Beltrami and mathematical physics in non-Euclidean spaces

Eugenio Beltrami (1835–1900) investigated by geometry and continuum mechanics if ambient space, where physical phenomena occur, is Euclidean. It was a trendy subject in the mathematical scientific community at the end of the nineteenth century, especially after the contributions in non-Euclidean geometry by Nikolaj Lobacevskij (1793–1856), Janos Bolyai (1802–1860), Bernhard Riemann (1826–1866), and Beltrami himself. The first two searched a geometry independent of Euclid’s fifth postulate on a purely mathematical basis, while Beltrami and Riemann were interested also on how non-Euclidean geometry affects elasticity and electro-magnetism. Beltrami opened new paths for mathematical physics; his pupils Ernesto Cesaro (1859–1906), Ernesto Padova (1845–1896), Carlo Somigliana (1860–1955) also studied elasticity in non-Euclidean spaces. Here we discuss Beltrami’s original results and comment on their influence.