Journal of the
Korean Mathematical Society JKMS

We consider a $3$-dimensional Riemannian manifold $M$ with a circulant metric $g$ and a circulant structure $q$ satisfying $q^{3}=\id$. The structure $q$ is compatible with $g$ such that an isometry is induced in any tangent space of $M$. We introduce three classes of such manifolds. Two of them are determined by special properties of the curvature tensor. The third class is composed by manifolds whose structure $q$ is parallel with respect to the Levi-Civita connection of $g$. We obtain some curvature properties of these manifolds $(M, g, q)$ and give some explicit examples of such manifolds.

Keywords: Riemannian metric, circulant matrix, almost Einstein manifold, Ricci curvature

MSC numbers: Primary 53B20, 53C15, 53C25; Secondary 15B05