The conjugate direction method for solving the unconstrained optimization problem is extended to solving the constrained optimization problem by method of differential geomtry.By inducing a new class of affine connections on a constrained sub-manifold, the primary constrched optilnhation problem is converted to a unconstrained local quadratic programming problem.Based on the definition and construction of a new class of generalized conjugate directions, it isproved that optimum value of the primary constrained optimization problem must be located on thegeodesic line which is formed by the conjugate directions mentioned above and can be reached withinfinite searching step. Therefore a new curve search algorithm with generalized conjugate directions isput forward.