A Notation for Infinite Manifolds

Abstract

I wish to introduce a notation for the content of a manifold of geometric objects depending on arbitrary functions. Our symbolism has been found useful in problems of geometry and physics, especially those depending on systems of partial differential equations. The notation was first mentioned in my review of Riquier's treatise on differential systems. (See Bulletin of the American Mathematical Society, 1912.) The content of a set of geometric objects, each uniquely determined by n arbitrary constants, is denoted by the familiar notation oon. Thus the number of points in ordinary space is oo 3, and the totality of events in Einstein space is oo 4. There are oo 4straight lines in space. The conics of the plane are oo I in number, whereas the number in space is 00 8. There are oo I quadric surfaces in space. The number of orbits of conceivable planets in the solar system obeying Kepler's laws is oo 5. A rigid body moving freely in space has 6 degrees of freedom. The number of possible positions is 0 6. Although the preceding sets are quite large in content, we soon meet manifolds which are very much more extensive. For example, consider the totality of curves in space; this depends on two arbitrary functions of a single variable, that is, y and z are arbitrary functions of x. This set would be of the type oo X since the number of arbitrary constants is endless. But this vague symbol would represent also the number of surfaces in space, which depends on one arbitrary function of two variables, that is, z is an arbitrary function of x and y. Therefore to distinguish between these two obviously distinct manifolds, we suggest that the number of curves in space be denoted by the symbol oo 2 f (1) and the number of surfaces by the symbol oo 1 f (2). As another example, the number of cylinders in space is oo 2+1 f(1), because a cylinder is determined by selecting an arbitrary curve in a fixed plane as base and an arbitrary direction in space for the generators. The 2 denotes two arbitrary constants and may be denoted by 2 f(0). Hence the totality of cylinders in space is oo 2 f(O)+1 f(l). Similarly, we find that the total set of cones in space is o3f (0)+1f(1)* Of course usually we may omitf(0) without ambiguity. We make the following definition: Let a manifold M depend on ro constants, r1functions of 1 variable, r2 functions of 2 variables, * * * , rkfunctions of k variables. Then the content of M is expressed by the symbol