  The existence of flat or bulgeless galaxies poses a challenge to the
hierarchical structure formation scenario advocated by modern cosmology. We
determine the geometrical environment of a sample of $315$ flat galaxies from
the Revised Flat Galaxy Catalog (RFGC) using `local dimension' $D$, which, on a
given length scale, quantifies the dimension of the cosmic structure in which a
galaxy is embedded. For galaxies residing in filaments, nodes and sheets, $D
\sim 1$, $D \sim 1.5$ and $D \sim 2$ respectively; $D \sim 3$ represents field
galaxies. We also determine the local dimensions of a sample of 15,622 non-flat
galaxies identified in the Galaxy Zoo project from the Sloan Digital Sky Survey
(SDSS). We find that the median values of $D$ for the flat and the non-flat
galaxies are $2.2$ and $1.8$ respectively, implying that flat galaxies are
located in a relatively sparser environment compared to non-flat galaxies; a
Kolmogorov-Smirnov (KS) test indicates that their geometrical environments are
different at $>99$\% confidence level. Further, using a group finding
algorithm, we study the local environment of a subset of $779$ flat galaxies
with major-to-minor axes ratio $a/b >10$ identified as superthin galaxies. We
find that the median clusterization index $k_{\rm{min}}$ for superthin flat
galaxies $\sim$ $2.3$ while $\sim$ $1.7$ for other flat galaxies, confirming
that the superthins reside in an under-dense environment compared to other flat
galaxies at $> 98 \%$ confidence level. Our results may therefore have
important implications for the formation and evolution models of flat galaxies
in the universe.
