Kernel principal component analysis (KPCA) has become a popular technique for process monitoring, owing to its capability of handling nonlinearity. Nevertheless, KPCA suffers from two major disadvantages. First, the underlying manifold structure of data is not considered in process modeling. Second, the selection of kernel parameters is problematic. To avoid such deficiencies, a manifold learning technique named maximum variance unfolding (MVU) is considered as an alternative. However, such method is merely able to deal with the training data, but has no means to handle new samples. Therefore, MVU cannot be applied to process monitoring directly. In this paper, an extended MVU (EMVU) method is proposed, extending the utilization of MVU to new samples by approximating the nonlinear mapping between the input space and the output space with a Gaussian process model. Consequently, EMVU is suitable to nonlinear process monitoring. A cross-validation algorithm is designed to determine the dimensionality of the EMVU output space. For online monitoring, three different types of monitoring indices are developed, including squared prediction error (SPE), Hotelling-T2, and the prediction variance of the outputs. In addition, a fault isolation algorithm based on missing data analysis is designed for EMVU to identify the variables contributing most to the faults. The effectiveness of the proposed methods is verified by the case studies on a numerical simulation and the benchmark Tennessee Eastman (TE) process.