  Semimetals exhibiting nodal lines or nodal surfaces represent a novel class
of topological states of matter. While conventional Weyl semimetals exhibit
momentum-space Dirac monopoles, these more exotic semimetals can feature
unusual topological defects that are analogous to extended monopoles. In this
work, we describe a scheme by which nodal rings and nodal spheres can be
realized in synthetic quantum matter through well-defined periodic-driving
protocols. As a central result of our work, we characterize these nodal defects
through the quantum metric, which is a gauge-invariant quantity associated with
the geometry of quantum states. In the case of nodal rings, where the Berry
curvature and conventional topological responses are absent, we show that the
quantum metric provides an observable signature for these extended topological
defects. Besides, we demonstrate that quantum-metric measurements could be
exploited to directly detect the topological charge associated with a nodal
sphere. We discuss possible experimental implementations of Floquet nodal
defects in few-level atomic systems, paving the way for the exploration of
Floquet extended monopoles in quantum matter.
