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	Riemann curvature tensor - Wikipedia, the free encyclopedia
		In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion.
		the curvature tensor measures noncommutativity of the covariant derivative.
		is the metric tensor and K is a function called the Gauss curvature.
	en.wikipedia.org /wiki/Riemann_tensor (399 words)

	Bernhard Riemann - Wikipedia, the free encyclopedia
		The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz.
		Riemann was born in Breselenz on September 17 1826, a village near Dannenberg in the Kingdom of Hanover in what is today Germany.
		Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous metric tensor.
	en.wikipedia.org /wiki/Bernhard_Riemann (990 words)

	Encyclopedia :: encyclopedia : Riemann hypothesis (Site not responding. Last check: 2007-11-01)
		The Riemann zeta function along the critical line is sometimes studied in terms of the Z function, whose real zeros correspond to the zeros of the zeta function on the critical line.
		Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof.
		The zeroes of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.
	www.hallencyclopedia.com /Riemann_hypothesis (1834 words)

	Johanneum Lüneburg Bernhard Riemann
		Bernhard Riemann was born on the 17th September 1826 in Breselenz/Dannenberg where his father was a vicar.
		The teachers' recommendation that Riemann was "because of his abilities definitely suitable for the study of mathematical sciences" was not appreciated by his father.
		I hope to have given you an idea of the person Bernhard Riemann, and to have imparted to you a presentiment of the depth of his mathematical thought, although his importance as a mathematician is mainly based upon abstract foundations, with no respect to intellegibility.
	rzserv2.fh-lueneburg.de /u1/gym03/englpage/chronik/riemann/riemann.htm (1706 words)

	Higher Dimensional Geometry
		Riemann examined surfaces of "positive curvature," like the surface of a sphere, where the angles of a triangle exceed 180°, parallel lines (defined as being great circles of the sphere, not latitudinal lines) always meet, and the shortest distance between two points goes through the surface.
		Riemann's version of wormholes consisted of two flat surfaces connected by a cut (now called a Riemann cut) that is topologically equivalent to the ordinary picture of a wormhole, but lacks a neck.
		Although Riemann himself saw these cuts as objects of geometrical interest and not as methods of traveling between universes or areas, the concept of wormholes as shortcuts through time and space is a popular one within the modern physics community and even today's popular culture.
	library.thinkquest.org /27930/geometry.htm (3058 words)

	What is a tensor?
		Tensors, defined mathematically, are simply arrays of numbers, or functions, that transform according to certain rules under a change of coordinates.
		A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points.
		A tensor may consist of a single number, in which case it is referred to as a tensor of order zero, or simply a scalar.
	www.physlink.com /Education/AskExperts/ae168.cfm (878 words)

	why Riemann tensor?...
		A doubt..why einstein Chose Riemann Tensor for GR?..i know its covariant derivative is zero and all that..but Why Riemann tensor?...was not other tensor avaliable or simpler than that?..i studied that and found that for Geodesic deviation (i didn,t understand that concept..sorry) the Riemann tensor was involved....
		All tensors transform under a change of coordinates in a specific way (in the case of relativity and most of modern physics they transform via the Lorentz transform, the transform that Einstein originally derived in his first paper on special relativity).
		The Riemann is R_abcd, the Ricci is formed by a process known as contraction which reduces the number of indices (the rank) of the tensor to two.
	www.physicsforums.com /showthread.php?t=120954 (1727 words)

	How does matter couple to space-time so that space-time becomes curved?
		According to Riemann's theory of curved manifolds, the geometry of space-time is completely described by the metric tensor g_(ab), which you can think of as a 4x4 symmetric matrix, so it has 10 algebraically independent components at each event (point).
		The components of the Riemann tensor identically satisfy a differential equation (the Bianchi identity), which is why the metric tensor (ten algebraically independent components at each event) can and does completely determine the Riemann curvature tensor (20 algebraically independent components at each event).
		The Riemann tensor R_(abcd) can be decomposed into two pieces, the Ricci tensor R_(ab) and the Weyl tensor C_(abcd), in a manner analogous to decomposing a matrix into trace and tracefree parts.
	www.physlink.com /Education/AskExperts/ae98.cfm (689 words)

	Riemannian Gaussian curvature Riemann Modern Relativity modernrelativity special general black hole mass energy ...
		The Riemann tensor is a tensor and the affine connections or Cristoffel symbols are not.
		The Riemann tensor being the field of spacetime curvature that is referred to by the name.
		Since the Riemann tensor is antisymetric in the last two indices, the Ricci tensor could just as well be defined as a contraction over the first and fourth indices.
	www.geocities.com /zcphysicsms/chap6.htm (3998 words)

	Maxima Manual: 29. ctensor
		As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor.
		Tensors of rank 2 are displayed as 2-dimensional matrices, while tensors of higher rank are displayed as a list of 2-dimensional matrices.
		Causes the contortion tensor to be included in the computation of the connection coefficients.
	maxima.sourceforge.net /docs/manual/en/maxima_29.html (2290 words)

	Special objects
		They are the metric tensor, the Kronecker delta, the Christoffel symbol, the Levi-Civita symbol and tensor, the Riemann tensor and the Weyl tensor.
		The metric tensor is symmetric and its covariant derivative is equal to zero.
		The inverse of the metric tensor is denoted by
	www.kfki.hu /(en,html3)/cnc/szhkpub/riccir/node7.html (483 words)

	GRTensorII demonstrations-General Relativity & Geometry.
		Demonstration 1 (ss1): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in curvature coordinates.
		Demonstration 2 (ss2): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in comoving coordinates.
		Demonstration 3 (ss3): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in Bondi coordinates.
	grtensor.phy.queensu.ca /NewDemo/demo.html (1569 words)

	Search Results for Riemann
		Riemann came close to proving the result, but the theory of functions of a complex variable was not sufficiently developed to enable him to complete the proof.
		This generalisation of the Riemann integral revolutionised the integral calculus.
		Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Riemann&CONTEXT=1 (9561 words)

	examples.html (Site not responding. Last check: 2007-11-01)
		Let us define the tensor K with the same symmetries of the Riemann tensor (no cyclic symmetry for now).
		gg is a generic tensor that wil be substituted by g (metric tensor) after the covariant differentiation.
		All invariants up to degree 4 of the Riemann tensor (with no Ricci tensor as factor) are different from zero (supposing a general metric or supposing that all components are positive).
	www.astro.queensu.ca /~portugal/ftp/Riegeom/examples.html (380 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Seventh, and lastly, recall that the Weyl tensor vanishes on a simply connected region iff for some chart on that region, the metric tensor is a scalar multiple of the metric tensor for Minkowski spacetime in the usual Cartesian chart.
		The Riemann tensor vanishes on a region iff that region is locally isometric to Minkowski spacetime ("is locally flat").
		Exercise: show that the extrinsic curvature tensor of the hyperslice T = T0 is K(X) = K_(ab) o^a & o^b, K_(ab) = a(T0) a'(T0)/z^2 diag(1,1,1) Use the Gauss equations to show that the three-dimensional Riemann tensor of this slice is r^2_(323) = r^2_(424) = r^3_(434) = -1/a(T0)^2 IOW, a hyperbolic space H^3 with "radius" a(T0).
	math.ucr.edu /home/baez/PUB/electromagneto (5847 words)

	Riemann Package
		Maple package for calculating components of user-defined or built-in tensors used in General Relativity.
		The user can add, multiply or contract tensors, and the result of any calculation can be assigned to a new tensor.
		Tensors be symmetric or antisymmetric in all indices, symmetric or antisymmetric in two indices, or they can have the symmetry of a symmetric bivector-tensor (Riemann tensor).
	www.cbpf.br /~portugal/Riemann.html (213 words)

	Re: Riemann-Christoffel Tensor?
		This means that we can find a coordinate system in which all the components of the metric tensor are globally constant.
		For the ordinary derivatives to also vanish, we require a frame in which all the Christoffel symbols, which are not tensors, also vanish.
		Both first and second derivatives of the metric tensor would need to vanish in order for the RC to be zero.
	www.lns.cornell.edu /spr/2004-03/msg0059020.html (346 words)

	ModPhy1
		The fourth rank Riemann curvature tensor tells how a vector changes as it is parallel transported from one point to another along different paths.
		The Ricci tensor is the symmetrical second order tensor obtained by contracting the Riemann tensor on its first and third vectors.
		The Einstein tensor is a symmetrical second rank tensor whose divergence is identically zero.
	physics.tamuk.edu /~hewett/ModPhy1/Unit1/GeneralRelativity/EinsteinGravity/EinsteinT/EinsteinT.html (724 words)

	SEP: A Modern Formulation of Riemann's Theory: A Supplement to Nineteenth Century Geometry
		The multidimensional continua that Riemann was concerned with are essentially instances of what is now known as a real n-dimensional smooth manifold.
		Riemann extended this concept of curvature to Riemannian n-manifolds.
		It is therefore known as the Riemann tensor.
	plato.stanford.edu /entries/geometry-19th/supplement.html (1125 words)

	Computations in Riemann Geometry - Curvature Tensors
		The resulting tensor is then multiplied by the matrix whose elements are the derivatives of the old coordinates as functions of the new.
		For tensors with multiple covariant indices, the matrix multiplication is performed on each original index: the transformation is a multilinear operation, the tensor transforming linearly in each index.
		This means that the Weyl Tensor is equal to the Riemann Tensor for such solutions and so we have the interpretation of the Weyl Tensor as that portion of the curvature which is not due to local stress-energy.
	www.rwc.uc.edu /koehler/crg/tensors.html (2017 words)

	outline1.html
		A TENSOR of "rank (0,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a number.
		A TENSOR of "rank (1,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a tangent vector at the point p.
		More generally, a TENSOR of "rank (j,k)" at a point p of spacetime is a function that takes as input a list of j cotangent vectors and k tangent vectors and returns as output a number.
	math.ucr.edu /home/baez/gr/outline2.html (3217 words)

	Search Results for tensor*
		He discovered a tensor, now called Weyl's conformal curvature tensor, whose vanishing is a necessary condition that the space be conformally flat, that is to say, that the space can be mapped conformally on the Euclidean space.
		The transition from the characteristic tensor to the dynamical variables is conveyed by an analysis of the physical meaning of the constituents.
		Schouten worked all his life on tensor analysis and although this seems quite far removed from the topics that van Kampen had been undertaking research on, nevertheless he collaborated with Schouten on three papers on tensor analysis, published in 1930, 1931 and 1933.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=tensor*&CONTEXT=1 (2237 words)

	PlanetMath: Sectional curvature determines Riemann curvature tensor
		But the cyclic sum is zero by (1).
		"Sectional curvature determines Riemann curvature tensor" is owned by kerwinhui.
		This is version 5 of Sectional curvature determines Riemann curvature tensor, born on 2006-05-22, modified 2006-05-24.
	planetmath.org /encyclopedia/SectionalCurvatureDeterminesRiemannCurvatureTensor.html (140 words)

	Riemannian Geometry & Tensor Calculus -- from Mathematica Information Center (Site not responding. Last check: 2007-11-01)
		Each tensor is stored as a nested list under an appropriate global name.
		Several examples of the use of these functions on tensors computed using different metrics are given.
		Beginning with version 2.5, tensor components can be calculated with respect to an arbitrary frame, and approximate calculations (series expansions) can be carried out.
	library.wolfram.com /infocenter/MathSource/4484 (225 words)

	Riemann curvature tensor derivation
		Riemann set up his geometry so it would look flat in the small.
		However, he was amazed that this difference resulting from taking a vector to nearby points could be described by an object (the full curvature tensor) that lived solely at the base point.
		This made him realize the importance of the curvature tensor and gave substance to his geometry.
	www.physicsforums.com /showthread.php?t=57670 (808 words)

	Cartan: a Mathematica package for tensor analysis
		Cartan is an easy-to-use tensor component package for interactive tensor calculations in general Riemann-Cartan spaces of arbitrary dimensions and signatures.
		The tensor concept is important in physics and has wide applications in such diverse fields as relativity theory, cosmology, high energy physics, field theory, thermodynamics, fluid dynamics and mechanics.
		The program allows the user to compute tensor calculations with hundreds or thousands of components and may, moreover, be extended by the addition of user-defined functions.
	www.adinfinitum.no /cartan (642 words)

	The Bianchi identities
		I first show that cycling among the first three terms and summing the results delivers zero: from this, we have that the triply-symmetric and triply-antisymmetric parts of the Riemann tensor are both zero.
		When we contract Riemann(Q) with a [R,R]-rank tensor, g, which Q (of rank R) annihilates, we get a [R,R,R,R]-rank tensor Riemann(Q)·g which is [1,0,2,3]-antisymmetric, [0,1,3,2]-antisymmetric and annihilated by (τ[0,1,2] +τ[1,2,0] +τ[2,0,1]).
		A length, however, is arrived at by integrating the square root of the result of applying a metric to the (tensor) square of displacement along the line.
	www.chaos.org.uk /~eddy/math/smooth/bianchi.html (1145 words)

	[No title]
		) ( :Warning: According to the standard mathematical notation for the Riemann tensor, the contravariant index will be always the first one, instead of being the last one as usual in Mathematica.
		This package and the definitions and conventions used are described in the paper entitled "Computing the Ricci and Einstein tensors", by the same author.
		The first index is the contravariant one." CovariantRiemann::usage = " CovariantRiemann[i,j,k,l] are the covariant components of the Riemann tensor." Riemann::usage = " Riemann[i,j,k,l] are the components of the Riemann tensor.
	www.physics.buffalo.edu /whkinney/PHY581/ricci.m (329 words)

	Springer Online Reference Works
		A four-valent tensor that is studied in the theory of curvature of spaces.
		The components (coordinates) of the Riemann tensor, which is once contravariant and three times covariant, take the form
		In an arbitrary space with an affine connection without torsion the coordinates of the Riemann tensor satisfy the first Bianchi identity
	eom.springer.de /R/r082070.htm (327 words)

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