by λdx + μδx, where the parameter λ : μ may have any value.
This pencil generates a two-dimensional series of points, which
may be regarded as a surface, and for which we may apply
Gauss’s formula for the measure of curvature at any point.
Thus at every point of our manifold there is a measure of curvature
corresponding to every such pencil; but all these can be found
when n·n − 1/2 of them are known. If figures are to be freely
movable, it is necessary and sufficient that the measure of
curvature should be the same for all points and all directions
at each point. Where this is the case, if α be the measure of
curvature, the linear element can be put into the form
If α be positive, space is finite, though still unbounded, and every straight line is closed—a possibility first recognized by Riemann. It is pointed out that, since the possible values of a form a continuous series, observations cannot prove that our space is strictly Euclidean. It is also regarded as possible that, in the infinitesimal, the measure of curvature of our space should be variable.
There are four points in which this profound and epoch-making work is open to criticism or development—(1) the idea of a manifold requires more precise determination; (2) the introduction of coordinates is entirely unexplained and the requisite presuppositions are unanalysed; (3) the assumption that ds is the square root of a quadratic function of dx1, dx2, ... is arbitrary; (4) the idea of superposition, or congruence, is not adequately analysed. The modern solution of these difficulties is properly considered in connexion with the general subject of the axioms of geometry.
The publication of Riemann’s dissertation was closely followed by two works of Hermann von Helmholtz,[1] again undertaken in ignorance of the work of predecessors. In these a Helmholtz. proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates. This proof has since been shown by Lie to stand in need of correction (see VII. Axioms of Geometry). Helmholtz’s remaining works on the subject[2] are of almost exclusively philosophical interest. We shall return to them later.
The only other writer of importance in the second period is Eugenio Beltrami, by whom Riemann’s work was brought into connexion with that of Lobatchewsky and Bolyai. As he gave, by an elegant method, a convenient Beltrami. Euclidean interpretation of hyperbolic plane geometry, his results will be stated at some length[3]. The Saggio shows that Lobatchewsky’s plane geometry holds in Euclidean geometry on surfaces of constant negative curvature, straight lines being replaced by geodesics. Such surfaces are capable of a conformal representation on a plane, by which geodesics are represented by straight lines. Hence if we take, as coordinates on the surface, the Cartesian coordinates of corresponding points on the plane, the geodesics must have linear equations.
Hence it follows that
where w2 = a2 − u2 − v2, and −1/R2 is the measure of curvature of our surface (note that k = γ as used above). The angle between two geodesics u = const., v = const. is θ, where
Thus u = 0 is orthogonal to all geodesies v = const., and vice versa. In order that sin θ may be real, w2 must be positive; thus geodesics have no real intersection when the corresponding straight lines intersect outside the circle u2 + v2 = α2. When they intersect on this circle, θ = 0. Thus Lobatchewsky’s parallels are represented by straight lines intersecting on the circle. Again, transforming to polar coordinates u = r cos μ, v = r sin μ, and calling ρ the geodesic distance of u, v from the origin, we have, for a geodesic through the origin,
Thus points on the surface corresponding to points in the plane on the limiting circle r = a, are all at an infinite distance from the origin. Again, considering r constant, the arc of a geodesic circle subtending an angle μ at the origin is
whence the circumference of a circle of radius ρ is 2πR sinh (ρ/R). Again, if α be the angle between any two geodesics
then
Thus α is imaginary when u, v is outside the limiting circle, and is zero when, and only when, u, v is on the limiting circle. All these results agree with those of Lobatchewsky and Bolyai. The maximum triangle, whose angles are all zero, is represented in the auxiliary plane by a triangle inscribed in the limiting circle. The angle of parallelism is also easily obtained. The perpendicular to v = 0 at a distance δ from the origin is u = a tanh (δ/R), and the parallel to this through the origin is u = v sinh (δ/R). Hence Π (δ), the angle which this parallel makes with v = 0, is given by
which is Lobatchewsky’s formula. We also obtain easily for the area of a triangle the formula R2(π − A − B − C).
Beltrami’s treatment connects two curves which, in the earlier treatment, had no connexion. These are limit-lines and curves of constant distance from a straight line. Both may be regarded as circles, the first having an infinite, the second an imaginary radius. The equation to a circle of radius ρ and centre u0v0 is
This equation remains real when ρ is a pure imaginary, and remains finite when w0 = 0, provided ρ becomes infinite in such a way that w0 cosh (ρ/R) remains finite. In the latter case the equation represents a limit-line. In the former case, by giving different values to C, we obtain concentric circles with the imaginary centre u0v0. One of these, obtained by putting C = 0, is the straight line a2 − uu0 − vv0 = 0. Hence the others are each throughout at a constant distance from this line. (It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity. All points describe, accordingly, one of the three types of circles.)
The above Euclidean interpretation fails for three or more dimensions. In the Teoria fondamentale, accordingly, where n dimensions are considered, Beltrami treats hyperbolic space in a purely analytical spirit. The paper shows that Lobatchewsky’s space of any number of dimensions has, in Riemann’s sense, a constant negative measure of curvature. Beltrami starts with the formula (analogous to that of the Saggio)
where
He shows that geodesics are represented by linear equations between x1, x2, ..., xn, and that the geodesic distance ρ between two points x and x′ is given by
| cosh | ρ | = | a2 − x1x′1 − x2x′2 − ... − xnx′n |
| R | {(a2 − x12 − x22 − ... − xn2) (a2 − x′12 − x′22 − ... − x′n2)}1⁄2 |
(a formula practically identical with Cayley’s, though obtained by a very different method). In order to show that the measure of curvature is constant, we make the substitutions
Also calling ρ the geodesic distance from the origin, we have
| cos h (ρ/R) = | a | , sinh (ρ/R) = | r | . |
| √(a2 − r2) | √(a2 − r2) |
we obtain
| ds2 = Σdz2 + | 1 | { ( | R | sinh | ρ | ) | 2 | − 1 } Σ (zidzk − zkdzi)2. |
| ρ2 | ρ | R |
Hence when ρ is small, we have approximately
| ds2 = Σdz2 +13R2Σ (zidzk − zkdzi)2 | (1). |
Considering a surface element through the origin, we may choose
our axes so that, for this element,
Thus
| ds2 = dz21+dz22+13R2 (z1dz2 − z2dz1)2 | (2). |
Now the area of the triangle whose vertices are (0, 0), (z1, z2), (dz1, dz2) is 12(z1, dz2 − z2dz1). Hence the quotient when the terms of
the fourth order in (2) are divided by the square of this triangle is- ↑ Wiss. Abh. vol. ii. pp. 610, 618 (1866, 1868).
- ↑ Mind, O.S., vols. i. and iii.; Vorträge und Reden, vol. ii. pp. 1, 256.
- ↑ His papers are “Saggio di interpretazione della geometria non-Euclidea,” Giornale di matematiche, vol. vi. (1868); “Teoria fondamentale degli spazii di curvatura costante,” Annali di matematica, vol. ii. (1868–1869). Both were translated into French by J. Hoüel, Annales scientifiques de l’École Normale supérieure, vol. vi. (1869).
