Through consider a Riemannian manifold as a classes space G/H, where G/H is a space of left and right invariant classes modulo an isotropy subgroup H, are studied and developed methods in integral geometry to measure the curvature of G/H, through their orbits, cycles, co-cycles, field ramifications, etcetera, obtaining geometrical invariants as images of integral transforms that define the topological nature of the space. Several of these geometrical invariants are product of embbedings in topological vector spaces on the Riemannian manifold. The Spectrum of curvature will be of sheaves with curved germs or that include the curvature tensor. Then can be established an algebraic scheme related with equivalences of the moduli problems in field theory between the geometrical invariants and spectral images of curvature. Finally, this can be generalized to the quantum context to obtain and design an advanced censorship to possible advanced devices to detect and measure quantum gravity using the second curvature.