- published: 11 Oct 2015
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In mathematics (specifically in geometry and topology), a manifold is a mathematical object that on a small enough scale resembles Euclidean space. For example, seen from far away, the surface of the planet Earth is not flat and Euclidean, but on a smaller scale, one may describe each region via a geographic map, a projection of the surface onto the Euclidean plane.
A precise mathematical definition of a manifold is given below.
Lines and circles (but not figure eights) are one-dimensional manifolds (1-manifolds). The plane and the sphere are examples of 2-manifolds. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds.
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. A manifold can be endowed with a differentiable structure that allows one to do calculus on manifolds and a Riemannian metric that allows one to measure 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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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