Connections and geodesics in the space of metrics

We argue that the exponential relation $g_{\mu\nu} = \bar{g}{\mu\rho}\big($\mathrm{e}$^h\big)^\rho{}\nu$ is the most natural metric parametrization since it describes geodesics that follow from the basic structure of the space of metrics. The corresponding connection is derived, and its relation to the Levi-Civita connection and the Vilkovisky-DeWitt connection is discussed. We address the impact of this geometric formalism on quantum gravity applications. In particular, the exponential parametrization is appropriate for constructing covariant quantities like a reparametrization invariant effective action in a straightforward way. Furthermore, we reveal an important difference between Euclidean and Lorentzian signatures: Based on the derived connection, any two Euclidean metrics can be connected by a geodesic, while this does not hold for the Lorentzian case.