A manifold is a mathematical object that, in geometrical terms, is nearly 'flat' on a small scale (though on a larger scale it may bend and twist into exotic and intricate forms). More precisely, a manifold is a topological space that looks locally like ordinary Euclidean space. Every manifold has a dimension, which is the number of coordinates needed to specify it in the local coordinate system.

 

A circle, although curved through two dimensions, is an example of a one-dimensional manifold, or one-manifold. A close-up view reveals that any small segment of the circle is practically indistinguishable from a straight line. Similarly, a sphere's two-dimensional surface, even though it curves through three dimensions, is an example of a two-manifold. Seen locally, the surface, like that of a small portion of Earth, appears flat.

 

A manifold that is smooth enough to have locally well-defined directions is said to be differentiable. If it has enough structure to enable lengths and angles to be measured, then it is called a Riemannian manifold. Differentiable manifolds are used in mathematics to describe geometrical objects, and are also the most natural and general settings in which to study differentiability. In physics, differentiable manifolds serve as the phase space in classical mechanics, while four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity.

 

Laminations

A lamination is a decoration of a manifold in which some subset is partitioned into sheets of some lower dimension, and the sheets are locally parallel. It may or may not be possible to fill the gaps in a lamination to make a foliation.