This paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part, subject to Dirichlet or Neumann boundary controls. The observability inequalities are established by the Riemannian geometry method under some geometric condition for the Dirichlet problem and for the Neumann problem, respectively. Next, a number of nontrivial examples are presented to verify the observability inequality. In particular, a counterexample is given without boundary exact controllability, where the control is exerted on the whole boundary.