In mathematics, a manifold of dimension n is a topological space that near each point resembles n-dimensional Euclidean space. More precisely, each point of an n-dimensional manifold has a neighbourhood (mathematics)|neighbourhood that is homeomorphic to the Euclidean space of dimension n. Line (geometry)|Lines and circles, but not Lemniscate|figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the Plane (geometry)|plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.
Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighbouring maps (in the context of manifolds they are called Atlas_(topology)#Charts|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 more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as Graph of a function|graphs of functions. Manifolds may have additional features. 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 mechanics|Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.

In mathematics, a manifold of dimension n is a topological space that near each point resembles n-dimensional Euclidean space. More precisely, each point of an n-dimensional manifold has a neighbourhood (mathematics)|neighbourhood that is homeomorphic to the Euclidean space of dimension n. Line (geometry)|Lines and circles, but not Lemniscate|figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the Plane (geometry)|plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.
Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighbouring maps (in the context of manifolds they are called...