008 



Starting from a point O a system of directions is now geodeticallj 

 moved along a closed curve. On returning (o the system will 

 generally appear to have lotated. Dependent on the choice of the 



o 



curve it is generally possible to obtain in this manner oo " positions 

 of the system. If this number is for one point and hence for 

 every point of the area oo'^', we call N the iminber of degrees of 

 freedom of the geodetically moving system. Now the following 

 theorem exists: 



I. The. number of dimensions of the etic/idian space, in ivhich a 

 given space X,, may be placed, tuithout changing its linear element, 

 is at most equal to the number of degrees of freedom belonging to 

 the geodetically moving system increased with n. 



We will prove this theorem. If the number of degrees of freedom 



n(n — 1) 

 is smaller than — , there will remain invariant /• mutually per- 



feotly perpendicular^directions of />,,/>,,/>,, — dimensions />i+. • • .-\-pr=n 

 (by direction of two dimensions or 2-direction we mean a plane 

 direction, etc.). The number of possibilities exactly corresponds to 

 the number of manners, in which n can be written as the sum of 

 whole positive numbers. We imagine the r invariant directions 

 marked once for all in (). The system then may be brought in 

 every point of A„, always by geodetically movitig. The invariant 

 /j^-direction, J =z\ . . . . , r, will then define at eveiy point a />^-direction, 

 and it is the question whether these directions will compose a system 



H—p . 



of OD "^ curved spaces Pj of pj dimensions. This is a Pfaffian 

 problem in a general space. 



We select a definite invariant direction, say the p^-direction, and for 

 convenience, sake we shall write p for pj. If wenow define the /^direction 

 belonging to this direction at every point by the simple /j-vectors, 



^v = V, . . . Yp, which all pass into one another by geodetically moving 

 and likewise the perfectly perpendicular {n — /j)-direction to this, 



by ,w = w, . . . Wy, q = 71 —p, then : ' 



d jj\ z= , d gV/ = 0, 

 hence : 



7 ^,v = , V ^w = 0. 

 It is worth mentioning, that the vectors Yi-, k = 1, . . . , p, do not 

 pass into each other by geodetically moving and hence (ivA; =|= 0. The 

 same holds good for W/, /=! q. If now the linear element 



