  In particle accelerators, transverse-longitudinal coupling (TLC) dynamics can
be invoked for efficient bunch compression or high harmonic generation when one
of the transverse eigenemittance is small. In this sense, complete or partial
transverse-to-longitudinal emittance exchange in optical wavelength range is
being actively studied, for example in free-electron lasers. Another example is
the recent work on generalized longitudinal strong focusing steady-state
microbunching, where TLC is exploited to take advantage of the ultrasmall
vertical emittance in a planar electron storage ring to lower the modulation
laser power for ultrashort microbunch generation on a turn-by-turn basis. For
this kind of schemes, we have proved before three theorems, invoking 4D phase
space dynamics, with their implications discussed. Here we generalize the
analysis to 6D phase space dynamics. Various TLC-based beam manipulation
scenarios, as listed in the references, are dictated by these theorems.
