In science though especially on math, but more specifically in topology and differential geometry, a Manifold is a tipical mathematical space which on a small enough scale resembles the Euclidean space of a certain dimension; it is called the Dimension of the Manifold.

Therefore a circle and a line are one-dimensional Manifolds, the surface of a ball and a plane are two-dimensional Manifolds, and so forth. Quite formally, every point of an any n-dimensional Manifold has a neighborhood homeomorphic to the n-dimensional space Rn.

Whereas Manifolds resemble Euclidean spaces near each point, or locally, the overall, or global; a structure of a manifold may be much more complicated.

For intance, any point on the usual two-dimensional surface of a sphere is surrounded by a circular region that can be flattened to a circular region of the plane, as in a geographical map. Yet, the sphere differs from the plane “in the large”: in the language of topology, they are not homeomorphic.

To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat.

In general, any object that is nearly “flat” on small scales is a Manifold, and so Manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.

This concept of a Manifold is key to many parts of geometry and modern mathematical physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.

So, a Manifold is typically provided with a differentiable structure that concedes one to do calculus, Riemannian metric that permits one to calculate distances and angles.

Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian Manifolds model space-time in general relativity.


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