We argue that the exponential relation $g_{μν} = \bar{g}_{μρ}\big(\mathrm{e}^h\big)^ρ_ν$ 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.
Integrability of geodesics of totally geodesic metrics
给出了在签名$(1,3)$的空间中的测地线的分析，所述签名$(1,3)$分裂成由Weyl张量本征空间（双曲和椭圆本征空间）产生的二维分布，所述Weyl张量本征空间在[V. Lychagin，V. Yumaguzhin，\emph{Differential invariants and exact solutions of the Einstein equations}，Anal.Math.Phys. 1664- 235 X 1-9（2016）]中描述。测地线方程是可积的情况下确定。对与[V. Lychagin，V. Yumaguzhi，\emph{Differential invariants and exact solutions of the Einstein-Maxwell equation}，Anal.Math.Phys. 1，19- 29，（2017）]中描述的电磁耦合的相同模型进行类似分析。
