Chern-Ricci invariance along G-geodesics

Over a compact oriented manifold, the space of Riemannian metrics and normalised positive volume forms admits a natural pseudo-Riemannian metric $G$, which is useful for the study of Perelman's $\mathcal{W}$ functional. We show that if the initial speed of a $G$-geodesic is $G$-orthogonal to the tangent space to the orbit of the initial point, under the action of the diffeomorphism group, then this property is preserved along all points of the $G$-geodesic. We show also that this property implies preservation of the Chern-Ricci form along such $G$-geodesics, under the extra assumption of complex aniti-invariant initial metric variation and vanishing of the Nijenhuis tensor along the $G$-geodesic. This result is useful for a slice type theorem needed for the proof of the dynamical stability of the Soliton-K\"ahler-Ricci flow.