Restricted Weyl invariance in four-dimensional curved spacetime

We discuss the physics of {\it restricted Weyl invariance}, a symmetry of dimensionless actions in four dimensional curved space time. When we study a scalar field nonminimally coupled to gravity with Weyl(conformal) weight of $-1$ (i.e. scalar field with the usual two-derivative kinetic term), we find that dimensionless terms are either fully Weyl invariant or are Weyl invariant if the conformal factor $\mathrm{\Omega }(x)$ obeys the condition $g^{\mu \nu }{\mathrm{\nabla }}_{\mu }{\mathrm{\nabla }}_{\nu }\mathrm{\Omega }=0$. We refer to the latter as {\it restricted Weyl invariance}. We show that all the dimensionless geometric terms such as $R^2$, $R_{\mu \nu }{R}^{\mu \nu }$ and $R_{\mu \nu \sigma \tau }{R}^{\mu \nu \sigma \tau }$ are restricted Weyl invariant. Restricted Weyl transformations possesses nice mathematical properties such as the existence of a composition and an inverse in four dimensional space-time. We exemplify the distinction among rigid Weyl invariance, restricted Weyl invariance and the full Weyl invariance in dimensionless actions constructed out of scalar fields and vector fields with Weyl weight zero.