  The quantum geometry of Bloch bands fundamentally affects a wide range of
physical phenomena. For example, the quantum Hall effect is governed by the
Chern number, and superconductivity by the distance between the Bloch states --
the quantum metric. Here, we show that key properties of a weakly interacting
Bose-Einstein condensate (BEC) depend on the underlying quantum geometry, and
in the flat band limit they radically depart from those of a dispersive system.
The speed of sound becomes proportional to the quantum metric of the condensed
state, and depends linearly on the interaction energy. The fraction of
particles depleted out of the condensate and the quantum fluctuations of the
density-density correlation obtain a finite value for infinitesimally small
interactions directly determined by the quantum distance, in striking contrast
to dispersive bands where they vanish with the interaction strength. Our
results reveal that non-trivial quantum geometry allows stability of a flat
band BEC and anomalously strong quantum correlation effects.
