Type III Einstein-Yang-Mills solutions

Exact solutions of Einstein equations have always attracted much attention. It is somewhat surprising to find exact solutions of such nonlinear equations. Many of them were collected in the by now classic book by Kramer et al. which has recently been revised [1]. Among others one finds the nondiverging spacetimes which were studied for the first time by Kundt in [2]. Solutions which admit a nonexpanding and twistfree null congruence are therefore said to belong to the Kundt’s class. Such metrics are algebraically special, i.e., of Petrov type III, N, O, II or D. If one restricts the solutions to those with plane wave surfaces they are constrained to be (at most) of type III; they can degenerate to type N and O. Conformally flat solutions include, for example, the Bertotti-Robinson [3,4] universe and the Edgar-Ludwig metric [5]. An Einstein- Yang-Mills solution of type N was given long time ago in [6] and recovered for D = 4 supergravity in [7]. Type III solutions have somehow attracted less attention. Yet, some references considering the type III Einstein-Maxwell system can be found [8,9]. To our knowledge, the solutions presented in this note are the first type III solutions of Einstein gravity coupled to a Yang-Mills field. We also want to point out that the Kundt’s class can be generalized to include a nonzero cosmological constant. Such solutions of type N were first considered in [10], followed by [11–14]. Type N Einstein-Maxwell solutions were generalized to an arbitrary number of dimensions in [15] and further generalized to Lovelock-Yang-Mills solutions in [16]. Not very long ago the ¿ ¿ 0 generalization of type III spacetimes in [1] was presented in [17].