riemannian
Definition
- Adjective:
- Relating to or characteristic of the mathematical theories, concepts, or geometry developed by the German mathematician Bernhard Riemann: Specifically, it pertains to the non-Euclidean geometric framework and the associated mathematical structures, such as manifolds and curvature, that he pioneered.
Usage Examples
- Adjective:
- The physicist studied the Riemannian geometry of spacetime.
- Understanding Riemannian manifolds is essential for general relativity.
Advanced Usage
"Riemannian manifold": A smooth manifold equipped with a Riemannian metric, which defines geometric notions like distance, angle, and curvature at every point.
- The surface of a sphere is a simple example of a Riemannian manifold.
"Riemannian geometry": The branch of differential geometry that studies smooth manifolds with a Riemannian metric, extending the concepts of classical geometry to curved spaces.
- Einstein's theory of general relativity is formulated within the framework of Riemannian geometry.
Variants and Related Words
Riemannian metric (n): A family of inner products defined on the tangent spaces of a manifold, allowing the measurement of lengths and angles.
- The Riemannian metric tensor is a fundamental object in this geometry.
Pseudo-Riemannian (adj): A generalization of Riemannian geometry where the metric tensor is not required to be positive-definite, crucial for the geometry of spacetime in relativity.
Synonyms
- Non-Euclidean: Pertaining to geometries that do not obey Euclid's parallel postulate, a category that includes Riemannian geometry.
Adjective
-
of or relating to Riemann's non-Euclidean geometry
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