§§ 8–10 & Concluding Remarks: Birational, Topological, Contact & Manifold Groups¶
A Comparative Review of Recent Researches in Geometry — §§ 8–10 & Concluding Remarks (Trans. M. W. Haskell, 1893, pp. 232–242)
§ 8. Enumeration of Other Methods Based on a Group of Point-Transformations¶
Elementary geometry, the geometry of reciprocal radii, and likewise projective geometry, if we disregard the dualistic transformations connected with the interchange of the space-element, are included as special cases among the large number of conceivable methods based on groups of point-transformations. We will here mention especially only the three following methods, which agree in this respect with those named. Though these methods are far from having been developed into independent theories in the same degree as projective geometry, yet they can clearly be traced in the more recent investigations [1].
1. The Group of Rational Transformations¶
In the case of rational transformations we must carefully distinguish whether they are rational for all points of the region under consideration, viz., of space, or of the plane, etc., or only for the points of a manifoldness contained in the region, viz., a surface or curve. The former alone are to be employed when the problem is to develop a geometry of space or of the plane in the meaning hitherto understood; the latter obtain a meaning, from our point of view, only when we wish to study the geometry on a given surface or curve. The same distinction is to be drawn in the case of the analysis situs to be discussed presently.
The investigations in both subjects up to this time have been occupied mainly with transformations of the second kind. Since in these investigations the question has not been with regard to the geometry on the surface or curve, but rather to find the criteria for the transformability of two surfaces or curves into each other, they are to be excluded from the sphere of the investigations here to be considered [2]. For the general synopsis here outlined does not embrace the entire field of mathematical research, but only brings certain lines of thought under a common point of view.
Of such a geometry of rational transformations as must result on the basis of the transformations of the first kind, only a beginning has so far been made. In the region of the first grade, viz., on the straight line, the rational transformations are identical with the linear transformations and therefore furnish nothing new. In the plane we know the totality of rational transformations (the Cremona transformations); we know that they can be produced by a combination of quadratic transformations. We know further certain invariant properties of plane curves [with reference to the totality of rational transformations], viz., their deficiency (genus), the existence of moduli; but these considerations have not yet been developed into a geometry of the plane, properly speaking, in the meaning here intended. In space the whole theory is still in its infancy. We know at present but few of the rational transformations, and use them to establish correspondences between known and unknown surfaces.
2. Analysis Situs¶
In the so-called analysis situs we try to find what remains unchanged under transformations resulting from a combination of infinitesimal distortions. Here, again, we must distinguish whether the whole region, all space, for instance, is to be subjected to the transformations, or only a manifoldness contained in the same, a surface. It is the transformations of the first kind on which we could found a space geometry. Their group would be entirely different in constitution from the groups heretofore considered. Embracing as it does all transformations compounded from (real) infinitesimal point-transformations, it necessarily involves the limitation to real space-elements, and belongs to the domain of arbitrary functions. This group of transformations can be extended to advantage by combining it with those real collineations which at the same time affect the region at infinity.
3. The Group of All Point-Transformations¶
While with reference to this group no surface possesses any individual characteristics, as any surface can be converted into any other by transformations of the group, the group can be employed to advantage in the investigation of higher configurations. Under the view of geometry upon which we have taken our stand, it is a matter of no importance that these configurations have hitherto not been regarded as geometric, but only as analytic, configurations, admitting occasionally of geometric application, and, furthermore, that in their investigation methods have been employed (these very point-transformations, for instance) which we have only recently begun to consciously regard as geometric transformations. To these analytic configurations belong, above all, homogeneous differential expressions, and also partial differential equations. For the general discussion of the latter, however, as will be explained in detail in the next section, the more comprehensive group of all contact-transformations seems to be more advantageous.
The principal theorem in force in the geometry founded on the group of all point-transformations is this: that for an infinitesimal portion of space a point-transformation always has the value of a linear transformation. Thus the developments of projective geometry will have their meaning for infinitesimals; and, whatever be the choice of the group for the treatment of the manifoldness, in this fact lies a distinguishing characteristic of the projective view.
Not having spoken for some time of the relation of methods of treatment founded on groups, one of which includes the other, let us now give one more example of the general theory of § 2. We will consider the question how projective properties are to be understood from the point of view of “all point-transformations,” disregarding here the dualistic transformations which, properly speaking, form part of the group of projective geometry. This question is identical with the other question, What condition differentiates the group of linear point-transformations from the totality of point-transformations? What characterizes the linear group is this, that to every plane it makes correspond a plane; it contains those transformations under which the manifoldness of planes (or, what amounts to the same thing, of straight lines) remains unchanged. Projective geometry is to be obtained from the geometry of all point-transformations by adjoining the manifoldness of planes, just as elementary geometry is obtained from projective geometry by adjoining the imaginary circle at infinity. Thus, for instance, from the point of view of all point-transformations the designation of a surface as an algebraic surface of a certain order must be regarded as an invariant relation to the manifoldness of planes. This becomes very clear if we connect, as Grassmann does (Crelle’s Journal 44, 1852, 1–25), the generation of algebraic configurations with their construction by lines.
§ 9. On the Group of All Contact-Transformations¶
Particular cases of contact-transformations have been long known; Jacobi has even made use of the most general contact-transformations in analytical investigations, but an effective geometrical interpretation has only been given them by recent researches of Lie's [3]. It will therefore not be superfluous to explain here in detail what a contact-transformation is. In this we restrict ourselves, as hitherto, to point-space with its three dimensions.
By a contact-transformation is to be understood, analytically speaking, any substitution which expresses the values of the variables $x, y, z$ and their partial derivatives $\frac{dz}{dx} = p$, $\frac{dz}{dy} = q$ in terms of new variables $x', y', z', p', q'$. It is evident that such substitutions, in general, convert surfaces that are in contact into surfaces in contact, and this accounts for the name. Contact-transformations are divided into three classes (the point being taken as space-element), viz., those in which points correspond to the $\infty^3$ points (the point-transformations just considered); those converting the points into curves; lastly, those converting them into surfaces. This classification is not to be regarded as essential, inasmuch as for other $\infty^3$ space-elements, say for planes, while a division into three classes again occurs, it does not coincide with the division occurring under the assumption of points as elements.
If a point be subjected to all contact-transformations it is converted into the totality of points, curves, and surfaces. Only in their entirety, then, do points, curves, and surfaces form a body of our group. From this may be deduced the general rule that the formal treatment of a problem from the point of view of all contact-transformations (e.g., the theory of partial differential equations considered below) must be incomplete if we operate only with point- (or plane-) co-ordinates, for the very reason that the chosen space-elements do not form a body.
If, however, we wish to preserve the connection with the ordinary methods, it will not do to introduce as space-elements all the individual configurations contained in the body, as their number is $\infty^\infty$. This makes it necessary to introduce in these considerations as space-element not the point, curve, or surface, but the “surface-element,” i.e., the system of values $x, y, z, p, q$. Each contact-transformation converts every surface-element into another; the $\infty^5$ surface-elements accordingly form a body.
From this point of view, point, curve, and surface must be uniformly regarded as aggregates of surface-elements, and indeed of $\infty^2$ elements. For the surface is covered by $\infty^2$ elements, the curve is tangent to the same number, through the point pass the same number. But these aggregates of $\infty^2$ elements have another characteristic property in common. Let us designate as the united position of two consecutive surface-elements $x, y, z, p, q$ and $x + dx, y + dy, z + dz, p + dp, q + dq$ the relation defined by the equation
Thus point, curve, and surface agree in being manifoldnesses of $\infty^2$ elements, each of which is united in position with the $\infty^1$ adjoining elements. This is the common characteristic of point, curve, and surface; and this must serve as the basis of the analytical investigation, if the group of contact-transformations is to be used.
The united position of consecutive elements is an invariant relation under any contact-transformation whatever. And, conversely, contact-transformations may be defined as those substitutions of the five variables $x, y, z, p, q$, by which the relation
is converted into itself. In these investigations space is therefore to be regarded as a manifoldness of five dimensions; and this manifoldness is to be treated by taking as fundamental group the totality of the transformations of the variables which leave a certain relation between the differentials unaltered.
First of all present themselves as subjects of investigation the manifoldnesses defined by one or more equations between the variables, i.e., by partial differential equations of the first order, and systems of such equations. It will be one of the principal problems to select out of the manifoldnesses of elements satisfying given equations systems of $\infty^1$, or of $\infty^2$, elements which are all united in position with a neighboring element. A question of this kind forms the sum and substance of the problem of the solution of a partial differential equation of the first order. It can be formulated in the following way: to select from among the $\infty^4$ elements satisfying the equation all the twofold manifoldnesses of the given kind. The problem of the complete solution thus assumes the definite form: to classify in some way the $\infty^4$ elements satisfying the equation into $\infty^2$ manifoldnesses of the given kind.
It cannot be my intention to pursue this consideration of partial differential equations further; on this point I refer to Lie's articles already cited. I will only point out one thing further, that from the point of view of the contact-transformations a partial differential equation of the first order has no invariant, that every such equation can be converted into any other, and that therefore linear equations in particular have no distinctive properties. Distinctions appear only when we return to the point of view of the point-transformations.
The groups of contact-transformations, of point-transformations, finally of projective transformations, may be defined in a uniform manner which should here not be passed over [4]. Contact-transformations have already been defined as those transformations under which the united position of consecutive surface-elements is preserved. But, on the other hand, point-transformations have the characteristic property of converting consecutive line-elements which are united in position into line-elements similarly situated; and, finally, linear and dualistic transformations maintain the united position of consecutive connex-elements. By a connex-element is meant the combination of a surface-element with a line-element contained in it; consecutive connex-elements are said to be united in position when not only the point but also the line-element of one is contained in the surface-element of the other. The term connex-element (though only preliminary) has reference to the configurations recently introduced into geometry by Clebsch [5] and represented by an equation containing simultaneously a series of point-coordinates as well as a series of plane- and a series of line-coordinates whose analogues in the plane Clebsch denotes as connexes.
§ 10. On Manifoldnesses of Any Number of Dimensions¶
We have already repeatedly laid stress on the fact that in connecting the expositions thus far with space-perception we have only been influenced by the desire to be able to develop abstract ideas more easily through dependence on graphic examples. But the considerations are in their nature independent of the concrete image, and belong to that general field of mathematical research which is designated as the theory of manifoldnesses of any dimensions, — called by Grassmann briefly “theory of extension” (Ausdehnungslehre). How the transference of the preceding development from space to the simple idea of a manifoldness is to be accomplished is obvious. It may be mentioned once more in this connection that in the abstract investigation we have the advantage over geometry of being able to choose arbitrarily the fundamental group of transformations, while in geometry a minimum group — the principal group — was given at the outset.
We will here touch, and that very briefly, only on the following three methods:
1. The Projective Method or Modern Algebra (Theory of Invariants)¶
Its group consists of the totality of linear and dualistic transformations of the variables employed to represent individual configurations in the manifoldness; it is the generalization of projective geometry. We have already noticed the application of this method in the discussion of infinitesimals in a manifoldness of one more dimension. It includes the two other methods to be mentioned, in so far as its group includes the groups upon which those methods are based.
2. The Manifoldness of Constant Curvature¶
The notion of such a manifoldness arose in Riemann's theory (1854/1867) from the more general idea of a manifoldness in which a differential expression in the variables is given. In his theory the group consists of the totality of those transformations of the variables which leave the given expression unchanged. On the other hand, the idea of a manifoldness of constant curvature presents itself when a projective measurement is based upon a given quadratic equation between the variables (Cayley, 1859; Klein, 1871). From this point of view as compared with Riemann’s the extension arises that the variables are regarded as complex; the variability can be limited to the real domain afterwards. Under this head belong the long series of investigations touched on in §§ 5, 6, 7.
3. The Plane Manifoldness¶
Riemann designates as a plane manifoldness one of constant zero curvature. Its theory is the immediate generalization of elementary geometry. Its group can, like the principal group of geometry, be separated from out the group of the projective method by supposing a configuration to remain fixed which is defined by two equations, a linear and a quadratic equation. We have then to distinguish between real and imaginary if we wish to adhere to the form in which the theory is usually presented. Under this head are to be counted, in the first place, elementary geometry itself, then for instance the recent generalizations of the ordinary theory of curvature, etc.
Concluding Remarks¶
In conclusion we will introduce two further remarks closely related to what has thus far been presented, — one with reference to the analytic form in which the ideas developed in the preceding pages are to be represented, the other marking certain problems whose investigation would appear important and fruitful in the light of the expositions here given.
Analytic geometry has often been reproached with giving preference to arbitrary elements by the introduction of the system of co-ordinates, and this objection applies equally well to every method of treating manifoldnesses in which individual configurations are characterized by the values of variables. But while this objection has been too often justified owing to the defective way in which, particularly in former times, the method of co-ordinates was manipulated, yet it disappears when the method is rationally treated. The analytical expressions arising in the investigation of a manifoldness with reference to its group must, from their meaning, be independent of the choice of the co-ordinate system; and the problem is then to clearly set forth this independence analytically. That this can be done, and how it is to be done, is shown by modern algebra, in which the abstract idea of an invariant that we have here in view has reached its clearest expression. It possesses a general and exhaustive law for constructing invariant expressions, and operates only with such expressions. This object should be kept in view in any formal (analytical) treatment, even when other groups than the projective group form the basis of the treatment [6]. For the analytical formulation should, after all, be congruent with the conceptions whether it be our purpose to use it only as a precise and perspicuous expression of the conceptions, or to penetrate by its aid into still unexplored regions.
The further problems which we wished to mention arise on comparing the views here set forth with the so-called Galois theory of equations.
In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness. But still the comparison may be pursued further owing to the identity of the group-idea [7], and I am the more ready to point it out here, as it will enable us to characterize the position to be awarded to certain investigations begun by Lie and myself [8] in accordance with the views here developed.
In the Galois theory, as it is presented for instance in Serret’s Cours d’Algèbre supérieure (1866) or in C. Jordan’s Traité des substitutions (1870), the real subject of investigation is the group theory or substitution theory itself, from which the theory of equations results as an application. Similarly we require a theory of transformations, a theory of the groups producible by transformations of any given characteristics. The ideas of commutativity, of similarity, etc., will find application just as in the theory of substitutions. As an application of the theory of transformations appears that treatment of a manifoldness which results from taking as a basis the groups of transformations.
In the theory of equations the first subjects to engage the attention are the symmetric functions of the coefficients, and in the next place those expressions which remain unaltered, if not under all, yet under a considerable number of permutations of the roots. In treating a manifoldness on the basis of a group our first inquiry is similarly with regard to the bodies (§ 5), viz., the configurations which remain unaltered under all the transformations of the group. But there are configurations admitting not all but some of the transformations of the group, and they are next of particular interest from the point of view of the treatment based on the group; they have distinctive characteristics. It amounts, then, to distinguishing in the sense of ordinary geometry symmetric and regular bodies, surfaces of revolution and helicoidal surfaces. If the subject be regarded from the point of view of projective geometry, and if it be further required that the transformations converting the configurations into themselves shall be commutative, we arrive at the configurations considered by Lie and myself in the article cited (Math. Ann. 4, 1871), and the general problem proposed in § 6 of that article. The determination (given in §§ 1, 3 of that article) of all groups of an infinite number of commutative linear transformations in the plane forms a part of the general theory of transformations named above [9].
Footnotes (1893 Edition)