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The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamicalsystem consisting of a $C^*$-algebra $A$ equipped with an ergodic action of acompact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$is used to interpret the Chevalley-Eilenberg complex with coefficients in thesmooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential formson the dynamical system. We conformally perturb the standard metric, which isassociated with the unique $G$-invariant state on $A$, by means of a Weylconformal factor given by a positive invertible element of the algebra, andconsider the Hermitian structure that it induces on the complex. A Hodgedecomposition theorem is proved, which allows us to relate the Eulercharacteristic of the complex to the index properties of a Hodge-de Rhamoperator for the perturbed metric. This operator, which is shown to beselfadjoint, is a key ingredient in our construction of a spectral triple on$\mathcal{A}$ and a twisted spectral triple on its opposite algebra. Theconformal invariance of the Euler characteristic is interpreted as anindication of the Chern-Gauss-Bonnet theorem in this setting. The spectraltriples encoding the conformally perturbed metrics are shown to enjoy the samespectral summability properties as the unperturbed case.

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^*$-Dynamical Systems