An essay on the foundations of geometry - Bertrand Russell

The method of determining the measure of curvature from within is, briefly, as follows: If any point on the surface be determined by two coordinates, u, v, then small arcs of the surface are given by the formula

ds² = Edu² + 2Fdu dv + Gdv²,

where E, F, G are, in general, functions of u, v.[27] From this formula alone, without reference to any space outside the surface, we can determine the measure of curvature at the point u, v, as a function of E, F, G and their differentials with respect to u and v. Thus we may regard the measure of curvature of a surface as an inherent property, and the above geometrical definition, which involved a reference to the third dimension, may be dropped. But at this point a caution is necessary. It will appear in [Chap. III. (§ 176)], that it is logically impossible to set up a precise coordinate system, in which the coordinates represent spatial magnitudes, without the axiom of Free Mobility, and this axiom, as we have just seen, holds on surfaces only when the measure of curvature is constant. Hence our definition of the measure of curvature will only be really free from reference to the third dimension, when we are dealing with a surface of constant measure of curvature—a point which Riemann entirely overlooks. This caution, however, applies only in space, and if we take the coordinate system as presupposed in the conception of a manifold, we may neglect the caution altogether—while remembering that the possibility of a coordinate system in space involves axioms to be investigated later. We can thus see how a meaning might be found, without reference to any higher dimension, for a constant measure of curvature of three-dimensional space, or for any measure of curvature of an n-dimensional manifold in general.

22. Such a meaning is supplied by Riemann's dissertation, to which, after this long digression, we can now return. We may define a continuous manifold as any continuum of elements, such that a single element is defined by n continuously variable magnitudes. This definition does not really include space, for coordinates in space do not define a point, but its relations to the origin, which is itself arbitrary. It includes, however, the analytical conception of space with which Riemann deals, and may, therefore, be allowed to stand for the moment. Riemann then assumes that the difference—or distance, as it may be loosely called—between any two elements is comparable, as regards magnitude, to the difference between any other two. He assumes further, what it is Helmholtz's merit to have proved, that the difference ds between two consecutive elements can be expressed as the square root of a quadratic function of the differences of the coordinates: i.e.

ds² = Σ₁ⁿ Σ₁ⁿ a_ik dx_i.dx_k ,

where the coefficients a_ik are, in general, functions of the coordinates x₁ x₂ ... x_n. [28] The question is: How are we to obtain a definition of the measure of curvature out of this formula? It is noticeable, in the first place, that, just as in a surface we found an infinite number of radii of curvature at a point, so in a manifold of three or more dimensions we must find an infinite number of measures of curvature at a point, one for every two-dimensional manifold passing through the point, and contained in the higher manifold. What we have first to do, therefore, is to define such two-dimensional manifolds. They must consist, as we saw on the surface, of a singly infinite series of geodesics through the point. Now a geodesic is completely determined by one point and its direction at that point, or by one point and the next consecutive point. Hence a geodesic through the point considered is determined by the ratios of the increments of coordinates, dx₁ dx₂ ... dx_n. Suppose we have two such geodesics, in which the i′th increments are respectively d′x_i and d″x_i. Then all the geodesics given by

dx_i = λ′d′x_i + λ″d″x_i

form a singly infinite series, since they contain one parameter, namely λ′: λ″. Such a series of geodesics, therefore, must form a two-dimensional manifold, with a measure of curvature in the ordinary Gaussian sense. This measure of curvature can be determined from the above formula for the elementary arc, by the help of Gauss's general formula alluded to above. We thus obtain an infinite number of measures of curvature at a point, but from n.(n – 1) 2 of these, the rest can be deduced (Riemann, Gesammelte Werke, p. 262). When all the measures of curvature at a point are constant, and equal to all the measures of curvature at any other point, we get what Riemann calls a manifold of constant curvature. In such a manifold free mobility is possible, and positions do not differ intrinsically from one another. If a be the measure of curvature, the formula for the arc becomes, in this case,

ds² = Σdx² _{_/} (1 + a 4 Σx²)^{^²}.