Differential geometric approach is useful for solving a special class of nonlinear control problems. The purpose of this paper is to present a linearization method of nonlinear systems by the Riemannian geometric approach, and to apply the linearization mapping to the design of nonlinear systems.<br>A geomeric model can be derived by replacing the orthogonal straight coordinate frame on the state space with a suitable curvilinear frame. Such a Riemannian geometric model has been proposed by the authors after the derivation of the geodesic curve on the gravitational gauge field in Einstein's principle of general relativity.<br>In this paper an attention is placed on a problem to decrease the dimension of the Riemannian space of the model by a proper choice of the construction of the space, which leads to decrease the computation time remarkably. For the design of control systems, a new quadratic-form performance index is introduced using Riemannian metric tensors. A nonlinear optimal regulator is constructed which is homeomorphic to the corresponding linear optimal regulator. A method to derive a curvilinear coordinates frame fitted to the nonlinear system is proposed by solving a partial differential equation with respect to the homeomorphism. A computational algorithm is proposed and numerical examples are shown.