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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.