Direct numerical simulation (DNS) is a very powerful tool to evaluate the validity of new models and theories for turbulent combustion, but the application of detailed chemistry is limited. In this paper, a dimension-reduction technique called the flamelet-generated manifold (FGM) method is considered. In this method a manifold is created by solving a set of one-dimensional flamelet equations. The use of low-dimensional FGMs in DNS of premixed turbulent flames in the thin reaction zones regime is investigated. A three-dimensional (3D) DNS is performed of a spherically expanding, premixed, turbulent, methane-air flame. 1D and 2D FGM's are created and used in simulations of flamelets which are subjected to stretch and curvature effects derived from the 3D DNS results. The results are compared with results from flamelet simulations with detailed chemistry. The results show that deviations from the 1D FGM due to stretch and curvature effects are significant, but they appear to be embedded in a 2D manifold. This 2D manifold corresponds well with 2D FGM's that are created in different ways, but it shows large differences with a 2D manifold based on chemical kinetics alone. This indicates that an attracting low-dimensional manifold exists which is not solely determined by chemical kinetics. As a consequence, the results of the flamelet simulations using 2D FGM's are more accurate than when a 1D FGM is applied: the mean error in the burning velocity is almost an order of magnitude smaller.