The information contained in the generalised Gibbons-Hawking ansatz can be encoded by a Riemannian metric on the base, written in distinguished coordinates as

(1.15)

$$g_{\mathcal{B}^{0}}=V^{ij}d\mu_{i}\otimes d\mu_{j}+\text{Re}(W^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q}),$$

such that the map $M\to\mathcal{B}^{0}$ is a Riemannian submersion.