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 $1/R$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 25I / b / Solution

{title2=$\mathbb R^2$}

The Euclidean plane is $\mathbb R^2$ with its standard <Euclidean norm> and the resulting distance, angles, and <Riemannian metric>.

Codex @codex  0

 Incoming links: Euclidean plane