A gravitational instanton $(M,g)$ is called ALF if there exists a compact set $K\subset M$, $R>0$ and a finite group $\Gamma<O(3)$ acting freely on $\mathbb{S}^{2}$ such that $M\setminus K$ is the total space of a circle fibration $\pi\colon\thinspace M\setminus K\rightarrow(\mathbb{R}^{3}\setminus B_{R})/\Gamma$ and the metric is asymptotically a Riemannian submersion

(3.2)

$$g=\pi^{\ast}g_{\mathbb{R}^{3}/\Gamma}+\theta^{2}+O(r^{-\tau})$$

for a connection $\theta$ on $\pi$ and some $\tau>0$. There are two possibilities for the finite group $\Gamma$: if $\Gamma=\text{id}$ we say that $M$ is an ALF gravitational instanton of cyclic type; if $\Gamma=\mathbb{Z}_{2}$ we say that $M$ is an ALF gravitational instanton of dihedral type.