Pseudospheres in geometry and physics: from Beltrami to De Sitter and   beyond
  We review the extraordinary fertility and proliferation in mathematics and physics of the concept of a surface with constant and negative Gaussian curvature. In his outstanding 1868 paper Beltrami discussed how non-Euclidean geometry is actually realized and displayed in a disk on the plane. This metric is intrinsically defined and definite but only if indefinite metrics are introduced it is possible to fully understand the structure of pseudospheres. In a 3D flat space R^3 the fundamental quadric is introduced, with the same signature as the metric of R^3; this leads to 3 kinds of surfaces with constant Gaussian curvature: the sphere, the single-sheet hyperboloid and the two-sheet hyperboloid; the last one is shown to be isomorphic to Beltrami's disk. The spacetime corresponding to the single-sheet case is de Sitter cosmological model, which, due to its symmetry, has an important role in cosmology. When two of the three fundamental quadrics are combined, a simple, yet deep, solution of Einstein-Maxwell equations corresponding to a uniform electromagnetic field is obtained with many applications in mathematical physics. One of them is a "no go" theorem: when one tries to frame a Riemannian four-dimensional manifold in a Kahlerian structure, it is found that, while this is generically possible with a definite signature, in spacetime only BR fulfills the requirement. The BR metric plays an important role in the exploration of new principles in fundamental physical theories; we briefly mention some examples related to the horizon of a black hole and the dilaton in string theory. 
B. Bertotti (U. Pavia), R. Catenacci (U. Piemonte Orientale), C.   Dappiaggi (U. Pavia and INFN - Sezione di Pavia)