  We analyze quantum-geometric bounds on optical weights in topological phases
with pairs of bands hosting non-trivial Euler class, a multi-gap invariant
characterizing non-Abelian band topology. We show how the bounds constrain the
combined optical weights of the Euler bands at different dopings and further
restrict the size of the adjacent band gaps. In this process, we also consider
the associated interband contributions to DC conductivities in the flat-band
limit. We physically validate these results by recasting the bound in terms of
transition rates associated with the optical absorption of light, and
demonstrate how the Euler connections and curvatures can be determined through
the use of momentum and frequency-resolved optical measurements, allowing for a
direct measurement of this multi-band invariant. Additionally, we prove that
the bound holds beyond the degenerate limit of Euler bands, resulting in nodal
topology captured by the patch Euler class. In this context, we deduce optical
manifestations of Euler topology within $\vec{k} \cdot \vec{p}$ models, which
include AC conductivity, and third-order jerk photoconductivities in doped
Euler semimetals. We showcase our findings with numerical validation in
lattice-regularized models that benchmark effective theories for real materials
and are themselves directly realizable in metamaterials and optical lattices.

Mesoscale and Nanoscale Physics;Quantum Gases;Strongly Correlated Electrons;Optics;Quantum Physics