  The central result about fast rotating-flow structures is the Taylor-Proudman
theorem (TPT) which connects various aspects of the dynamics. Taylor's
geometrical proof of TPT is reproduced and extended substantially, with Lie's
theory for general frozen-in laws and the consequent generalized invariant
circulation theorems, to compressible flows and to $d$-dimensional Euclidean
space ($\mathbb{E}^{d}$) with $d\ge 3$. The TPT relatives, the reduced models
(with particular interests on passive-scalar problems), the inertial (resonant)
waves and the higher-order corrections, are discussed coherently for a
comprehensive bird view of rotating flows in high spatial dimensions.
