Riemannian geometry

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Definition
  1. Noun:
    • A non-Euclidean geometry: Riemannian geometry is a branch of mathematics that studies curved spaces, specifically smooth, differentiable manifolds equipped with a Riemannian metric. It generalizes the geometry of surfaces to higher dimensions and is fundamental to modern physics, particularly Einstein's theory of general relativity.
Usage Examples
  • Noun:
    • Einstein's theory of general relativity is formulated within the framework of Riemannian geometry.
    • Understanding Riemannian geometry is essential for studying the curvature of spacetime.
Advanced Usage
  • "Riemannian manifold": A smooth manifold equipped with a positive-definite inner product (a Riemannian metric) on each tangent space, allowing for the measurement of lengths and angles.
    • A sphere is a simple example of a two-dimensional Riemannian manifold.
Variants and Related Words

  • Riemannian metric (n): A mathematical object that defines the geometric structure of a Riemannian manifold, enabling the calculation of distances and angles.

    • The Riemannian metric allows us to compute the length of a curve on the manifold.

  • Pseudo-Riemannian geometry (n): A generalization of Riemannian geometry where the metric tensor is not required to be positive-definite; it is the mathematical foundation of general relativity.

    • The geometry of spacetime in general relativity is described by pseudo-Riemannian geometry.
Synonyms
  • Elliptic geometry: A specific type of non-Euclidean geometry where, given a line and a point not on that line, no parallel lines exist. Riemannian geometry can encompass elliptic geometries on spaces of constant positive curvature.
  • Differential geometry: The broader field of mathematics that uses calculus and linear algebra to study geometric problems, of which Riemannian geometry is a major subfield.
Related Phrases

  • "Sectional curvature": In Riemannian geometry, a way to describe the curvature of a manifold by considering the curvature of two-dimensional subspaces (sections) of the tangent space.

    • The sign of the sectional curvature determines whether the geometry is spherical, flat, or hyperbolic locally.

  • "Geodesic": In Riemannian geometry, the generalization of a straight line to curved spaces; it is the shortest path between two points on a manifold.

    • On a sphere, the geodesics are segments of great circles.
Related Concepts
  • Non-Euclidean geometry: A term for geometries that do not obey Euclid's parallel postulate, encompassing both Riemannian (elliptic/spherical) and hyperbolic geometries.
    • Riemannian geometry is a primary example of a non-Euclidean geometry.
Noun
  1. (mathematics) a non-Euclidean geometry that regards space as like a sphere and a line as like a great circle
    • Bernhard Riemann pioneered elliptic geometry

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