Closed manifold

(Redirected from Compact manifold)

In mathematics, a closed manifold is a manifold without boundary that is compact.

In comparison, an open manifold is a manifold without boundary that has only non-compact components.

Examples

The only connected one-dimensional example is a circle. The torus and the Klein bottle are closed. A line is not closed because it is not compact. A closed disk is compact, but is not a closed manifold because it has a boundary.

Open manifolds

For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

Abuse of language

Most books generally define a manifold as a space that is, locally, diffeomorphic to Euclidean space, thus by this definition, every manifold does not include its boundary. However, this definition is too specific as it doesn’t cover even basic objects such as a closed disk, so authors usually define a manifold with boundary and abusively say manifold without reference to the boundary. Due to this, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifolds if the definition is taken to be original definition.

The notion of a closed manifold is unrelated with that of a closed set. A disk with its boundary is a closed subset of the plane, but not a closed manifold

Use in physics

The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.

References

• Michael Spivak: A Comprehensive Introduction to Differential Geometry. Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, ISBN 0-914098-70-5.