Il~.srrz/c~t: We consider four-dimensional D'Atri spaces. which is to say Riemannian spaces for which ever!' canonical geodesic involution preserves the volume element up to sign. We show that four-dimensional D' Atri spaces which are also Einstein are necessarily locally symmetric. The same methods enable us to \how also that four-dimensional D'Atri spaces with anti-self-dual Weyl tensor are locally symmetric. Kc,vr~~ords: D' Atri spaces, Einstein spaces. self-dual spaces. .MS c./trt,sific.rrtiorl: 53ASO. 5X25. D' Atri spaces have been a topic of interest in Riemannian geometry since they were in.. traduced by D'Atri and Nickerson [2,3]. Many examples are known and in particular all arc known in dimension 3 151. In this paper, our aim is to prove a conjecture about D'Atri spaces in dimension 4 due to Sekigawa and Vanhecke [lo], namely that four-dimensional Einsteinian D'Atri spaces are locally symmetric. The case of dimension 4 lends itself particularly well to spinor calculus and that will be our method. Along the way the method shows easily that a D'Atri space in dimension 4 is locally symmetric if it is Kahler, a result due to Sekigawa and Vanhecke [ 1 1 I. Finally we are able to show that a four-dimensional D'Atri space is locally sy mmetric if it has anti-self-dual (or self-dual) Weyl tensor. At a point /3 in a Riemannian manifold M it is possible to define a map s,, known in the literature variously as 'the local geodesic symmetry at p' [6] or 'the canonical geodesic involution at p' [ 1,131. The definition is as follows:
.s,) : q,,(X) + exp,,(-X)