A brief note on the limit $\omega\rightarrow\infty$ in Weyl geometrical scalar-tensor theory

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Abstract

We obtain vacuum solutions in the presence of a cosmological constant in the context of the Weyl geometrical scalar-tensor theory. We investigate the limit when $\omega$ goes to infinity and show by working out the solutions that in this limit there are some cases in which the scalar field tends to a constant (with the implicit consequence of the geometry becoming Riemannian), although the solutions do not reduce to the corresponding Einstein solutions. We have also extended a previous result, known in the literature, by showing that in the case of vacuum with cosmological constant the field equations of the Weyl geometrical scalar-tensor theory are formally identical to Brans-Dicke field equations, even though these theories are not physically equivalent.