= Solution
Let $\pi:\widetilde M\to M$ be the <universal cover> and equip it with the pullback <Riemannian metric> $\widetilde g=\pi^*g$. The <covering map> is a <local isometry>, so $\widetilde M$ has constant <sectional curvature> one.
We must check completeness of the lifted <Riemannian metric>. A <geodesic> on $\widetilde M$ projects to a <geodesic> on $M$. By <geodesic completeness> of $M$, the projected <geodesic> extends for all real time. Its unique lift with a specified initial point extends the original lifted <geodesic>, because $\pi$ is a <local isometry>. Hence $\widetilde M$ is geodesically complete, and the <Hopf-Rinow theorem> also gives metric completeness.
The allowed constant-curvature form of <Cartan-Ambrose-Hicks theorem> says that a linear tangent-space <isometry> between complete simply connected <Riemannian manifolds> of the same constant <sectional curvature> extends to a global <isometry>. The unit round <sphere> $S^n$ is complete, simply connected for $n\geq2$, and has <sectional curvature> one. Comparing it with $\widetilde M$ therefore yields
$$
\boxed{(\widetilde M,\widetilde g)\cong S^n.}
$$
This is the covering step in the <classification of complete positive constant-curvature manifolds>. As usual for this assertion, $n\geq2$ is understood: in dimension one there are no sectional two-planes, and the <universal cover> of the circle is the line, not the circle.
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