Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time

The Carroll group was originally introduced by Levy-Leblond [1] by considering the limit of the Poincar\'e group as $c\to0$. In this paper an alternative definition, based on the geometric properties of a non-Minkowskian, non-Galilean but nevertheless boost-invariant, space-time structure is proposed. A "duality" with the Galilean limit $c\to\infty$ is established. Our theory is illustrated by Carrollian electromagnetism.