 Compact manifold

A compact manifold is a smooth manifold whose underlying topological space is compact. Every smooth vector field on a compact manifold without boundary is complete: its integral curves exist for all real time. A finite-time endpoint obstruction would require the curve to escape every compact subset of the domain; compactness and local existence prevent that escape.

With boundary, the direction matters. A vector field tangent to the boundary gives a two-sided flow remaining in the manifold. A field pointing inward at the boundary can instead give a forward flow, while the negative-time curve leaves the manifold. On the compact interval $[0,1]$, the field $x(1-x){\partial }_x$ is tangent at both endpoints and is complete. The field $(1/2-x){\partial }_x$ points inward at both endpoints, but its backward flow from zero immediately exits the interval.