  The recently introduced matrix group SE2(3) provides a 5x5 matrix
representation for the orientation, velocity and position of an object in the
3-D space, a triplet we call "extended pose". In this paper we build on this
group to develop a theory to associate uncertainty with extended poses
represented by 5x5 matrices. Our approach is particularly suited to describe
how uncertainty propagates when the extended pose represents the state of an
Inertial Measurement Unit (IMU). In particular it allows revisiting the theory
of IMU preintegration on manifold and reaching a further theoretic level in
this field. Exact preintegration formulas that account for rotating Earth, that
is, centrifugal force and Coriolis force, are derived as a byproduct, and the
factors are shown to be more accurate. The approach is validated through
extensive simulations and applied to sensor-fusion where a loosely-coupled
fixed-lag smoother fuses IMU and LiDAR on one hour long experiments using our
experimental car. It shows how handling rotating Earth may be beneficial for
long-term navigation within incremental smoothing algorithms.
