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CONFORMAI, GEOMETRY
B Y
EDWARD
KASNER.
INTRODUCTION.
The connection of conformai transformations with the theory of functions of
a complex variable is so important that the geometry based on the group of
conformai transformations has been studied almost exclusively as an auxiliary to
the theory of functions.
Conformai geometry, that is the study of those properties of geometric configurations which are invariant under all conformai transformations, has therefore not
yet been developed into a systematic theory, comparable, for example, with projective
geometry. The former theory is naturally more difficult than the latter, since it is
based on a larger group. It wrould seem that it deserves study for its own sake, and
also as an example of a geometry based on an infinite continuous group, instead of
(in Lie's terminology), a finite continuous group.
The simplest configurations to investigate in conformai geometry are curves and
sets of curves. The curves considered will throughout be assumed to be analytic : we
shall confine ourselves in fact to regular real analytic arcs. The conformai transformations are assumed to be regular in a region including the curves considered.
A single curve has no invariants; any curve may be transformed conformally
into any other curve, in particular into a straight line.
The first configuration of real interest is composed of two curves having a
common point, that is, a curvilinear angle. The conformai transformations operating
on the angle are assumed to be regular in a region including the vertex of the angle.
The result of such a transformation is a new curvilinear angle, having the same
magnitude as the original. This common magnitude we shall denote by 6 : this is
the radian measure of the angle formed by the tangent lines at the vertex. It is
a differential invariant of the first order since it depends only on the slopes of the
curves.
If two curvilinear angles are conformally equivalent, they certainly have the
same 6, but is this sufficient ? If two angles are equal in magnitude, that is have
the same 6, will they necessarily be conformally equivalent? This is certainly a
fundamental question in conformai geometry. We shall show that the answer is not
always in the affirmative. Curvilinear angles exist, which while equal in magnitude,
are not conformally equivalent. For example : it is possible to construct a curvilinear
angle formed by two analytic arcs, intersecting orthogonally, which can not be
transformed conformally into an angle formed by a pair of perpendicular straight
lines (the transformation to be of course regular at the vertex).
6
M. c. n.
82
EDWARD KASNER
This fact indicates the existence of other invariants besides 0. The principal
problem of the present paper is to find these absolute invariants of higher order. The
invariants considered are differential invariants involving curvatures and derivatives
of the curvatures of the sides of the curvilinear angle. Only invariants of finite
order are discussed. The existence of invariants of infinite order, that is, those
involving all the coefficients in the power series representing the sides of the angle,
is not settled. It is shown that if 0 is a rational part of ir (in which case we shall
term the angle rational), there exist higher invariants. If the angle is irrational, it
is shown that no higher invariants exist.
Among the rational angles we have to include the case where 0 is zero : such an
angle is termed a horn angle. Each type of horn angle has a unique invariant whose
order depends on the degree of contact of the sides.
The question as to when two curvilinear angles are conformally equivalent is not
yet completely solved. In order that two angles shall be equivalent, it is necessary
that not simply 0, but the higher invariants obtained in this paper shall be equal.
Whether this is also sufficient depends on the existence of invariants of infinite order.
At the end of the paper we consider briefly the corresponding (dual) problem of
equilong geometry.
CONFORMAL INVARIANTS OF CURVILINEAR ANGLES.
THEOREM I. An irrational curvilinear angle has no conformai invariants
higher order ; that is, the magnitude 0 of the angle is its only invariant.
oj
To prove this consider two angles of the same magnitude 0, where 0/TT is
irrational. Without loss of generality we may assume the angles in reduced form,
that is, one side a straight line. Thus the angle in the plane z = x + iy is formed say
by the axis of reals and the curve
y = aix + a2x2 +
(1);
and the angle in the plane Z — X + iY is formed by the axis of reals and the curve
Y = A1X + A9X* +
(2).
By assumption a1=A1 = tan 0. To shew that there is no other invariant relation
involving a finite number of higher coefficients of (1) and (2), we prove that a regular
conformai transformation
Z=c1z + c 2 2 2 4
(3),
(where the c's are real and cx =f 0), converting (1) into (2), can be (formally) found.
For the determination of the c's, we have an infinite number of equations. The
first is an identity in virtue of a1 = A1 and allows Cj to be arbitrary. The second
involves c1 and c2 and is linear in c2. The /eth involves c1 up to cK and is linear in cK.
The coefficient of cK can be put into the form
sin K0 — tan 0 cos /c0
(4).
This cannot vanish for any integer K > 1 ; for if it did 0 would be a rational part
of 7T. Hence for arbitrary values of a1} a2, ..., an and Aly A2, ..., An values of
Cj, c2, ..., cn can be determined: in fact in GO 1 ways since cx is arbitrary.
CONFORMAL GEOMETRY
83
For arbitrary curves (1) and (2) the series (3) can be formally calculated, the
coefficients being rational functions of the parameter c2. Probably for some value of
c1 the corresponding series (3) will be convergent. This, however, the author has not
succeeded in proving. If there are actually cases where (3) is divergent for all values
of d, this would indicate the existence of invariant relations involving all the
coefficients of the curves, that is, invariants of infinite order.
If the angle is rational, 0/7r=p/q, where the fraction is supposed to be reduced to
lowest terms, some of the expressions (4) will vanish, namely those for which K is
1 + 1 or 2q + 1 or 3q + 1, etc. The series (3) cannot usually be found even formally,
Conditions of consistency arise. The discussion of the system of equations for the c's
is complicated, but by a certain device we can prove
THEOREM II. Every curvilinear angle of rational magnitude has a conformai
invariant of higher order.
For this purpose, we consider the process of general symmetry, or reflexion,
with respect to an analytic curve. The function-theoretic definition of Schwarz may
be stated in purely geometric language as follows: two points are symmetric with
respect to a given curve provided the pairs of minimal lines determined by the points
intersect on the given curve. For an analytic arc a neighbourhood (region) can be
found such that each point in the neighbourhood has a unique symmetric point or
image in the neighbourhood. The process is covariant under the conformai group.
It is fundamental in our geometry.
Consider then an angle of magnitude 0 with vertex V and curved sides a and b.
The image of a with respect to b is a definite curve c passing through V. The
magnitude of the angle b, c is of course 0. Take the image of b with respect to c, etc.
If 0 is irrational we shall obtain curves passing through V so that their tangents are
everywhere dense. But if 0 is rational we shall after a finite number of steps arrive
at a curve having the same direction as a. Hence every rational angle determines a
unique horn angle, i.e. angle of magnitude zero, formed by two curves in contact at
the vertex. Any invariant of this angle will be an invariant of the original rational
angle. I t now remains to prove
THEOREM
III.
Every horn angle has one and only one higher conformai
invariant.
If the order of contact of the two sides of the horn angle is h — 1, so that they
have h consecutive points in common, we shall say that the angle is of type h and
denote it by Hh. Of course the integer h is invariant under conformai transformation
as it is under all contact transformation. The simplest type (ordinary contact) is H2.
Consider two angles of type h. Taking one side in each plane to be the axis of
reals, and the vertex as origin, the curved sides are of the form
h
y==ahx
-j-ah+1xh+1
h
Y = AhX
+ ...
+ Ah+1X^
+
(5),
(6).
Expressing the fact that the transformation (3) converts (4) into (5), we are led to a
system of equations for the determination of the coefficients c1} c2, — I t is sufficient
6—2
84
EDWARD KASNER
to note that the /cth equation involves the first K coefficients and is linear with
respect to cK, the coefficient of this term being
(/c-h)ah
(7).
By assumption ah does not vanish, therefore expression (7) vanishes when and only
when fc = h. The first h equations thus involve only cY, c2, ..., ch^l : eliminating these
we find a relation involving ah, ..., a2Ä_2 and Ah, ..., A2ll_Y.
There are no other
relations since no other eliminations are possible.
The order of thé unique invariant of a horn angle of type his 2h — l.
the invariant by I^-i*
We denote
For a horn angle H2 (contact of first order), the invariant in question is
'•-£
«»
This is, of course, for the reduced form, in which one side is the axis of reals.
the general form where both sides of the angle are curved, we find
For
dji _ c?72
^3 — 7
rz
{&),
(7i - ^y
where y1} <y2 denote the curvatures of the sides (at the vertex), and sl9 s2 denote the
arc lengths. This is the simplest example of a conformai invariant of higher order.
Every horn angle can be reduced conformally (so far as terms of finite order are
concerned) to the normal form in which the sides are y = 0 and y = xh + Xx271"1. The
constant \ is then essential, being equivalent to the invariant I2h~iReturn now to the discussion of rational angles. Such an angle determines a
horn angle by the process of successive symmetry described above. If this angle is
of type h, its invariant I2n-i will be an invariant of the rational angle. This proves
Theorem II. In general if the ratio of 0 to nr is p to q (in lowest terms) then the
type h equals q + 1 so that the invariant is of order 2^ + 1. However, for some
rational angles, h may have a greater value. It is necessary to classify rational angles
of a given magnitude first according to the values of the arithmetic invariant h, and
then according to the values of the geometric (differential) invariant I2h~iA question remains. Can a rational angle have more than one higher invariant ?
The related horn angle has only one invariant as stated in Theorem II. The horn
angle is determined by the rational angle, but is the converse true ? We are led to
the following general problem of conformai geometry: given two curves a and b
through a point V, find n — 1 curves through V so that the image of a in the first is
the second, the image of the first in the second is the third, and so on until finally
the image of the penultimate in the last is 6. This may be termed the equipartition
problem for curvilinear angles. Obviously if the solution exists the magnitudes of
the angles formed by the successive curves will all be equal.
The simplest case, n = 2, is the problem of bisection of a curvilinear angle : to
construct a curve c so that the given curves a and b shall be symmetric with respect
to c. Judging from ordinary angles, where the sides are straight lines, we should
expect two solutions, an internal and an external bisector. This is, however, not
CONFORMAL GEOMETRY
85
always true. We state only the following results. If the given angle is a horn angle
for which the sides a and b have contact of exactly the first order, then there will be
an internal bisector, but no external. If the sides of the horn angle have contact
of exactly the second order, then both internal and external bisectors exist*.
Consider now a curvilinear angle of magnitude 7r/2. The image of the first side
in the second will be a curve having contact of even order with the first side. For a
general right angle the contact will be of second order, that is, the related horn angle
is of type 3. Right angles such that the type of the horn angle is 5, 7, 9, ... are of
greater and greater specialty. In the extreme exceptional case, the order of contact
is infinite, that is, the right angle is such that each side is its own image with respect
to the other side : this kind of right angle is conformally reducible to a pair of
perpendicular straight lines.
A right angle of general type has a unique invariant ; this is of the fifth order.
This follows because the related horn angle, of type 3, uniquely determines the
right angle. The invariant in question is the I5 of the related horn angle. If the
original right angle is given in reduced form so that one side is y = 0 and the
other is
x = ß2y2 + ß,y?'+ ...,
the invariant is found to be
The exceptional right angles arise when ßt vanishes.
rational angle, there is only one higher * invariant.
Probably for each type of
EQUILONG GEOMETRY.
This theory is a sort of dual (not however the direct projective dual) of the
conformai theory. It is based on the infinite group of equilong transformations
introduced by Scheffers in 1905. These are related to the theory of functions of dual
numbers u +jv, where j is an imaginary unit whose square is zero, just as the
conformai transformations are related to functions of the ordinary complex numbers
x 4 iy, where the square of i equals — 1.
Dual numbers are interpreted geometrically by directed straight lines, using
Hessian line coordinates ; curves are considered as envelopes of straight lines and are
also oriented. Any curve may be transformed into any other curve ; in particular,
any curve may be transformed into a point (of course, a point is considered as the
envelope of all the straight lines passing through it). A single curve, therefore, has
no invariant.
The first configuration to be studied consists of two curves having a common
tangent line. This is in fact the dual of the figure discussed in the first part,
namely: a curvilinear angle composed of two curves having a common point.
The configuration now to be studied has not been given any special name. It
possesses an obvious invariant with respect to the equilong group, namely: the
* In general if the type of the horn angle is even, no external bisector exists.
an external bisector probably exists.
If the type is odd,
86
EDWARD KASNER
distance measured on the common tangent between the two points of contact. The
question arises as to whether the configuration has additional invariants, that is,
invariants of higher (but finite) order. A special case arises when the distance
between the points of contact is zero. In this case the curves have a common
tangent and a common point, and therefore form a horn angle. A horn angle arises
in both theories since it is in fact a self-dual configuration. The following results are
obtained :
IV. The figure formed by two curves having a common tangent and
distinct points of contact has no higher equilong invariant.
THEOREM
THEOREM V. If the points of contact coincide, so that the figure is a horn angle,
then there will be one, and only one, equilong invariant of higher order.
In this, as well as in the previous discussion, the existence of invariants
of infinite order is left unsettled. The decision as to whether such invariants exist
depends on questions of convergence of certain power series which can be formally
constructed.
For a horn angle of type h, the unique invariant is of order 2A — 1. It is of
course different from the corresponding conformai invariant I2h-i and is here denoted
by J2h-i.
For the simplest type, h = 2, the equilong invariant is
drx dr2
d0
d02
J,= 1
(Ti - r2f
where i\, r2 denote the radii of curvature and 0l9 02 denote the inclinations of the two
curves to any fixed initial line. Of course the values of the radii and the derivatives
are taken at the vertex of the angle.
CHARACTERIZATION OF THE TWO GROUPS.
From the usual points of view conformai transformations and equilong transformations are characterized in entirely different ways. Conformai transformations
are singled out from all point transformations by the requirement that the angle
between two curves at a common point shall have its magnitude preserved. Equilong
transformations, on the other hand, are singled out from all line transformations by
requiring that the distance between two curves measured on a common tangent shall
be preserved. We shall show that both groups may be characterized in the domain
of all contact transformations in terms of their behaviour with respect to horn
angles.
Every contact transformation turns a horn angle into a horn angle. It is sufficient
to consider the simplest type, where the contact is of first order. Conformai transformations leave unaltered a certain invariant of the third order, discussed in the first
part and denoted by Is. Equilong transformations leave unaltered the invariant
denoted by J3. Both of these quantities are differential expressions of the third
order ; they are in fact combinations of the curvatures of the sides of the horn angle,
and of the rates of variation of the curvatures.
CONFORMAL GEOMETRY
87
THEOREM VI. Conformai transformations are the only contact transformations
for which the Iò of every horn angle is unaltered. Equilong transformations are the
only contact transformations for which the Js of every horn angle is unaltered.
It thus appears that the conformai group may be characterized without
mentioning angular magnitude, and the equilong group may be characterized without
mentioning distance.
We note that the two invariants may be put into the form
d201 _ cM)2
T
_ ds,2
-*3 —
d2Sj
2
J,=
dsi
fd0x _d£1\2>
\dsj
ds2)
d2s2
W
d02
(ds1
ds2\2 '
U#r
d0J
which differ only by the interchange of the letters s and 0.
It is to be remembered that there is no (known) automatic principle by means
of which we can pass from the results of conformai geometry to those of equilong
geometry; we have to deal with a general analogy rather than a strict duality. What
is, for example, the analogue of isothermal systems of curves, which are so important
in conformai geometry ? The resulting systems are defined in terms of Hessian line
coordinates by linear differential equations of the first order and their theory is
essentially simpler than that of isothermal systems.