  A 3-dimensional Riemannian manifold equipped with a tensor structure of type
$(1,1)$, whose third power is the identity, is considered. This structure and
the metric have circulant matrices with respect to some basis, i.e., these
structures are circulant. An associated manifold, whose metric is expressed by
both structures, is studied. Three classes of such manifolds are considered.
Two of them are determined by special properties of the curvature tensor of the
manifold. The third class is composed by manifolds whose structure is parallel
with respect to the Levi-Civita connection of the metric. Some geometric
characteristics of these manifolds are obtained. Examples of such manifolds are
given.
