24 
MA THEM A TICS: L. P. EISENHART Proc. N. A. S. 
RICCrS PRINCIPAL DIRECTIONS FOR A RIEMANN SPACE 
AND THE EINSTEIN THEORY 
L. P. ElSENHART 
Department of Mathematics, Princeton University 
Communicated by Oswald Veblen, January 18, 1922 
In 1904 Ricci (Atti R. 1st. Veneto, 62, 1230) developed the idea of prin- 
cipal directions in a Riemann space of dimensions, and in doing so in- 
troduced the contracted curvature tensor, which is fundamental in the 
Einstein theory, and gave a geometrical interpretation to it. A space 
in which these principal directions are completely indeterminate may 
be thought of as possessing a homogeneous character. We derive Ricci 's 
results by a slightly different method, and then show that the three types 
of space, chosen by Einstein in 1914, 1917 and 1919, as spaces free from 
matter are of this homogeneous character, and include all types of such 
spaces. 
Consider a Riemann space Vn of ?z-dimensions with the hnear element 
ds^ = gij dx' dx' (gij = gji) . (1) 
The right-hand member represents the sum of terms as i and / take on 
the values 1 , n, in accordance with the usual convention in such 
formulas that any term represents a summation with respect to each 
letter which appears in it both as a subscript and a superscript. 
Suppose that we have in Vn ^ orthogonal unit vectors and let \\ {i = 
1 , n) denote the contra variant components of the vector (h) . Then 
we have 
gij^'h^i = ^hkj (2) 
where 
_1 ior h = k 
If we take the surface consisting of the geodesies tangent at a point 
P to the pencil of directions determined by the lines of two congruences 
(h) and (k) through P, the gaussian curvature of this surface at P is given 
by 
^hk — I^Pq,rs ^k^h ^1' (4) 
where Rpq^rs is the Riemann tensor of the first kind. By definition 
Titk is the Riemann curvature of F„ at P for the directions (h) and (k) 
(cf. Bianchi, 1, 342). 
Since the n vectors are mutually orthogonal, we have 
2 MXI = g«^ (5) 
k 
