Seismic data can be described in a low-dimensional manifold. Thus, the low dimensionality of the seismic data patch manifold can serve as a good regularizer for seismic data interpolation. However, we have found that having only low-dimensional manifold regularization is not sufficient for interpolating seismic data with large data gaps or spectral aliasing. Therefore, we propose the application of a curvature-regularized low-dimensional manifold method for seismic data interpolation. A windowed version of the method is proposed, which provides adaptability and efficiency for seismic data interpolation. Numerical experiments on the synthetic and field data indicate that the proposed method outperforms the f-x, curvelet, and low-dimensional manifold methods in many missing data cases, especially in the presence of large gaps or spectral aliasing.