  We used the Atacama Large Millimeter/submillimeter Array (ALMA) to map the
CO(3-2) and the underlying continuum emissions around the type 1 low-luminosity
active galactic nucleus (LLAGN; bolometric luminosity $\lesssim 10^{42}$
erg~s$^{-1}$) of NGC 1097 at $\sim 10$ pc resolution. These observations
revealed a detailed cold gas distribution within a $\sim 100$ pc of this LLAGN.
In contrast to the luminous Seyfert galaxy NGC 1068, where a $\sim 7$ pc cold
molecular torus was recently revealed, a distinctively dense and compact torus
is missing in our CO(3-2) integrated intensity map of NGC 1097. Based on the
CO(3-2) flux, the gas mass of the torus of NGC 1097 would be a factor of
$\gtrsim 2-3$ less than that found for NGC 1068 by using the same CO-to-H$_2$
conversion factor, which implies less active nuclear star formation and/or
inflows in NGC 1097. Our dynamical modeling of the CO(3-2) velocity field
implies that the cold molecular gas is concentrated in a thin layer as compared
to the hot gas traced by the 2.12 $\mu$m H$_2$ emission in and around the
torus. Furthermore, we suggest that NGC 1097 hosts a geometrically thinner
torus than NGC 1068. Although the physical origin of the torus thickness
remains unclear, our observations support a theoretical prediction that
geometrically thick tori with high opacity will become deficient as AGNs evolve
from luminous Seyferts to LLAGNs.
