We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate
the theory of the classical r-matrix for integrable systems. In this picture
the r-matrix is related to a fourth rank tensor, named the r-tensor, on the
configuration space. The r-matrix itself carries one connection type index and
three tensorial indices. Being defined on the configuration space it has no
momentum dependence but is dynamical in the sense of depending on the
configuration variables. The tensorial nature of the r-matrix is used to derive
its transformation properties. The resulting transformation formula turns out
to be valid for a general r-matrix structure independently of the geometric
framework. Moreover, the entire structure of the r-matrix equation follows
directly from a simple covariant expression involving the Lax matrix and its
covariant derivative. Therefore it is argued that the geometric formulation
proposed here helps to improve the understanding of general r-matrix
structures. It is also shown how the Jacobi identity gives rise to a
generalized dynamical classical Yang-Baxter equation involving the Riemannian
curvature.