= Solution
Take the <circular cylinder>
$$
S=\{(R\cos\theta,R\sin\theta,z):\theta\in\mathbb R, z\in\mathbb R\},
\qquad R>0.
$$
Its <first fundamental form> is $R^2d\theta^2+dz^2$. The universal covering parametrization $\mathbb R^2\to S$ is therefore a <local isometry> from the <Euclidean plane>, after scaling the first coordinate. By <geodesic preservation by a local isometry>, every geodesic on the cylinder lifts to a straight line defined for all $t\in\mathbb R$. Thus \b[the cylinder is geodesically complete]. It is noncompact because $z$ is unbounded, and it is not a plane because one of its <principal curvatures> is $1/R$.
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