524 ALGEBRA AND ANALYSIS 



Spaces of constant Riemann curvature have been the object of 

 numerous interesting investigations, but these are more or less of 

 a specific geometric character. 



If in particular R is zero, then all the Riemann symbols vanish 

 and it can easily be shown that ds 2 can be transformed into the 

 sum of n squares 



The converse is true. In this case the hyperspace of n dimensions 

 is called a flat or also Euclidean space. 

 In every case the quadratic 



can be transformed into 



n+r 



Z dy\ 



i=l 



where r has the maximum value ?l(/ o~ ^ We might say then that the 

 given hyperspace of n dimensions is always contained in an 

 Euclidean space of n+r dimensions, where r is one of the numbers, 

 0, 1. *^~ 



The number r is evidently characteristic for the hyperspace the 

 square of the arc-element of which is the given quadratic. This 

 number r has been called by Ricci the class of the given differential 

 quadratic quantic. It is evident that this class is an invariant num- 

 ber, and the condition that a given differential quadratic be of class 

 r must certainly be an invariantive condition. For r = we have 

 just seen that the condition is R = 0. For higher values of r no at- 

 tempt has yet been made, so far as I know, to establish this invari- 

 antive condition though this problem is certainly one of fundamental 

 interest. 



Beltrami, in his paper, Teoria generale dei parametri differenziali. 

 has extended the definition of his differential parameters to the 

 case of n variables. The definition, for instance, of the first differ- 

 ential parameters is 



where A ik denotes the minor of the element a ik in the determinant 



\a ik \ =a. 



Beltrami shows that by means of the geodesies emanating from one 

 point and of the hypersurfaces orthogonal to them he can choose 

 his parameters such that ds 2 is transformed into 



n-l 



ds 2 =dr*+ V b ik dyidy k , 

 i,/t=i 



where r satisfies the equation A^r = 1 , and that thus Gauss's theorems 



