![]() ![]() ![]() ![]() So, by projecting onto the first coordinate, one obtains a continuous mapping between the semicircle and the open interval (−1, 1): Any point of this semicircle can be uniquely described by its x-coordinate. Consider, for instance, the top half of the unit circle, x 2 + y 2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Topology ignores bending, so a small piece of a circle is exactly the same as a small piece of a line. The circle is the simplest example of a topological manifold after a line. The origin is understood to be at the centre of the circle. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding sets and functions, and helpful to have a working knowledge of calculus and topology.įigure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.Ī technical mathematical definition of a manifold is given below. Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.Īdditional structures are often defined on manifolds. Examples include a plane, the surface of a sphere, and the surface of a torus. In a two-manifold, every point has a neighbourhood that looks like a disk. Examples of one-manifolds include a line, a circle, and two separate circles. In a one-dimensional manifold (or one-manifold), every point has a neighbourhood that looks like a segment of a line. For example, lines are one-dimensional, and planes two-dimensional. In discussing manifolds, the idea of dimension is important. A sphere can be represented by a collection of two dimensional maps, therefore a sphere is a manifold.Ī manifold is an abstract mathematical space in which every point has a neighbourhood which resembles Euclidean space, but in which the global structure may be more complicated. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. On a sphere, the sum of the angles of a triangle is not equal to 180°. ![]()
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