Curved space
Curved space often refers to a spatial geometry which is not “flat” where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in General Relativity where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.
A very familiar example of a curved space is the surface of a sphere. While to our familiar outlook the sphere looks three-dimensional, if an object is constrained to lie on the surface, it only has two dimensions that it can move in. The surface of a sphere can be completely described by two dimensions since no matter how rough the surface may appear to be, it is still only a surface, which is the two-dimensional outside border of a volume. Even the surface of the Earth, which is fractal in complexity, is still only a two-dimensional boundary along the outside of a volume.
One of the defining characteristics of a curved space is its departure with the Pythagorean theorem. In a curved space
The Pythagorean relationship can often be restored by describing the space with an extra dimension. Suppose we have a non-euclidean three-dimensional space with coordinates . Because it is not flat
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