## manifoldA mathematical object that, in geometrical terms, is nearly "flat" on a small scale (though on a larger scale it may bend and twist into exotic and intricate forms). More precisely, a manifold is a topological space that looks locally like ordinary Euclidean space. Every manifold has a dimension, which is the number of coordinates needed to specify it in the local coordinate system. A circle, although curved through two dimensions, is an example of a one-dimensional manifold, or one-manifold.
A close-up view reveals that any small segment of the circle is practically
indistinguishable from a straight line. Similarly, a sphere's
two-dimensional surface, even though it curves through three dimensions,
is an example of a two-manifold. Seen locally, the surface,
like that of a small portion of the Earth, appears flat. A manifold that
is smooth enough to have locally well-defined directions is said to be differentiable.
If it has enough structure to enable lengths and angles to be measured,
then it is called a Riemannian manifold. Differentiable
manifolds are used in mathematics to describe geometrical objects, and are
also the most natural and general settings in which to study differentiability.
In physics, differentiable manifolds serve as the phase
space in classical mechanics, while four dimensional pseudo-Riemannian
manifolds are used to model spacetime
in general relativity.## Related category• TOPOLOGY | |||||

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