  A band-projection formalism is developed for calculating the superfluid
weight in two-dimensional multi-orbital superconductors with an
orbital-dependent pairing. It is discovered that, in this case, the band
geometric superfluid stiffness tensor can be locally non-positive-definite in
some regions of the Brillouin zone. When these regions are large enough or
include nodal singularities, the total superfluid weight becomes
non-positive-definite due to pairing fluctuations, resulting in the transition
of a BCS state to a pair-density wave (PDW). This geometric BCS-PDW transition
is studied in the context of two-orbital superconductors, and proof of the
existence of a geometric BCS-PDW transition in a generic topological flat band
is established.
