PROBLEMS OF ALGEBRA AND ANALYSIS 523 



together with the condition of the vanishing of one simultaneous 

 invariant proper and one simultaneous covariant, dominates then, 

 in a certain sense, the whole of differential geometry. 



Passing now to the case of n variables we may consider the differ- 

 ential quadratic form 



^aikdxfdxk =ds 2 



i,A-=l 



as the square of the arc in a hyperspace of n dimensions. 



The fundamental role which the Gaussian curvature plays in the 

 case n = 2 is here represented by -an invariant expresson of ds 2 which 

 - in a certain sense might be regarded as a generalization of the 

 Gaussian curvature, namely, the Riemann curvature of the hyper- 

 space. Riemann 's investigations on this subject are found in his 

 paper, Ueber die Hypothesen, welche der Geometric zu Grunde liegen, 

 and in the mathematical supplement to it Commentatio mathematica, 

 etc. in the prize-problem of the Parisian Academy, 1861. 



The geometrical definition of the Riemann curvature is briefly the 

 following: Starting from any point P with the coordinates x^ we 

 consider two linear directions defined by the increments dxi and dxi. 

 If we remain in the vicinity of P these two directions define a plane 

 of two dimensions and the determinants 



may be considered as the coordinates of this plane. If now we 

 draw geodesic lines from the point P whose initial arc-elements 

 lie all in this plane, then these geodesies define a surface of two di- 

 mensions and the Gaussian curvature of this geodesic surface at the 

 point P is the Riemann curvature. The analytic expression for it is 



2 (ikrs) (dx { dx s dx s dx { ) (dx k dx r dx r dx k ) 



R = $ 



2(a ik a rs a ir a ks )(dx i ox s dx s ox i )(dx k ox r dx r dx lc , 



where the sum is to be taken over all values of i, k, r, s from 1 to n 

 with the exception of those for which i =k or r =s. 



The coefficients (ikrs) are certain quantities depending on the 

 coefficients a ik , their first and second derivatives; they occur in the 

 literature mostly under the name of the " Christoffel quadruple index 

 symbols." A better, certainly shorter, notation would be the one 

 used by Ricci, namely, " Riemann symbols." 



The Riemann curvature R is an invariant expression, and as its 

 form shows it is a covariant of two sets of differentials. For n = 2 

 it is identical with the Gaussian curvature. For greater numbers 

 n the value of R depends, at a given point, on the plane-direction 

 at that point and in general varies with the plane. If it should be 

 constant for all plane-directions through one point, and if this is 

 so for all the points, then R is, as Schur has shown, altogether con- 

 stant that is, for every point. 



