from $x=0$, with $x\in[-1,1]$, and where we set $s_\pm:=s(\pm1)$. The functions $f$ and $g$ depend only on $x$ (or equivalently on $s$); hence these metrics are cohomogeneity one. In [@IKS14], we choose boundary conditions on $f$ and $g$ that result in these metrics inducing geometries on ${\mathcal}S^3\times{\mathcal}S^1$. Here, we instead choose boundary conditions on $f$ and $g$ that result in smooth cohomogeneity geometries on ${\mathcal}S^2\tilde\times{\mathcal}S^2$, thereby defining the class of \[${\mathcal}S^2\tilde\times{\mathcal}S^2$\]-warped Berger geometries. We do this as follows.

It is a standard result in Riemannian geometry that one may smoothly close the boundary at $s_-$, provided that the functions $f_-(s):=f(s_-+s)$ and $g_-(s):=g(s_-+s)$ defined for $0\leq s\leq s_+-s_-$ satisfy $$\label{close-}
f_-^{({\mathrm}{even})}(0)=0,\quad
f_-'(0)=1,\qquad\mbox{and}\qquad
g_-(0)>0,\quad g_-^{({\mathrm}{odd})}(0)=0.$$ The topology then locally becomes that of the disc bundle ${\mathcal}D^2\hookrightarrow{\mathcal}D_1^4\twoheadrightarrow{\mathcal}S^2$ with Euler class $1$ and boundary $\partial{\mathcal}D_1^4\approx{\mathcal}S^3$ that appears in the handlebody construction of ${\mathbb}{CP}^2$. Note that the $2$-sphere here is the base of the Hopf fibration on ${\mathcal}S^3\approx{\mathrm}{SU}(2)$. If one repeats this construction at $s_+$, with $f_+(s):=f(s_++s)$ and $g_+(s):=g(s_++s)$ defined for $s_--s_+\leq s\leq 0$ satisfying $$\label{close+}
f_+^{(\mathrm{even})}(0)=0,\quad
f_+'(0)=-1,\qquad\mbox{and}\qquad
g_+(0)>0,\quad g_+^{({\mathrm}{odd})}(0)=0,$$ one obtains a closed $4$-manifold with the topology of ${\mathcal}S^2\tilde\times{\mathcal}S^2$. We denote by ${\mathcal}S^2_\pm$ the distinguished $2$-spheres that appear as the base spaces in the 