# Manifold vs. Manyfold

By Jaxson • Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

One-dimensional manifolds include lines and circles, but not figure eights (because they have crossing points that are not locally homeomorphic to Euclidean 1-space). Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.

Although a manifold locally resembles Euclidean space, meaning that every point has a neighborhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts). When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions.

Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds; this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

Wikipedia
• Manifold (noun)

A copy made by the manifold writing process.

• Manifold (noun)

A pipe fitting or similar device that connects multiple inputs or outputs.

• Manifold (noun)

The third stomach of a ruminant animal, an omasum.

• Manifold (noun)

A topological space that looks locally like the “ordinary” Euclidean space mathbb{R}^n and is Hausdorff.

• Manifold (noun)

A polygon mesh representing the continuous, closed surface of a solid object

Various in kind or quality, diverse.

“The manifold meanings of the simple English word ‘set’ are infamous among dictionary makers.”

Many in number, numerous; multiple, multiplied.

Complicated.

Exhibited at diverse times or in various ways.

“c1384 … the manyfold grace of God. — I Petre 4:10 ([http://wesley.nnu.edu/biblical_studies/wycliffe/ Wycliffe’s Bible])”

“1611 The manifold wisdom of God. — [http://en.wikisource.org/wiki/Bible_%28King_James%29/Ephesians?oldid=448367#Chapter_3 Ephesians 3:10]. (w:King James Bible)”

Many times; repeatedly.

• Manifold (verb)

To make manifold; multiply.

• Manifold (verb)

To multiply or reproduce impressions of by a single operation.