  Weyl conformal geometry is a gauge theory of scale invariance that naturally
brings together the Standard Model (SM) and Einstein gravity. The SM embedding
in this geometry is possible without new degrees of freedom beyond SM and Weyl
geometry, while Einstein gravity is generated by the broken phase of this
symmetry. This follows a Stueckelberg breaking mechanism in which the Weyl
gauge boson becomes massive and decouples, as discussed in the past
(arXiv:1812.08613, 1904.06596, 2104.15118). However, Weyl anomaly could break
explicitly this gauge symmetry, hence we study it in Weyl geometry. We first
note that in Weyl geometry {\it metricity} can be restored with respect to a
new differential operator ($\hat \nabla$) that also enforces a Weyl-covariant
formulation. This leads to a metric-like Weyl gauge invariant formalism that
enables one to do quantum calculations directly in Weyl geometry, rather than
use a Riemannian (metric) geometry picture. The result is the Weyl-covariance
in $d$ dimensions of all geometric operators ($\hat R$, etc) {\it and} of their
derivatives ($\hat\nabla_\mu\hat R$, etc), including the Euler-Gauss-Bonnet
term. A natural, Weyl-invariant dimensional regularisation of quantum
corrections exists and Weyl gauge symmetry is then maintained and manifest at
the quantum level. This is related to a non-trivial current of this symmetry,
the divergence of which cancels the trace of the energy-momentum tensor. The
"usual" Weyl anomaly and Riemannian geometry are recovered in the
(spontaneously) broken phase. The relation to holographic Weyl anomaly is
discussed.
