riemannian

Thân thiện

A mathematician draws a riemannian manifold on a chalkboard.

Definition

1. 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.

Học thuật

Adjective

1. of or relating to Riemann's non-Euclidean geometry

Từ đồng nghĩa

• riemannian