Riemannian Curvature Tensor

Riemannian most often refers to

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

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Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

Pseudo-Riemannian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

Sub-Riemannian manifold In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal s ...

** Riemannian submanifold **

Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

Riemannian circle In mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-p ...

Riemannian submersion In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let ( ...

Riemannian Penrose inequality In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemanni ...

Riemannian holonomy In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...

Riemann curvature tensor Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

Riemannian connection In mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along ...

Riemannian connection on a surface In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel t ...

Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geomet ...

Riemannian volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...

Riemannian bundle metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric. Definition If ''M'' is a topological manifold ...

List of topics named after Bernhard Riemann The German mathematician Bernhard Riemann (1826–1866) is the eponym of many things. "Riemann" (by field) * Riemann bilinear relations * Riemann conditions * Riemann form * Riemann function * Riemann–Hurwitz formula * Riemann matrix * Riemann ...

but may also refer to

Hugo Riemann Karl Wilhelm Julius Hugo Riemann (18 July 1849 – 10 July 1919) was a German musicologist and composer who was among the founders of modern musicology. The leading European music scholar of his time, he was active and influential as both a mus ...

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Neo-Riemannian theory Neo-Riemannian theory is a loose collection of ideas present in the writings of music theory, music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harm ...

(music) {{disambiguation