This paper studies exact controllability properties of two coupled wave equations with variable coefficients by a Riemannian geometrical approach. One of the PDEs holds on the interior of a bounded open domain Omega in R-m and the other on a piece Gamma(0) of the boundary partial derivative Omega. First, an exact controllability result is established by assuming the existence of an escape vector field with two boundary controls, the first of which is a Neumann control and the second a distributed control for the boundary equation. Moreover, a geometric structure between the boundary and the variable-coefficient principal part is considered in order to implement one control on the boundary segment Gamma(0) only. Concrete examples are given for such geometrical configurations.