Given this result, let us show the following corollary:
hx 0 h constant,
2. IIAPII = IIPII•
3. AT=A-1.
4. A = {a1 a2] such that .\/< az, > = ö,.
Definition A.6 The real orthogonal group 0(2, R) (of order 2) is defined to be the set of all matrices:
O(2,R) := {A E M2R I AT =
where the group product is matrix multiplication.
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Proposition A.7 Let f : R2 x R2 R2 be a function,. Then f E E(2) such that 1(0) = 0 if and only if 3! A E 0(2, R) such that for all p E I''2, f (p) = Ap.
Remark A.8 By Lemma A.5.4, if A E 0(2,R), then 3 8 E [0, 2x) such that either:
A= cos 8 — sin 8 sin 8 cos 8Re,
which is (counterclockwise) rotation by 0, or:
cos 0 sin() A _
sin8 —cos° '
which is reflection across the line making an angle of 8/2 with the horizontal.
If we recall that:
SO(2, R) :_ {A E 0(2,R) I det(A) = 1}, then by the above:
SO(2,R) = {Ro I E [0, 27r)},
where RB is rotation by 0, so SO(2, R) is also known as the 2-dimensional rota-tion group. Incidentally, because of the identities for cos(0 + 0) and sin(0 + 0), 0(2, R) is abelian.
We now state the following characterization theorem for E(2):
Theorem A.9 f E E(2) iff 3! A E 0(2,R) and b E 11)2 such that V p E R2:
.f(P) = Ap + b.
Proof: Let f E E(2). If we define the transformation g : R2 -f R2 by g(p) f (P) — f (0), then:
d (g(P), g(q)) = d (f (P), .f (q)) = d(P, q),
since f is an isometry, so g is an isometry that fixes the origin. Hence, by Proposition A.7, g defines an A E 0(2, R), which implies that:
f(p)=Ap+f(0) for all pER2.
(~) Conversely, let f : R2 -÷ R2 such that f (p) = Ap + b for some A E 0(2, 11; ), b E R2. Then:
d(f(P),f(q)) = d(AP,Aq)
= IIAP — AgII
= IIA(P — c1)II IIP — qlI
= d(P, q)
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Consequently, f E E(2). •
In other words, we have shown that:
E(2) = { f : R2 II82 I d (f (p), f(q)) = d(p, q) V p,q E R2}
= {(A, b) I A E 0(2,R), ), b E R21.
By being able to express the elements of E(2) in a particularly concise, algebraic form, we can now prove that it forms a group under composition.
Theorem A.10 Let o : E(2) x E(2) -* E(2) be a binary operation on E(2) defined by:
o((A', b'), (A, b)) := (A', b') o (A, b) (A'A, A'b + b').
Then (E(2), o) is a group.
Proof: It is not difficult to check that E(2) satisfies the group axioms with respect to o; for reference, we note that (A, b)-1 = (A~', —A-'b). •
Since the symmetry group of a set is the group of all isometries with respect to composition, we conclude that the 2-dimensional Euclidean group, E(2), is the symmetry group of R2. Note that E(2) has the subgroup:
SE(2) := {(A, b) E E(2) I A E SO(2, IR)},
which is known as the special Euclidean group of motions of R2.
We conclude this appendix with a comment regarding the group structure of SE(2):
Definition A.11 Let G be a group with identity e, a normal subgroup N a G, and a subgroup H C. G. Then G is a semi-direct product of H and N, written G = H x N (note the direction of the symbol!), whenever G = NH and NfH={e}.
In our case, we have:
G = E(2),
N =R2 = {(12,b) E E(2)}, and H = O(2, R) = {(A, 0) E E(2)},
so then:
O(2, R) o R2 = {(A, 0) o (I2, b)}
= {(A, Ab)}
= E(2),
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and:
0(2,R) n i^2 = {(A, 0)} n {(12,0)}
={(12,0)}.
Therefore:
E(2) = 0(2,R) v 1R2, and similarly, we see that:
SE(2) = SO(2,R) ix R2. *
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B The Schwarzian Derivative and GL(2, R)
In this appendix, we provide some propositions and lemmas necessary for discussion in Section 4.
Proposition B.1 Let A, B E GL(2, R), where:
A = ail a12
a21 a22
and B =b11 b b12 21 b22 S
and TA :R-+R be the linear fractional transformation defined as:
a11x + a12 TA(x) :=
a21 x + a22 1. TA is one-to-one.
2. TA o TB = TAB
3. TA=T<;A for all cElR*.
4. TA=T12 A=cI2 for all cEIR*.
Proof:
1. Let x,y E IR; then:
Txallx A() =7 + a12=ally + a12= 7A(l)
a21 x + a22a2l y + a22 a11a22x + ai2a2ly = a2lal2x + alla22y
det (A) (x - y) = 0
4 x = y since det(A) 0. A 2. Let x E IR; then:
=bll x + b12
TA(TB(x))TA ,0
21X+ b22
a buz+biz llbzix+bzz + a12
bj 1 x+b12 a21b21 x+b22+a22
_ (allbll + a12b21)x + (allb12 + a12b22) (a21b11 + a22b21)x + (a21b12 + a22b22
= TAB (x) • A
the
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3. Let x E R; then for c E R*:
a11x + a12
TA(x) =
a21 x + a22 ca11x + ca12 ca21 x + Ca22
= TcA(x)•
4. If Tc12 (x) = TA(x) for all x E IR, then:
cx+0 aux+ai2
= x =
Ox+c a21x+a
/22
<=> 0 = a21x2 +(a22 — aii)x — a12.
Hence:
a21 = 0 = a12, and all = a22 0.
The converse holds trivially as a special case of 3. •
Remark B.2 The transformation TA can be extended to a transformation on RP1 in the following way:
a11x1 + a12x2
TA ([xi : x2])•
a21x 1 + a22x2
Thus, when we are in a neighbourhood of 0 E RP1, we can write this as:
TA([x: 1]) =
a11x + a12 a21 x + a22
and when we are in a neighbourhood of oo E RP', we can write this as:
all+ a12y
TA([1: y]) =
a21 + a22Y!
so we see that linear fractional transformations are well-defined on RP1, as well.
Lemma B.3 Let f : M -* P: be a smooth function with non-zero derivative.
Then, 3 A E GL(2, IR) such that:
au + b f(
u) = TA(u) = cu + d if and only if Su (f) = 0.
Proof: (=) Suppose f = TA for some A E GL(2, IR). Then by the calcula-tion given in the (=) direccalcula-tion of Theorem 4.3, we have:
Su,(TA)=Su(f)=Su(u)=0.
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O Suppose Su( f) = 0. Then by definition of Su( f) (Equation (4.1)) : f_„ 1 f„ 2
Su(f) = 0 (T)=2f,.
Letting g(u) = f,(u) for the moment, we can re-write this as:
dg=1du g2 and integrate to obtain:
where c, d E , c 0 are integration constants. Since f' 0 0 for all u:
f,/(lnif'~)'
is well-defined (i.e. either f' > 0 V u or f' < 0 b' u), so we can integrate again to obtain:
i1
,1j)-2) lad ln~fI=In((u++In(c2bci ),
where a, b E R such that ad - bc 0 are also integration constants. Exponenti-ating yields:
ad—bc
so integrating one last time gives us:l bc—ad
f(u)= ---c2a .f
u+~C c+
u+d
au + b
_
cu -F- d
Hence, for a, b, c, d E lR such that ad - bc 0, where:
f (0) =b(0) _ad - bcandf"(0)-2c dd2f'(0) d '
we can conclude that there is a A E GL(2,R) such that f (u) = TA(u) for all uEM. ^
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Lemma B.4 If v is another coordinate on M and u = u(v), then for a projective curve p(u) = [f (u) : 11, its Schwarzian satisfies:
Sv (f) dv2 = Su(f)du2 + Su (u) dv2.
In particular, Sv(u) dv2 = —Su(v) du2.
Proof: By direct calculation, we see that:
Sv (.f)—a(.1ou) (v)3dv2(tou)
d3(v) 2 aU (f ° u)(v) 2av(f o u) (v)
_.f"(u)(du)3+3 fu(u).2du+fi(u)d~3f'~(u)(du)2+.f`('a)d2u2
_ az,d2 \dvaU1dv~^vd~,z
f/ (u) av 2f(u) aut ii~2ii2'STY-2"2d2z~2
_ fdu3 f dud—„33fdu3 f du 3
f' dv 7- dv2 +du dvdv 2 f' dv f' dv2 2 chi.
fiii 3f"22dv3d2
---- 2l (dudvdu—23du
f''f1\
rdv dv
= Su(f)ld^ v)2 +'Sv(s)-
Hence, we obtain:
Sv(.f) dv2 = Su(f) du2 + Su(v) dv2.
In particular, for f (v) = v, Sv (v) = 0, so:
0 = Su (v) du2 + Sv (u) dv2
Su (V)du2 = -SU(u) dv2. •
As a corollary to the above lemmas, we observe the following:
Corollary B.5 Let p : M -+ IRP1 be a curve with inhomogeneous coordinate expression p(u) = [f(u) : 1]. If there is an A E GL(2, R) such that:
v(u) = TA(u) =au + b
cu+d then:
Su( f) du2 = So(f) dv2.
Remark B.6 We conclude with a remark on our method of associating the Schwarzian with the projective curve. Suppose we had instead used the change of variables method introduced at the beginning of Section 6: for
x"+alx'+aax =0, where:
xlx2 —x2x1xlx2—x2x1
a--- 1 and ac ---
xix2 — x2x1xlx2,— x,2x1
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let:
x -* x exp [ — z f al du]
= x exp [ f 2 au lnJxlx2 — x2xi I du]
= x1/'xcx2 — x2xil . Then:VV
x"+Qx=0, where:
Q=a0 —al
12
— 2 a13 4x2 — 4xi 1 xi x2 - x2 xi 3(x`-7 ,x2 — x2xi 2 xix2 — x2xi + 2 xix2 — x2xi4 xix2 — x2xi • However, by direct calculation, we see that:
(2)„, 1(Z)//)„2 S u x2=(xi ), — 2 (7t2---),
xix2 -
+ xi x2 - x2 xi 3 xix2 — x2 xl
xix2 —x2:X;1 xix2 — x<2x1 2 xix2 — :X:2:X:1
„
} 2(xix2---—x2xix2,—2x2
\xix2 —x2xi x2 X2
2
3xix2-x2xix',/ ./x2 - xxi3xix2 - xix2 — x2xi+xix2 — x2xi- 2xix2 — x2xi where we have used the fact that x2 is a solution of Equation (4.3), so:
1
x:X;1:X;"x2—x"'X;1 :X:,:X'12
-212 2
/2—2 1 = —ai -- ao=
X2 x2 xix2 — x2xi X2 xix2 — x2x1 • Therefore, we conclude that:
Q(u) = 25u(2)(u),
so either approach to bringing the ODE into its normal form is equivalent. 4
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Acknowledgements
I owe my deepest gratitude, first and foremost, to my parents, for letting me travel overseas to continue my studies in mathematics, and my wife, for supporting me in my studies and work. Of course, it goes without saying that I could not have completed this thesis without the guidance and supervision of my advisor, Dr. Martin Guest, who first suggested that I compare the material in Dr. Inoguchi's book with his own work in D-modules. I would also like to thank Dr. Inoguchi, himself, for taking the time to answer various questions of mine about his book and the topic. Last but not least, I would like to express my gratitude to the Tokyo Metropolitan University, for granting me a scholarship in my first year and a tuition exemption in my second year, and the Ishimori Memorial North American Friendship Scholarship Fund, for granting me a scholarship in my second year.
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