  A debate has appeared in the literature on loop quantum gravity and spin
foams, over whether the secondary simplicity constraints, reducing the
connection to be Levi-Civita, should imply the shape matching conditions,
reducing twisted geometries to Regge geometries. We address the question using
a simple model with a flat dynamics, in which secondary simplicity constraints
arise from a dynamical preservation of the primary ones. We find that shape
matching conditions arise, thus providing support to an affirmative question.
The origin of these extra conditions is to be found in the different graph
localisation of the Hamiltonian and primary simplicity constraints. Our results
are consistent with previous claims by Dittrich and Ryan, and extend their
validity to Lorentzian signature and arbitrary cellular decompositions. We show
in particular how the (gauge-invariant version of the) twist angle xi featuring
in twisted geometries equals on-shell the Regge dihedral angle multiplied by
the Immirzi parameter, thus recovering the discrete extrinsic geometry from the
Ashtekar-Barbero holonomy. Finally, we confirm that flatness implies both the
Levi-Civita and the shape-matching conditions using twisted geometries and a
4-dimensional version of the vertex condition appearing in 't Hooft's polygon
model.
