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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an ${\displaystyle n}$-dimensional manifold, or ${\displaystyle n}$-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of ${\displaystyle n}$-dimensional Euclidean space.

One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. 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.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. 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).

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 study of manifolds requires working knowledge of calculus and topology.

Motivating examples

Circle

After a line, a circle 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. Considering, for instance, the top part of the unit circle, x² + y² = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous and invertible mapping from the upper arc to the open interval (−1, 1):

$${\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,}$$

Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle:

$${\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}}$$

Together, these parts cover the whole circle, and the four charts form an atlas for the circle.

The top and right charts, ${\displaystyle \chi _{\mathrm {top} }}$ and ${\displaystyle \chi _{\mathrm {right} }}$ respectively, overlap in their domain: their intersection lies in the quarter of the circle where both ${\displaystyle x}$ and ${\displaystyle y}$-coordinates are positive. Both map this part into the interval ${\displaystyle (0,1)}$Zdroj:https://en.wikipedia.org?pojem=Manifold_(mathematics)
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