For all $p \in \mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$, let $\displaystyle M(p)=\frac{1}{p^{2}(p-1)^{2}}$. Then, let $g_{p} \, : \, (u,v) \, \longmapsto \, uM(p)v$.
I am not sure about the following assertions :
- Let $M=\mathbb{R} \smallsetminus \lbrace 0,1 \rbrace$. $\big( g_{p} \big)_{p \in M}$ defines a Riemannian metric on $M$. Hence, $M$ equipped with this metric is a Riemannian manifold.
- $M$ cannot be geodesically complete for this metric because, according to Hopf-Rinow theorem, $(M,\vert \cdot \vert)$ would be a complete metric space (which it is not).
I think both are true but I'd feel better if someone could validate or invalidate these assertions.
You have modified the standard metric by a conformal factor function which is always positive so this certainly defines a Riemannian metric.
On the other hand, the metric resembles a hyperbolic metric on the y-axis near the poles of the conformal factor, so it is actually complete "near" these points. Whether or not it is complete "at infinity" I leave to you as an exercise (hint: integrate).