Metric spaces with geodesic Ricci curves. II

Jack Levine
1939 Bulletin of the American Mathematical Society  
1. Introduction. In this paper we give a partial classification of four-dimensional metric spaces admitting geodesic Ricci curves. The results of I* will be assumed known, along with the notations of that paper. We assume a given set of independent vectors f Xjj such that £^ = 0 (i not summed), so as to obtain geodesic curves, and impose conditions (24), (25), (26) of I on the 0 a . From I, (25) we see that if M&& = 0 for any k, then by I, (24), we have (since |X*|| 9*0) d/z&/dx a = 0, or jdk =
more » ... const. If, however, for any given k, fikk^O, then from I, (25), c£ = 0, for all i and ƒ This gives us a means of classifying the spaces according to the number of fXkk which equal zero. For w = 4 there are five cases, which, without loss of generality, we may take in the form :
doi:10.1090/s0002-9904-1939-06913-9 fatcat:446tun5zujayvetaabwc6uno2u