  We investigate Floquet engineering in a sawtooth chain -- a minimal model
hosting a nearly flat band endowed with nontrivial quantum geometry -- coupled
to driven surface polaritons. In this paradigmatic flat band model,
light-matter coupling to a flat band is enabled by quantum geometry despite the
vanishing band velocity and band curvature. We show that light polarization and
finite momentum transfer in polaritonic settings provide sufficient tunability
to flatten or unflatten bands, with sometimes drastic band structure
modifications beyond what is attainable with laser pulses in free space.
Possible implications for light-driven phenomena in prototypical flat-band
moir\'e or kagome materials are discussed.
