Projective symmetries of five-dimensional spaces

A review of invariant-group methods in 5-dimensional theories of electromagnetic, gravitational and other physical fields is presented. The symmetries of the five-dimensional curved spaces in the form of Lie groups of infinitesimal transformations, in particular, in the form of projective motions which preserve geodesics are discussed. The 5-dimensional rigid ℎ-spaces 𝐻221, 𝐻32, 𝐻41 and 𝐻5, i.e. pseudo-Riemannian manifolds (𝑀5, 𝑔) of arbitrary signature with (non-degenerate) Segre characteristic = {𝑟1, ..., 𝑟𝑘}, 𝑟1, ..., ∈ 𝑁, 𝑟1+...+𝑟𝑘 = 5, and real eigenvalues of the Lie derivative of the metric in the direction of the infinitesimal transformation are investigated, which admit (non-homothetic) infinitesimal projective and affine transformations, and for each of them the structure of the corresponding maximal projective and affine Lie algebras are determined; the classification of ℎ-spaces 𝐻221 of type {221} on maximal Lie algebras of projective and affine transformations, wider than the Lie algebras of homotheties, is obtained