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A simple example of a non-complete manifold is given by the punctured plane (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.
There exists non geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. It is the case for example of the Clifton–Pohl torus.
Path-connectedness, completeness and geodesic completeness
It can be shown that a finite-dimensional path-connected Riemannian manifold is a complete metric space (with respect to the Riemannian distance) if and only if it is geodesically complete. This is the Hopf–Rinow theorem. This theorem does not hold for infinite-dimensional manifolds. The example of a non-complete manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.