We present for a polyatomic reaction system a reaction-path Hamiltonian described with a reaction coordinate and quasirectilinear vibrational coordinates that are constructed from a nonlinear combination of curvilinear internal coordinates. To determine the vibrational coordinates we use a quasipotential-energy expression in which, in a Taylor-series expansion of the potential energy around the reaction path, the usual derivatives with respect to the internal coordinates are replaced by the corresponding covariant derivatives. The vibrational coordinates are determined so that (1) when the angular momentum is assumed to be zero, the respective expressions for the quasipotential energy and the kinetic energy have diagonal forms in the second-order terms and the first-order terms with respect to the vibrational coordinates in the internal configuration subspace perpendicular to the reaction path, and (2) the covariant second derivatives of the potential energy with respect to the vibrational coordinates coincide with the usual second derivatives. The reaction and vibrational coordinates determined presently not only are separated from rotations but are also invariant under the transformation of the original internal coordinates. The present reaction-path Hamiltonian is expected to be of advantage for obtaining an appropriate picture of the reaction-path dynamics under the zero-angular momentum assumption.