In , a series of results on compact manifolds are included.

In higher dimensions, classification theory normally focuses only on compact connected manifolds.

Every vector field on a compact manifold without boundary is complete.

The space of smooth functions on any compact manifold is nuclear.

These turns allow for a more compact manifold, with denser packaging of the whole engine, as a result.

All compact, orientable manifolds of dimension 3 or less are spin.

Let's return to the general case that is a compact manifold.

Note that the closure of each fiber is a compact manifold with boundary.

Let be a diffeomorphism of a compact smooth manifold .

The triple torus is a three-dimensional compact manifold with no boundary.