  Flatbands appear in many condensed matter systems, such as in high magnetic
fields, correlated materials and moire heterostructures. They are characterized
by intrinsic geometric properties such as the Berry curvature and Fubini-Study
metric. In general the band geometry is nonuniform in momentum space, making
its influence on electron-electron interactions a difficult problem to
understand analytically. In this work, we study this problem in a topological
flatband of Chern number C=1 with the ideal properties that the Berry curvature
is positive definite and fluctuates in sync with Fubini-Study metric. We derive
an exact correspondence between such ideal flatbands and Landau levels by
showing how the band geometry fluctuation in ideal flatbands gives raise to a
new type of interaction in Landau levels which depends on the center-of-mass of
two particles. We characterize such interaction by generalizing the usual
Haldane pseudopotentials. This mapping gives exact zero-energy ground states
for short-ranged repulsive generalized pseudopotentials in flatbands, in
analogy to fractional quantum Hall systems. Driving the center-of-mass
interactions beyond the repulsive regime leads to a dramatic reconstruction of
the ground states towards gapless phases. The generalized pseudopotential could
be a useful basis for future numerical studies.
