Completeness and global shortest paths

Try

Remove the origin from the Euclidean plane and connect (−1,0) to (1,0). Every allowed path has length at least two, but the straight segment attaining that bound is missing a point. Detour around a tiny semicircle of radius ε: the path length is 2−2ε+πε, tending to two without reaching it.

Thus the intrinsic distance can be an unattained infimum. Smooth local geometry and local geodesic existence do not guarantee a global shortest path between every pair.

Predict

Predict what happens to a straight geodesic aimed at the missing origin. It reaches the absent point in finite time and cannot continue as that geodesic inside the punctured plane.

Predict whether compactness of a smooth boundaryless surface prevents such finite-time escape. Constant speed confines its lifted position and velocity to a compact region of the tangent bundle, allowing ODE continuation.

Experiment

Separate global properties.

Surface with its intrinsic metric	Complete?	Typical global feature
Euclidean plane	Yes	Unique straight minimizer
Punctured plane	No	Some distance infima unattained
Round sphere	Yes	Antipodal minimizers nonunique
Flat cylinder	Yes	Multiple routes from wrapping

The table is about the intrinsic length metric, not merely closedness of a drawing in a chosen viewport. Verify the punctured-plane detour formula and explain why no allowed path can have length exactly two.

Understand

For the punctured-plane example, equality in the endpoint-distance bound would require the tangent to point monotonically along the displacement direction almost everywhere. The path would follow the straight segment and pass through the missing origin. Hence no allowed path attains two, while the explicit detours show the infimum is two.

Now consider a compact smooth surface without boundary. A geodesic has constant speed. Its lifted curve (γ,γ′) stays in the fixed-speed sphere bundle over the compact base, a compact subset of the tangent bundle. The geodesic equation is a smooth vector field there. The ODE continuation theorem says a solution that remains in a compact subset of its domain cannot have a finite maximal endpoint. Thus every geodesic extends for all time.

The global Hopf–Rinow theorem states that for a connected finite-dimensional Riemannian manifold, metric completeness, geodesic completeness and compactness of closed bounded metric balls are equivalent. Under these conditions every pair of points is joined by a minimizing geodesic. This is a substantive theorem beyond the local ODE and compact-continuation argument just proved.

Apply

Completeness ensures existence of minimizing paths, not uniqueness. Antipodal sphere points have infinitely many great-circle semicircle minimizers, each of length πR. Nor does completeness make every long geodesic segment minimizing; a great circle can continue far beyond its cut point.

Repair “The coordinate chart has a finite edge, so the manifold is incomplete.” A chart can end while the surface and geodesic continue in another chart. Incompleteness concerns the manifold and its metric, not the parameter rectangle. The sphere’s angular seam and poles are coordinate limitations, not missing points.

For optional depth, noncompact complete manifolds such as the plane show that completeness is weaker than compactness. The compactness conclusion in Hopf–Rinow applies to closed bounded metric balls, not the whole manifold unless its diameter is bounded.

Continue

In a shortest-path claim distinguish local minimality, global attainment, uniqueness and extension for all time. Continue to geodesic curvature and boundary turning to measure the bending of boundaries within a surface.