  Quantum particles under geometric constraints are sensitive to the geometry
and topology of the underlying space. We analytically study the laser-driven
nonlinear dynamics of a quantum particle whose motion is constrained to a
two-dimensional Riemannian manifold embedded in a three-dimensional hyperspace.
The geometry of space results in a potential-like term that supports bound
states on the manifold. In the presence of a laser field, we derive expressions
for a generalized Kramers-Henneberger-type unitary transformation which is
shown to be generally space- and time-dependent, and deduce a
Schr\"odinger-like equation in the Kramers-Henneberger frame. Compared to a
flat (geometrically trivial) space, new time-averaged coefficients of
differential operators and operator-valued perturbation terms appear which
determine the geometry-dependent laser-dressed states on Riemannian manifolds.
