Classification of simply connected four-dimensional $RR$-manifolds

Abstract:

Let (M, g) be a Riemannian manifold. We assume that there is a mapping $s:M \to I(M)$, where $I(M)$ is the group of isometries of (M, g), such that ${s_x} = s(x),\forall x \in M$, has x as a fixed isolated point, then (M, g) is called a Riemannian s-manifold. If the tensor field S on M defined by the relation ${S_x} = {(d{s_x})_x},\forall x \in M$, is differentiable and invariant by each isometry ${s_x}$, then the manifold (M, g) is called a regularly s-symmetric Riemannian manifold. The aim of the present paper is to classify simply connected four-dimensional regularly s-symmetric Riemannian manifolds.