  The band connectivity as imposed by the compatibility relations between the
irreducible representations of little groups can give rise to the exotic
hourglass-like shape composed of four branches of bands and five band crossings
(BCs). Such an hourglass band connectivity could enforce the emergence of
nontrivial excitations like Weyl fermion, Dirac fermion or even beyond them. On
the other hand, the bosons, like phonons, magnons, and photons, were also shown
to possess nontrivial topology and a comprehensive symmetry classification of
the hourglass bosonic excitations would be of great significance to both
materials design and device applications. Here we firstly list all concrete
positions and representations of little groups in the Brillouin zone (BZ)
related with the hourglass bosonic excitations in all the 1651 magnetic space
groups and 528 magnetic layer groups, applicable to three dimensional (3D) and
two dimensional (2D) systems, respectively. 255 (42) MSGs (MLGs) are found to
essentially host such hourglass BCs: Here ``essentially'' means that the
bosonic hourglass BC exists definitely as long as the studied system is
crystallized in the corresponding MSG/MLG. We also perform first-principles
calculations on hundreds of 3D nonmagnetic materials essentially hosting
hourglass phonons and propose that the 2D material AlI can host hourglass
phonons. We choose AuX (X=Br and I) as illustrative examples to demonstrate
that two essential hourglass band structures can coexist in the phonon spectra
for both materials while for AuBr, an accidental band crossing sticking two
hourglasses is found interestingly. Our results of symmetry conditions for
hourglass bosonic excitations can provide a useful guide of designing
artificial structures with hourglass bosonic excitations.
