Maxwell electrodynamics in media, geometry effect on constitutive relations

The problem of constitutive relations in Maxwell electrodynamics, their possible form and role in physical manifestation of electromagnetic fields, its behavior under the motion of the reference frame and its connection with Special Relativity theory, interplay between electrodynamics constitutive relations and gravity theory, and so on, has a long history. The main accent in our treatment is the known possibility to simulate material media by geometrical methods. This review includes the following items: Riemannian geometry and Maxwell theory; Maxwell equations in Riemannian space and effective media; metrical tensor and constitutive relations; inverse constitutive equations; geometric simulation of inhomogeneous media; geometrical simulation of uniform media; geometrical modeling of anisotropic uniform media; the moving medium and anisotropy; geometry effect on material equations in arbitrary linear media; the plane wave in the Lobachevsky space, simulating a special medium; arbitrary metrics, etc.