Manifold
A topological space locally resembling Euclidean space near each point.
In mathematics, a manifold is a space that, if you zoom in close enough to any point, looks like ordinary flat space. More formally, an n-dimensional manifold (or n-manifold) is a topological space where every point has a neighborhood that can be stretched and bent into an open region of n-dimensional Euclidean space without tearing or gluing. One-dimensional examples include lines and circles—but not a figure-eight curve, which crosses itself. Two-dimensional manifolds are called surfaces; the plane, sphere, torus, Klein bottle, and real projective plane are all examples. Manifolds are fundamental in geometry and modern physics because they let complex structures be studied using the well-understood properties of simpler spaces. They naturally show up as solution sets of equations or as graphs of functions, and they have practical uses in computer graphics, such as linking coordinates to images in CT scans. Manifolds can also carry extra structure. Differentiable manifolds allow calculus to be performed on them. A Riemannian metric lets you measure distances and angles. Symplectic manifolds appear as phase spaces in classical mechanics, and four-dimensional Lorentzian manifolds model spacetime in general relativity. Studying manifolds requires a working knowledge of calculus and topology.
- Field
- Mathematics
- Known for
- Topological space locally resembling Euclidean space; central to geometry and modern mathematical physics
- Examples
- One-dimensional: lines and circles; Two-dimensional: plane, sphere, torus, Klein bottle, real projective plane
- Additional structure
- Differentiable manifolds (allow calculus), Riemannian metric (measures distances and angles), Symplectic manifolds (phase spaces in Hamiltonian mechanics), Lorentzian manifolds (model spacetime in gen
Lore & Background
Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer graphics given the need to associate pictures with coordinates (e.g., CT scans). One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure-eight. Two-dimensional manifolds are also called surfaces; examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
Reader's Guide
The study of manifolds requires working knowledge of calculus and topology. Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. 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. The circle serves as a motivating example: after a line, it is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Charts, such as those for the top, bottom, left, and right parts of the circle, together form an atlas. The transition function between overlapping charts, such as the top and right charts, maps an interval to itself via a specific formula.
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