On the geometry of almost Golden Riemannian manifolds

An almost Golden Riemannian structure $(\varphi ,g)$ on a manifold is given by a tensor field $\varphi $ of type (1,1) satisfying the Golden section relation $\varphi ^{2}=\varphi +1$, and a pure Riemannian metric $g$, i.e., a metric satisfying $g(\varphi X,Y)=g(X,\varphi Y)$. We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of $\varphi $ and the integrability of the $G$-structure corresponding to $(\varphi ,g)$.