
map T˜α defined by
0
T˜α(f)= Tα(f) ....
 
 
0 0
 ··· 
Then (T˜α,~) is a matrix regularization which is not unital, since
lim T˜α(1) 1 1.
α→∞(cid:12)(cid:12) − Nα+1(cid:12)(cid:12)≥
(cid:12)(cid:12) (cid:12)(cid:12)
Proposition 3.5. Let (Tα,~) be(cid:12)(cid:12)a unital matrix(cid:12)(cid:12)regularization. Then
(3.7) lim 2πN ~ = ω.
α α
α→∞ Z
Σ
Proof. Let us use formula (3.4) with f =1.
ω = lim 2π~ TrTα(1)= lim 2π~ Tr Tα(1)+1 1
ZΣ α→∞ α α→∞ α h Nα − Nαi
= lim 2π~ N +2π~ Tr(Tα(1) 1 ) = lim 2π~ N
α→∞(cid:16) α α α − Nα (cid:17) α→∞ α α
4 JOAKIMARNLIND,JENSHOPPE,ANDGERHARDHUISKEN
since
lim 2π~ Tr(Tα(1) 1 ) lim 2π~ N Tα(1) 1 =0,
α→∞| α − Nα |≤α→∞ α α|| − Nα||
due to the fact that the matrix regularizationis unital. (cid:3)
Proposition 3.6. Let (Tα,~) be a matrix regularization of (Σ,ω) and let fˆα be
{ k}
a sequence of matrices converging to f C∞(Σ) for k =1,...,n. Then a fˆα+
k ∈ { 1 1
+a fˆα converges to a f + +a f for any a ,...,a R and
··· n n} 1 1 ··· n n 1 n ∈
n
(3.8) lim fˆα fˆα lim Tα(f )
α→∞(cid:12)(cid:12)(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12)≤kY=1α→∞|| k ||
(cid:12)(cid:12) (cid:12)(cid:12)
(3.9) lim fˆα fˆα Tα(f f ) =0
α→∞(cid:12)(cid:12) 1 ··· n − 1··· n (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
(3.10) lim 2(cid:12)(cid:12)π~ Tr fˆα fˆα = f(cid:12)(cid:12) f ω.
α→∞ α 1 ··· n Z 1··· n
(cid:0) (cid:1) Σ
Proof. The first statement about a fˆα+ a fˆα follows directly from the lin-
{ 1 1 ··· n n}
earity of Tα. Let us prove (3.8) by induction on n. Thus, we assume that (3.8)
holds and compute
lim fˆα fˆα lim fˆα fˆα fˆα Tα(f )+Tα(f )
α→∞(cid:12)(cid:12) 1 ··· n+1(cid:12)(cid:12)≤α→∞(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12) n+1− n+1 n+1 (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) lim fˆ(cid:12)α(cid:12) fˆα (cid:12)(cid:12)fˆα T(cid:12)α(cid:12)((cid:12)f(cid:12) ) + fˆα fˆα Tα(f (cid:12)(cid:12))
≤α→∞(cid:16)(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12) n+1− n+1 (cid:12)(cid:12) (cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)|| n+1 ||(cid:17)
(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) n+1 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
= lim fˆα fˆα Tα(f ) lim Tα(f ) .
α→∞(cid:12)(cid:12)(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12)|| n+1 ||≤ kY=1α→∞|| k ||
(cid:12)(cid:12) (cid:12)(cid:12)
To prove (3.9) we again proceed by induction and assume that (3.9) holds for any
given n, and then compute
lim fˆα fˆα Tα(f f )
α→∞(cid:12)(cid:12) 1 ··· n+1− 1··· n+1 (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
lim(cid:12)(cid:12) fˆα fˆα fˆα Tα(f(cid:12)(cid:12) ) + fˆα fˆαTα(f ) Tα(f f )
≤α→∞(cid:16)(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12) n+1− n+1 (cid:12)(cid:12) (cid:12)(cid:12) 1 ··· n n+1 − 1··· n+1 (cid:12)(cid:12)(cid:17)
(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
lim (cid:12)(cid:12)fˆα fˆα(cid:12)(cid:12)(cid:12)(cid:12)fˆα Tα(f )(cid:12)(cid:12)+(cid:12)(cid:12)fˆα fˆαTα(f ) (cid:12)(cid:12)
≤α→∞(cid:16)(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12) n+1− n+1 (cid:12)(cid:12) (cid:12)(cid:12) 1 ··· n n+1
(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12)Tα(f (cid:12)(cid:12)f(cid:12)(cid:12) ) Tα(f f(cid:12)(cid:12))Tα(cid:12)((cid:12)f )+Tα(f f )Tα(f )
1 n+1 1 n n+1 1 n n+1
− ··· − ··· ··· (cid:12)(cid:12)(cid:17)
(cid:12)(cid:12)
lim fˆα fˆα fˆα Tα(f ) + Tα(f ) fˆα fˆα Tα(f (cid:12)(cid:12) f )
≤α→∞(cid:16)(cid:12)(cid:12) 1 ··· n(cid:12)(cid:12)(cid:12)(cid:12) n+1− n+1 (cid:12)(cid:12) || n+1 ||(cid:12)(cid:12) 1 ··· n − 1··· n (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
+ Tα(f (cid:12)(cid:12) f )Tα((cid:12)f(cid:12)(cid:12)(cid:12) ) Tα(f f (cid:12)(cid:12) ) =0. (cid:12)(cid:12) (cid:12)(cid:12)
1 n n+1 1 n+1
|| ··· − ··· ||(cid:17)
Finally, we prove the trace formula:
lim 2π~ Tr fˆα fˆα
α→∞ α 1 ··· n
(cid:0) (cid:1)
= lim 2π~ TrTα(f f )+Tr fˆα fˆα Tα(f f )
α→∞ αh 1··· n (cid:0) 1 ··· n − 1··· n (cid:1)i
= f f ω,
1 n
Z ···
Σ
DISCRETE CURVATURE AND THE GAUSS–BONNET THEOREM 5
since
lim ~ Tr fˆα fˆα Tα(f f )
α→∞(cid:12) α 1 ··· n − 1··· n (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) lim ~ N fˆα fˆα Tα(f (cid:12) f ) =0
≤α→∞ α α(cid:12)(cid:12) 1 ··· n − 1··· n (cid:12)(cid:12)
by formula (3.9). (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:3)
Theaboveresultallowsoneto easilyconstructsequencesofmatricesconvergingto
any sum of products of functions and Poisson brackets. Namley, simply substitute
for every factor in every term of the sum, a sequence converging to that function,
where Poisson brackets of functions may be replaced by commutators of matrices.
Proposition 3.6 then guarantees that the matrix sequence obtained in this way
converges to the sum of the products of the corresponding functions.
Proposition 3.7. Let (Tα,~) be a matrix regularization and assume that fˆ
α
{ }
converges to f. Then fˆ† converges to f.
{ α}
Proof. Due to the fact that A = A† one sees that
|| || || ||
lim fˆ† Tα(f) = lim fˆ Tα(f) † = lim fˆ Tα(f) =0,
α→∞(cid:12)(cid:12) α− (cid:12)(cid:12) α→∞(cid:12)(cid:12) α− (cid:12)(cid:12) α→∞(cid:12)(cid:12) α− (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:0) (cid:1) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
since fˆ (cid:12)(cid:12)converges to(cid:12)(cid:12)f. (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:3)
α
{ }
Ifthematrixregularizationisunital,onecanrelatethematrixsequenceconverging
to the function 1/f, to the inverse of a sequence converging to f.
Proposition 3.8. Let (Tα,~) be a unital matrix regularization and assume that f
is a nowhere vanishing function and that fˆ converges to f. If fˆ−1 exists and
{ α} α
fˆ−1 is uniformly bounded for all α, then fˆ−1 converges to 1/f.
|| α || { α }
Proof. One calculates
lim fˆ−1 Tα(1/f) lim fˆ−1 1 fˆ Tα(1/f)
α→∞(cid:12)(cid:12) α − (cid:12)(cid:12)≤α→∞(cid:12)(cid:12) α (cid:12)(cid:12)(cid:12)(cid:12) Nα − α (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)=(cid:12) lim fˆ−1 (cid:12)1(cid:12) fˆ T(cid:12)(cid:12)α(1/(cid:12)f(cid:12))(cid:12)(cid:12)+Tα(1) Tα(1) (cid:12)(cid:12)
α→∞(cid:12)(cid:12) α (cid:12)(cid:12)(cid:12)(cid:12) Nα − α − (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)
lim (cid:12)(cid:12)fˆ−1(cid:12)(cid:12)(cid:12)(cid:12) 1 Tα(1) + fˆ Tα(1/f) T(cid:12)α(cid:12)(1)
≤α→∞(cid:12)(cid:12) α (cid:12)(cid:12)(cid:16)|| Nα − || (cid:12)(cid:12) α − (cid:12)(cid:12)(cid:17)
=0, (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
since the matrix regularization is unital and fˆ−1 is assumed to be uniformly
bounded. || α || (cid:3)
Recallthat Σ is embedded in a m=2+p dimensional manifold M via the embed-
dingcoordinatesx1(u1,u2),...,xm(u1,u2),andthatporthonormalnormalvectors
are given with components ni . By Xi and Ni we will denote arbitrary
A { α} { A,α}
sequences converging to xi and ni respectively. Moreover, given the metric g¯
A ij
andthe Christoffelsymbols Γ¯i ofM,we let Gˆ and Γˆi denote sequences
jk { ij,α} { jk,α}
converging to g¯ and Γi respectively. To avoid excess of notation, we suppress
ij jk
the index α whenever all matrices are considered at a fixed (but arbitrary) α.
In analogy with (2.3) we define
(3.11) (SˆA)jk = i1~[Xj,NAk′]Gˆk′k+ i1~[Xj,Xl]Γˆklm′ NAmGˆk′k,
6 JOAKIMARNLIND,JENSHOPPE,ANDGERHARDHUISKEN
and
(3.12) tr ˆ2 = (ˆ )j †(ˆ )k.
SA SA k SA j
(cid:0) (cid:1)
Let gab be the induced metricbon Σ, and g its determinant. We set
√g
(3.13) γ = ,
ρ
and denote by γˆ an arbitrary sequence of invertible matrices converging to γ.
α
{ }
By defining
1
ˆj = [Xj,Xl]Gˆ ,
Pk i~ lk
it follows from (2.7) that
1 1
(3.14) (ˆi)†ˆk = Gˆ† [Xi,Xj][Xk,Xl]Gˆ
−2 Pk Pi 2~2 jk li
converges to γ2.
If the embedding space is Rm, the above formulas reduce to
m
1
(3.15) tr ˆ2 = [Xi,Nj][Xj,Ni],
SA −~2 A A
iX,j=1
b
m
1 1
(3.16) (ˆi)†ˆk = [Xi,Xj]2,
−2 Pk Pi −~2
Xi<j
and in R3 one obtains
3
(3.17) trSˆ2 = 41~4 εjklεik′l′(γˆ†)−1 Xi,[Xk,Xl] Xj,[Xk′,Xl′] γˆ−1,
X (cid:2) (cid:3)(cid:2) (cid:3)
since b
1
(3.18) ni = εi xj,xk ,
2γ jk{ }
definesaunitnormalvectortothesurface(cp. [AHH10],where(3.17)isalsogiven
for arbitrary codimension).
We are now ready to define and present formulas for the discrete curvature in a
matrix regularizationof Σ.
Definition 3.9. Let (Tα,~) be a matrix regularization of (Σ,ω) and let K be
the Gaussian curvature of Σ. A Discrete Curvature of Σ is a matrix sequence
Kˆ ,Kˆ ,Kˆ ,... converging to K, and a Discrete Euler Characteristic of Σ is a
1 2 3
{ }
sequence χˆ ,χˆ ,χˆ ,... such that lim χˆ =χ.
1 2 3 α
{ } α→∞
Fromthe classicalGauss-Bonnettheorem,it is immediate to derive a discrete ana-
logue for matrix regularizations.
Theorem3.10. Let(Tα,~)beamatrixregularizationof(Σ,ω),andlet Kˆ ,Kˆ ,...
1 2
{ }
be a discrete curvature of Σ. Then the sequence χˆ ,χˆ ,... defined by
1 2
(3.19) χˆ =~ Tr γˆ Kˆ ,
α α α α
h i
is a discrete Euler characteristic of Σ.
DISCRETE CURVATURE AND THE GAUSS–BONNET THEOREM 7
Proof. To prove the statement, we compute lim χˆ and show that it is equal
α→∞ α
to χ(Σ). Thus
1
lim χˆ = lim 2π~ Tr γˆ Kˆ ,
α α α α
α→∞ α→∞2π h i
and by using Proposition 3.6 we can write
1 √g 1 √g 1
lim χˆ = K ω = K ρdudv = K√gdudv =χ(Σ),
α
α→∞ 2π Z ρ 2π Z ρ 2π Z
Σ Σ Σ
where the last equality is the classical Gauss-Bonnet theorem. (cid:3)
Theorem 3.11. Let (Tα,~) be a unital matrix regularization of (Σ,ω) and let
K¯α be a matrix sequence converging to g¯ R¯(e ,e )e ,e /g (the sectional curva-
{ 12} 1 2 2 1
ture of TΣ in M). Then the sequence Kˆ of(cid:0)matrices defin(cid:1)ed by
p
1
(3.20) Kˆ =K¯ (γˆ†)−1 tr ˆ2 γˆ−1
12− 2 SA
AX=1 (cid:0) (cid:1)
b
is a discrete curvature of Σ. Thus, a discrete Euler characteristic is given by
~ p
(3.21) χˆ=~Tr γˆK¯ Tr γˆ−1tr ˆ2 .
(cid:0) 12(cid:1)− 2AX=1 h SAi
b
Proof. By using the way of constructing matrix sequences given through Proposi-
tion 3.6, the result follows immediately from Theorem 2.1. (cid:3)
Note thatifρ=√g, thenγ =1whichimplies thatone canchooseγˆα =1Nα when
the matrix regularizationis unital.
4. Two simple examples
4.1. The round fuzzy sphere. For the sphere embedded in R3 as
(4.1) ~x=(x1,x2,x3)=(cosϕsinθ,sinϕsinθ,cosθ)
with the induced metric
1 0
(4.2) (g )= ,
ab (cid:18)0 sin2θ(cid:19)
itiswellknownthatonecanconstructamatrixregularizationfromrepresentations
of su(2). Namely, let S ,S ,S be hermitian N N matrices such that [Sj,Sk]=
1 2 3
×
iǫjk Sl, (S1)2+(S2)2+(S3)2 =(N2 1)/4, and define
l
−
2
(4.3) Xi = Si.
√N2 1
−
ThenthereexistsamapT(N) (whichcanbedefinedthroughexpansioninspherical
harmonics)suchthatT(N)(xi)=Xi and(T(N),~=2/√N2 1)isaunitalmatrix
regularization of (S2,√gdθ dϕ) [Hop82]. A unit normal −of the sphere in R3 is
∧
given by N TR3 with N =xi∂ , which gives Ni =Xi, and one can compute the
i
∈
discrete curvature as
m
1
(4.4) Kˆ = Tr[Xi,Xj]2 =1
N −~2 N
i<Xj=1
8 JOAKIMARNLIND,JENSHOPPE,ANDGERHARDHUISKEN
which gives the discrete Euler characteristic
2N
(4.5) χˆ =~TrKˆ =~N = ,
N N
√N2 1
−
converging to 2 as N .
→∞
4.2. The fuzzy Clifford torus. The Clifford torus in S3 can be regarded as
embedded in R4 through
1
~x=(x1,x2,x3,x4)= (cosϕ ,sinϕ ,cosϕ ,sinϕ ),
1 1 2 2
√2
with the induced metric
1 1 0
(g )= ,
ab 2(cid:18)0 1(cid:19)
and two orthonormal vectors, normal to the tangent plane of the surface in TR4,
can be written as
N =x1∂ +x2∂ x3∂ x4∂ .
± 1 2 3 4
± ±
ToconstructamatrixregularizationfortheCliffordtorus,oneconsiderstheN N
×
matrices g and h with non-zero elements
g =ωk−1 for k =1,...,N
kk
h =1 for k =1,...,N 1
k,k+1
−
h =1,
N,1
where ω = exp(i2θ) and θ = π/N. These matrices satisfy the relation hg = ωgh.
The map T(N) is then defined on the Fourier modes
Y =eim~·ϕ~ =eim1ϕ1+im2ϕ2
m~
as
T(N)(Ym~)=ω21m1m2gm1hm2,
and the pair (T(N),~=sinθ) is a unital matrix regularizationof the Clifford torus
with respect to √gdϕ1 dϕ2 [FFZ89, Hop89]. Thus, using this map one finds that
∧
1 1
X1 =T(x1)= T(cosϕ )= (g†+g)
1
√2 2√2
1 i
X2 =T(x2)= T(sinϕ )= (g† g)
1
√2 2√2 −
1 1
X3 =T(x3)= T(cosϕ )= (h†+h)
2
√2 2√2
1 i
X4 =T(x4)= T(sinϕ )= (h† h)
2
√2 2√2 −
which implies that N1 = X1, N2 = X2, N3 = X3 and N4 = X4. By a
± ± ± ± ± ±
straightforwardcomputation one obtains
4
1
[Xi,Xj]2 =21
−~2
iX,j=1
DISCRETE CURVATURE AND THE GAUSS–BONNET THEOREM 9
and therefore
4 4
1 1
[Xi,Nj][Xj,Ni]= [Xi,Xj]2 =1,
2~2 + + −2~2
iX,j=1 iX,j=1
and since [X1,X2]=[X3,X4]=0 it follows that
4 4
1 1
[Xi,Nj][Xj,Ni]= [Xi,Xj]2 = 1.
2~2 − − 2~2 −
iX,j=1 iX,j=1
This implies that the discrete curvature vanishes, i.e.
4 4
1 1
Kˆ = [Xi,Nj][Xj,Ni]+ [Xi,Nj][Xj,Ni]=1 1=0,
N 2~2 + + 2~2 − − −
iX,j=1 iX,j=1
which immediately gives χˆ =0.
N
5. Axially symmetric surfaces in R3
Recall the classical description of general axially symmetric surfaces:
(5.1) ~x= f(u)cosv,f(u)sinv,h(u)
(cid:0) 1 (cid:1)
~n= ± h′(u)cosv,h′(u)sinv, f′(u) ,
h′(u)2+f′(u)2 −
(cid:0) (cid:1)
p
which implies
f′2+h′2 0 1 h′f′′ h′′f′ 0
gab =(cid:18) 0 f2(cid:19) hab = h′±2+f′2 (cid:18) −0 fh′(cid:19),
(cid:0) (cid:1) (cid:0) (cid:1) −
p
where h are the components of the second fundamental form. The Euler charac-
ab
teristic can be computed as
1 u+ h′ h′f′′ h′′f′ f′ u+
(5.2) χ= K√g = − du= ,
2π Z −Zu− (cid:0)f′2+h′2 3/2(cid:1) − f′2+h′2(cid:12)(cid:12)(cid:12)u−
(cid:0) (cid:1) p (cid:12)
which is equal to zero for tori (due to periodicity) and equal to +2 for(cid:12)spherical
surfaces (due to f′(u )= ).
±
∓∞
While a generalprocedure for constructing matrix analogues of surfacesembed-
dedinR3 wasobtainedin[ABH+09b,ABH+09a](cp. also[Arn08b]),letusrestrict
nowto h(u)=u=z,hence describe the axiallysymmetric surface Σ as a levelset,
C =0, of
1
(5.3) C(~x)= x2+y2 f2(z) ,
2 −
(cid:0) (cid:1)
to carry out the construction in detail, and make the resulting formulas explicit.
Defining
(5.4) F(~x),G(~x) R3 = C F G ,
{ } ∇ · ∇ ×∇
(cid:0) (cid:1)
one has
(5.5) x,y = ff′(z), y,z =x, z,x =y,
{ } − { } { }
respectively
(5.6) [X,Y]=i~ff′(Z), [Y,Z]=i~X, [Z,X]=i~Y
10 JOAKIMARNLIND,JENSHOPPE,ANDGERHARDHUISKEN
for the “quantized” (“non-commutative”)surface. In terms of the parametrization
given in (5.1), the above Poissonbracket is equivalent to
(5.7) F(u,v),G(u,v) =εab ∂ F ∂ G
a b
{ }
(cid:0) (cid:1)(cid:0) (cid:1)
where ∂ =∂ and ∂ =∂ . By finding matrices of increasing dimension satisfying
1 v 2 u
(5.6),onecanconstructamapTα havingtheproperties(3.2)and(3.3)ofamatrix
regularizationrestricted to polynomial functions in x,y,z (cp. [Arn08a]).
For the round 2-sphere, f(z)=1 z2, (5.6) gives the Lie algebra su(2), and its
−
celebrated irreducible representations satisfy
2
(5.8) X2+Y2+Z2 =1 if ~= .
√N2 1
−
When f is arbitrary,one canstill findfinite dimensionalrepresentationsof(5.6) as
follows: rewrite (5.6) as
(5.9) [Z,W]=~W
(5.10) [W,W†]= 2~ff′(Z)
−
implyingthatz z =~wheneverW =0andZ diagonal. AssumingW =X+iY
i j ij
− 6
with non-zeromatrix elements W =w for k =1,...,N 1, one thus obtains
k,k+1 k
−
(with w =w =0)
0 N
~
Z = N +1 2k
kk
2 −
(cid:0) (cid:1)
w2 w2 = 2~ff′ ~(N +1 2k)/2 Q ,
k− k−1 − − ≡ k
(cid:0) (cid:1)
which implies that
k
w2 = Q
k l
Xl=1
andtheonlynon-trivialproblemistofindtheanalogueof(5.8). Tothisend,define
1
(5.11) fˆ2 =X2+Y2 = WW†+W†W ,
2
(cid:0) (cid:1)
withW givenasabove. AsZ haspairwisedifferenteigenvalues,thediagonalmatrix
givenin(5.11)canbethoughtofasafunctionofZ;henceasfˆ2(Z). Itthentrivially
holds that
(5.12) Cˆ =X2+Y2 fˆ2(Z)=0,
−
for the representation defined above. The quantization of ~ comes through the
requirement that fˆ2 should correspond to f2. While for the round 2-sphere fˆ2
equalsf2, provided~ is chosenasin (5.8), it is easyto see thatin generalthey can
not coincide, as
[X2+Y2 f(Z)2,W]=[(WW†+W†W)/2 f(Z)2,W]
− −
1 1
= W[W†,W]+ [W†,W]W f(Z)[f(Z),W] [f(Z),W]f(Z)
2 2 − −
= =f(Z) ~f′(Z)W [f(Z),W] + ~f′(Z)W [f(Z),W] f(Z)
··· − −
(cid:0) (cid:1) (cid:0) (cid:1)
with off-diagonalelements
f(z )+f(z ) ~f′(z ) (f(z ) f(z ))
k k−1 k k k−1
− −
(cid:0) (cid:1)(cid:0) (cid:1)