  Recently it has been established that torsional Newton-Cartan (TNC) geometry
is the appropriate geometrical framework to which non-relativistic field
theories couple. We show that when these geometries are made dynamical they
give rise to Horava-Lifshitz (HL) gravity. Projectable HL gravity corresponds
to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL
gravity corresponds to dynamical NC geometry with twistless torsion
(hypersurface orthogonal foliation). We build a precise dictionary relating all
fields (including the scalar khronon), their transformations and other
properties in both HL gravity and dynamical TNC geometry. We use TNC invariance
to construct the effective action for dynamical twistless torsional
Newton-Cartan geometries in 2+1 dimensions for dynamical exponent 1<z\le 2 and
demonstrate that this exactly agrees with the most general forms of the HL
actions constructed in the literature. Further, we identify the origin of the
U(1) symmetry observed by Horava and Melby-Thompson as coming from the Bargmann
extension of the local Galilean algebra that acts on the tangent space to TNC
geometries. We argue that TNC geometry, which is manifestly diffeomorphism
covariant, is a natural geometrical framework underlying HL gravity and discuss
some of its implications.
