- published: 11 Oct 2015
- views: 5916
In mathematics, a manifold is a topological space that 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.
One-dimensional manifolds include lines and circles, but not figure eights (because they have singularities called crossing points). 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 cannot.
Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, 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.
Supernova
Million years the time has come
Million years are on the ground
The stars are always there at night
Shining on the earth so bright
Endless systems in the space
Endless silence around this place
Million years the time has come