NECESSARY AND SUFFICIENT CONDITIONS FOR (I) WEYL, (II) RIEMANN-CARTANCONNECTIONS

Spaces with semi-metric connections (which include metric, Weyl and Ri emann-Cartan connections), defined by del(c)h(ab) = h(ab)lambda(c), ne cessarily satisfy an algebraic relationship of the type h(ai)R(i)bcd h(bi)R(i)acd = 0, where h(ab) is a metric tensor, and R(a)bcd is rela ted to the curvature tensor R(a)bcd of the connection by R(a)bcd = R(a )bcd - 1/4delta(b)(a)R(i)icd. It is shown-in a four-dimensional space- time, for almost all curvature tensors-that this algebraic relationshi p is also a sufficient condition for the local existence of a curvatur e tensor of a semi-metric connection. Generalisations of this result, involving a tensor more general than the curvature tensor, are also gi ven.