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Riemannian Wavefield Extrapolation (music)
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Riemannian most often refers to Bernhard Riemann: * Riemannian geometry *Riemannian manifold ** Pseudo-Riemannian manifold **Sub-Riemannian manifold **Riemannian submanifold ** Riemannian metric *Riemannian circle * Riemannian submersion *Riemannian Penrose inequality *Riemannian holonomy * Riemann curvature tensor * Riemannian connection **Riemannian connection on a surface *Riemannian symmetric space * Riemannian volume form *Riemannian bundle metric * List of topics named after Bernhard Riemann but may also refer to Hugo Riemann: *Neo-Riemannian theory Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Biography Early years Riemann was born on 17 September 1826 in Breselenz, a village near Dannenb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Holonomy
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features. Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry (called Riemannian holonomy), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy of the connection can be identified with a Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 operator * Riemann singularity theorem ** Riemann-Kempf singularity theorem *Riemann surface **Compact Riemann surface ** Planar Riemann surface * Cauchy–Riemann manifold ** The tangential Cauchy–Riemann complex *Zariski–Riemann space Analysis *Cauchy–Riemann equations *Riemann integral ** Generalized Riemann integral ** Riemann multiple integral *Riemann invariant *Riemann mapping theorem **Measurable Riemann mapping theorem *Riemann problem *Riemann solver *Riemann sphere *Riemann–Hilbert correspondence *Riemann–Hilbert problem *Riemann–Lebesgue lemma *Riemann–Liouville integral *Riemann–Roch theorem **Arithmetic Riemann–Roch theorem ** Riemann–Roch theorem for smooth manifolds **Riemann–Roch theorem for surfaces ** ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 and : ''E'' → ''M'' a vector bundle on ''M'', then a metric on ''E'' is a bundle map ''k'' : ''E'' ×''M'' ''E'' → ''M'' × R from the fiber product of ''E'' with itself to the trivial bundle with fiber R such that the restriction of ''k'' to each fibre over ''M'' is a nondegenerate bilinear map of vector spaces.. Roughly speaking, ''k'' gives a kind of dot product (not necessarily symmetric or positive definite) on the vector space above each point of ''M'', and these products vary smoothly over ''M''. Properties Every vector bundle with paracompact base space can be equipped with a bundle metric. For a vector bundle of rank ''n'', this follows from the bundle charts \phi:\pi^(U)\to U\times\mathbb^n: the bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form'' or ''pseudo-volume form''. It als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Symmetric Space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections. Historical overview After the classical work of Gauss on the differential geometry of surfaces and the subsequent emergence of the concept of Riemannian manifold initiated by Bernhard Riemann in the mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Connection
In mathematics, a metric connection is a connection (vector bundle), 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 any curve. .(''Third edition: see chapter 3; Sixth edition: see chapter 4.'') This is equivalent to: * A connection for which the connection (vector bundle), covariant derivatives of the metric on ''E'' vanish. * A connection (principal bundle), principal connection on the bundle of orthonormal frames of ''E''. A special case of a metric connection is a #Riemannian connection, Riemannian connection; there is a unique such which is torsion tensor, torsion free, the Levi-Civita connection. In this case, the bundle ''E'' is the tangent bundle ''TM'' of a manifold, and the metric on ''E'' is induced by a Riemannian metric on ''M''. Another special case of a metric connection is a #Yang–Mills connection, Yang–M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Curvature Tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is ''flat'', i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Riemannian Penrose inequality is an important special case. Specifically, if (''M'', ''g'') is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass ''m'', and ''A'' is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts : m \geq \sqrt. This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (''M'', ''g'') having nonnegative scalar curvature is equivalent to the spacetime obeying the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (''M'', ''g'') and (''N'', ''h'') be two Riemannian manifolds and f:M\to N a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution \mathrm(df)^ is a sub-bundle of the tangent bundle of TM which depends both on the projection f and on the metric g. Then, ''f'' is called a Riemannian submersion if and only if the isomorphism df : \mathrm(df)^ \rightarrow TN is an isometry. Examples An example of a Riemannian submersion arises when a Lie group G acts isometrically, freely and properly on a Riemannian manifold (M,g). The projection \pi: M \rightarrow N to the quotient space N = M /G equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on S^3 \subset \mathbb^2 by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
