  It is well known that the geometrical framework of Riemannian geometry that
underlies general relativity and its torsionful extension to Riemann-Cartan
geometry can be obtained from a procedure known as gauging the Poincare
algebra. Recently it has been shown that gauging the centrally extended Galilei
algebra, known as the Bargmann algebra, leads to a geometrical framework that
when made dynamical gives rise to Horava-Lifshitz gravity. Here we consider the
case where we contract the Poincare algebra by sending the speed of light to
zero leading to the Carroll algebra. We show how this algebra can be gauged and
we construct the most general affine connection leading to the geometry of
so-called Carrollian space-times. Carrollian space-times appear for example as
the geometry on null hypersurfaces in a Lorentzian space-time of one dimension
higher. We also construct theories of ultra-relativistic (Carrollian) gravity
in 2+1 dimensions with dynamical exponent z<1 including cases that have
anisotropic Weyl invariance for z=0.

High Energy Physics - Theory;General Relativity and Quantum Cosmology