Transformations of Einstein spaces

Abstract

In a paper which is to be published elsewhere are obtained all Einstein manifolds whose line elements are determined by a quadratic differential form of the type ds^2 = f(xyzt)(dx^2 + dy^2 + dz^2) + g(xyzt)dt^2 (1) where f is really a function of t. Of the ten apparently distinct solutions of the cosmological equations for an element of this type one represents a hypersphere,(1) two are characterized by the fact-that f and g involve an essentially complex argument and in the remaining seven, f is in each case related to a Weierstrass p-function. This suggests that only three of the ten solutions are distinct and that the relations between the various solutions of each group are to be found by transformation of coordinates. It is the purpose of this note to develop a theorem which will establish these relations.