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Patrick Sossoumihen
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Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?
Hi Patrick:Patrick Sossoumihen said:Some of proofs without words have been described using Euclidian Geometry, do those proofs still hold alike in Riemmanian Geometry?
Euclidian geometry is a branch of mathematics that deals with the properties and relationships of points, lines, angles, and figures in space. It was developed by the Greek mathematician Euclid and is based on five postulates that are used to prove geometric theorems.
Riemannian geometry is a branch of mathematics that deals with the properties and relationships of geometric figures on curved surfaces. It was developed by the German mathematician Bernhard Riemann and is based on the concept of curvature, which allows for the study of non-Euclidian spaces.
While Euclidian geometry is based on the concept of flat, or Euclidian, space, Riemannian geometry allows for the study of curved spaces. This means that the postulates and theorems in Euclidian geometry may not hold true in Riemannian geometry.
Yes, some aspects of Euclidian geometry can hold in Riemannian geometry, but not all. For example, the first four postulates of Euclid are still valid in Riemannian geometry, but the fifth postulate, also known as the parallel postulate, does not hold true in all curved spaces.
The validity of Euclidian geometry in Riemannian geometry is determined by the curvature of the space in question. In spaces with zero curvature, Euclidian geometry is valid. However, in spaces with non-zero curvature, Euclidian geometry may not hold true and Riemannian geometry must be used instead.