We develop a supervised dimensionality reduction method, called Lorentzian discriminant projection(LDP), for feature extraction and classification. Our method represents the structures of sample data bya manifold, which is furnished with a Lorentzian metric tensor. Different from classic discriminantanalysis techniques, LDP uses distances from points to their within-class neighbors and globalgeometric centroid to model a new manifold to detect the intrinsic local and global geometricstructures of data set. In this way, both the geometry of a group of classes and global data structures canbe learnt from the Lorentzian metric tensor. Thus discriminant analysis in the original sample spacereduces to metric learning on a Lorentzian manifold. We also establish the kernel, tensor andregularization extensions of LDP in this paper. The experimental results on benchmark databasesdemonstrate the effectiveness of our proposed method and the corresponding extensions.