Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

39

upon the surface. When K is the same for all points of the surface, the surface is said to be one of constant curvature. The importance of the curvature lies in the two theorems:
A necessary condition that two portions of sutfaces may be brought
into point for point correspondence with preservation of distance is
that they have the same curvature at corresponding points.
If the two portions of surfaces are of constant curvature, the condition is also sufficient.
The Gaussian measure of curvature of a surface is extended by
Riemann to space of n dimensions. For three dimensions consider
a point (z,, 3) and two directions (a, a21 a3) and (a, /32, /3.),
taken from that point. Then the Riemann curvature is a function
K(z1 2 Z3; V a7, aX3; 30 2) 383)
which gives the Gaussian curvature of the geodesic surface determined by the point and the directions. The Riemann curvature
of a general space is accordingly dependent both on the point
of space for which it is reckoned and on the directions of the
lines taken through that point to define a geodesic surface. But
if the space satisfies our third hypothesis, the curvature is a function of the point only. For by this hypothesis, any two geodesic
pencils with their vertices at the same point P may be brought
into point for point correspondence with preservation of distance.
Hence by the surface theorems above quoted, the two geodesic
surfaces formed by the pencils must have the same curvature at
corresponding points and in particular at P. Schur * has proved
that when the curvature is thus constant at each point, it does not
change as we pass from point to point. The space is then said to
be of constant curvature. A new proof of Schur's theorem will
be given in the following paragraph.
4. THE LINE-ELEMENT.
Take any point 0 at which the functions a i are single-valued
and continuous. Then, as we have seen, there exists around 0
* Schur, F., "Ueber den Zusammenhang der Raume constanten Riemann' -
schen Kriimmungsmasses," Math. Annalen, vol. 27 (1886), p. 593.