In the paper "Asymptotically flat self-dual solutions to euclidean gravity" by T.Eguchi and A.J. Hanson, the author constructed a Ricci flat metric on $\mathbb{R}\times S^3$. Let $$ \sigma_x=\frac12(-\cos \psi d\theta-\sin \theta \sin \psi d\phi), $$ $$ \sigma_y=\frac12(\sin \psi d\theta-\sin \theta\cos \psi d\phi), $$ $$ \sigma_z=\frac12(-d\psi-\cos \theta d\phi). $$ They obey the structure equations of the exterior algebra, $$ d\sigma_x=2\sigma_y \Lambda \sigma_z, \quad etc. $$ Here $\theta, \psi$ and $\phi$ are Euler angles on $S^3$ with ranges $0\leqslant \theta \leqslant \pi, 0\leqslant \phi \leqslant 2\pi, 0\leqslant \psi \leqslant 4\pi$. Let me list one of the two types of metrics having axial symmetry in this paper: $$ ds^2=f^2(r)dr^2+r^2(\sigma_x^2+\sigma_y^2)+r^2 g^2(r)\sigma_z^2, $$ where $$ g(r)=f^{-1}(r)=[1-(\frac{a}{r})^4]^{\frac12}, $$ $a$ is an integration constant.

The sectional curvature are one of $\pm \frac{2a^4}{r^6}, \pm \frac{4a^4}{r^6}$. The curvatures are regular everywhere for $r\geqslant a$. Although the metrics have an apparent singularity at $r=a$, it can be eliminated by a change of variable.

**Question 1: I find in some other references, Eguchi-Hanson metric is a metric on the tangent bundle of $S^2$, not $\mathbb{R} \times S^3$, why?**

**Question 2: In the metric we should consider $r\geqslant a$ or $r\geqslant 0$? Is the tangent cone at infinity $(\eta M, p)\to (\mathbb{R}^4/Z_2,0)$ as $\eta \to 0$?**