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Topic: Ricci curvature tensor

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Ricci tensor

Curvature of Riemannian manifolds

Scalar curvature

Sectional curvature

Riemannian geometry

Einstein manifold

Christoffel symbols

Ricci flow

Geometrization conjecture

Uniformization theorem

Tensor

Riemannian manifold

Tangent bundle

Chern class

Riemannian metric

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	General relativity - Wikinfo
		Mathematically, Einstein models space-time by a four-dimensional pseudo-Riemannian manifold, and his field equation states that the manifold's curvature at a point is directly related to the stress energy tensor at that point; the latter tensor being a measure of the density of matter and energy.
		General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law.
		where $R_{ik}$ is the Ricci curvature tensor, $R$ is the Ricci curvature scalar, $g_{ik}$ is the metric tensor, $\Lambda$ is the cosmological constant, $T_{ik}$ is the stress-energy tensor, $\pi$ is pi, $c$ is the speed of light and $G$ is the gravitational constant which also occurs in Newton's law of gravity.
	wikinfo.org /wiki.php?title=General_relativity (7204 words)

	Ricci curvature
		The Ricci curvature is proportional to the metric tensor in this case.
		Ricci curvature is also used in Ricci flow, where a metric is deformed in the direction of the Ricci curvature.
		Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
	publicliterature.org /en/wikipedia/r/ri/ricci_curvature_1.html (571 words)

	PlanetMath: Ricci tensor (Site not responding. Last check: 2007-10-30)
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		The Einstein field equations assert that the energy-momentum tensor is proportional to the Einstein tensor.
		This is version 6 of Ricci tensor, born on 2005-02-16, modified 2006-09-07.
	planetmath.org /encyclopedia/RicciTensor.html (372 words)

	Tensor Information - tensor lamps
		Physicists and engineers are among the first to recognise that vectors and tensors have tensor lamps a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated.
		A tensor may be expressed as the sequence of values represented by a function with a vector valued domain and a scalar valued range.
		Examples of physical tensors are the tensor analysis tensor fasciae latae energy-momentum tensor, the inertia tensor and the polarization tensor.
	www.inanot.com /Ina-Electronics_Topics_T-/Tensor.html (1961 words)

	Wikinfo \| Tensor
		In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independent of any chosen frame of reference.
		The tensor calculus achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915, which is formulated completely in the language of tensors.
		Note that the word "tensor" is often used as a shorthand for tensor field, which a tensor value defined at every point in a manifold.
	www.wikinfo.org /wiki.php?title=Tensor (1140 words)

	physics - Ricci curvature
		Furthermore, the Ricci tensor on a Riemannian manifold is symmetric in its arguments
		Ricci curvature can be also explained in terms of the sectional curvature in the following way: for a unit vector v, Ric(v,v) is sum of the sectional curvatures of all the planes spanned by the vector v and a vector from an orthonormal frame containing v (there are n−1 such planes).
		An explicit expression for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols.
	www.physicsdaily.com /physics/Ricci_curvature_tensor (640 words)

	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.
		Note that the Gauss curvature coincides with the sectional curvature of the surface.
	en.wikipedia.org /wiki/Curvature_tensor (399 words)

	Ricci curvature - Wikipedia, the free encyclopedia
		Roughly speaking, the Ricci tensor is a measure of volume distortion; that is, it encapsulates the degree to which n-dimensional volumes of regions in the given n-dimensional manifold differ from the volumes of comparable regions in Euclidean n-space.
		Because the Levi-Civita connection is torsion-free, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that
		Ricci curvature also appears in the Ricci flow equation, where a time-dependent Riemannian metric is deformed in the direction of minus its Ricci curvature.
	en.wikipedia.org /wiki/Ricci_tensor (1163 words)

	Curved Space Geometry
		Ricci tensor itself is diagonally symmetric, so in 4D time-space it has 10 independent components out of its 16 components.
		Weyl tensor has the same form as Riemann tensor but in addition to all the component dependencies in Riemann tensor all the components of its Ricci tensor are zeros.
		Ricci tensor can describe everything about 3D curvature and all the 81 elements of the Riemann tensor can be calculated from the Ricci tensor.
	www.rafimoor.com /english/GRE2.htm (3550 words)

	curvature@Everything2.com
		In geometry, the curvature of a surface at a point is the value of the second derivative of the function describing the surface, at that point.
		Curvature for a surface, then, might be defined as the extent to which curves within the surface are forced to accelerate, simply due to the fact that they are restricted to the surface.
		Curvature over a principal bundle gives the basis for a classical theory of everything, in which the curvature F is equated to the force field strength, and A is the gauge potential.
	www.everything2.com /index.pl?node_id=39386 (7748 words)

	How does matter couple to space-time so that space-time becomes curved?
		Furthermore, the curvature of space-time at each event is completely described by a multilinear operator (a generalization of a linear operator) called the Riemann curvature tensor, which has 20 algebraically independent components at each event.
		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).
		Thus, the Ricci curvature is directly coupled to the immediate presence of matter at a given event.
	www.physlink.com /Education/AskExperts/ae98.cfm (689 words)

	Demystifying Einstein’s Field Equations
		Ricci Tensor in the field equation defines the deviation of the n-dimensional volume of the space in a curved space-time from the flat Euclidean space.
		Ricci tensor defines this amount of deviation in terms of volume in a curved space from that of flat space.
		Ricci Scalar is the result of the contraction of Ricci tensor and Metric tensor.
	www.hitxp.com /phy/rel/gr/210906.htm (2858 words)

	S on Flickr - Photo Sharing!
		The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.
		Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature.
		Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface, whereas mean curvature and the Weingarten map are not detectable to someone who can't study the three-dimensional space surrounding the surface on which he resides.
	www.flickr.com /photos/27153340@N00/495243436 (420 words)

	PhysOrgForum Science, Physics and Technology Discussion Forums -> Color Charges Curve Space
		R is the Ricci scalar (the tensor contraction of the Ricci tensor)
		The EFE is understood to be an equation for the metric tensor gμν (given a specified distribution of matter and energy in the form of a stress-energy tensor).
		This is because both the Ricci tensor and Ricci scalar depend on the metric in a complicated nonlinear manner.
	forum.physorg.com /index.php?showtopic=6600&view=getlastpost (1612 words)

	Riemann Tensor -- from Wolfram MathWorld
		Other important general relativistic tensors such that the Ricci curvature tensor and curvature scalar can be defined in terms of
		The Riemann tensor is in some sense the only tensor that can be constructed from the metric tensor and its first and second derivatives,
		Petrov Notation, Ricci Curvature Tensor, Riemannian Geometry, Riemannian Metric, Weyl Tensor.
	mathworld.wolfram.com /RiemannTensor.html (297 words)

	The Ricci Tensor
		The Ricci tensor arises as a trace of the Riemann tensor which, in turn, is obtained by taking the second derivatives of tensor fields and antisymmetrising over the gradient-rank factors added to the tensor fields' ranks in so doing.
		Now, it so happens that the construction, starting with a differential operator and yielding its Ricci tensor, can actually be applied to a broad class of tensor operators, which I describe as Leibniz operators.
		By construction, this tensor is antisymmetric under interchange of its first two components: this slightly reduces the collection of options to be tried.
	utter.chaos.org.uk /~eddy/math/smooth/ricci.html (723 words)

	The Field Equation and More
		To make the equation look simpler, it is not the Ricci tensor itself but a new tensor that is called Einstein tensor.
		The stress-energy tensor on the right side of the equation holds all the information about the distribution of energy and mass in space-time, but Einstein tensor on the left, has only information about Ricci curvature which is only part of the picture.
		For calculation of Weyl curvature we need some other equations based only on the geometry and also knowledge about the curvature in the neighborhood of the points that cause this curvature.
	www.rafimoor.com /english/GRE3.htm (2245 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)

	The Weyl Tensor
		Combining this result with the previous symmetries, it then follows that the Weyl tensor is trace-free, in other words, it vanishes for any pair of contracted indices. One can think of the Weyl tensor as that part of the curvature tensor for which all contradictions vanish.
		In General Relativity the source of the Ricci tensor is the energy-momentum of the local matter distribution.
		However the spacetime is not necessarily flat in this case since the Weyl tensor contributes curvature to the Riemann curvature tensor and so the gravitational field is not zero in spacetime void situations.
	io.uwinnipeg.ca /~vincent/4500.6-001/Cosmology/WeylTensor.htm (482 words)

	Search Results for tensor* (Site not responding. Last check: 2007-10-30)
		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.
		Struik decided to change to the topic he was studying with Schouten, tensor analysis, for his doctoral thesis and he presented his dissertation on applications of tensor methods to Riemannian manifolds in 1922.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=tensor*&CONTEXT=1 (2237 words)

	The Field Equations
		Of course, a tensor of rank four can be contracted in six different ways (the number of ways of choosing two of the four indices), and in general this gives six distinct tensors of rank two.
		In tensor calculus the divergence generalizes to the covariant derivative, so we expect that the covariant derivatives of the metrical field equations must identically vanish. The Ricci tensor R
		Noting that partial differentiation is commutative, and the metric tensor is symmetrical, we see that the sum of these three tensors vanishes at the origin of Riemann normal coordinates, and therefore with respect to all coordinates.
	www.mathpages.com /rr/s5-08/5-08.htm (1713 words)

	Riemannian Geometry and Tensor Calculus @ Mathematica
		Each tensor is stored as a nested list (of its components) under an appropriate global name.
		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.
		In addition, it introduces three new functions for calculating Lie derivatives and the Laplacian of tensors as well as the norm of the gradient of scalars.
	www.inp.demokritos.gr /~sbonano/RGTC/RiemannTensorCalculus.html (419 words)

	Ricci Flow, 3-Manifolds and Geometry — Program
		The topics include: Analysis of large curvature part of Ricci flow solutions, Ricci flow with surgery, basic properties of solutions with surgery, long time behavior of solutions with surgery, applications to geometrization.
		Hamilton's 3-manifolds with positive Ricci curvature theorem: background and basic techniques used in its proof - linearization of the Ricci tensor, short time existence, basic evolution equations, maximum principles, curvature pinching estimates, convergence criteria.
		Hamilton's trace Harnack estimate for the Ricci flow on surfaces and its consequences
	www.claymath.org /programs/summer_school/2005/program.php (535 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		Generally, arguments of the commands are mainly a metric tensor metrictensor[x1,...,xn] that must be a symmetric n x n -matrix of functions of n variables.
		Not structured so easy because of difficulties with the format of input data of covariantDerivative: n variables x1,...,xn versus vector {x1,...,xn} (only one variable)." curvatureTensor::usage="curvatureTensor[metrictensor][point] is the 4-dimensional matrix of all entries of the curvature tensor of a local Riemannian metric 'metrictensor' on R^n given in terms of functions of the coordinate vector 'point'.
		The first index i is the covariant one.\n Example:\n cc=curvatureTensorWithIndizes[{{ee[#1,#2],ff[#1,#2]},{ff[#1,#2],gg[#1,#2]}}&][{x,y}];\n Select[cc,#[[1]]=={1,2,1,2}&]\n cc1=Transpose[cc][[2]];\n cc2=cc1//FullSimplify//Union;\n cc1/cc2[[2]]//FullSimplify\n (giving the Gaussian curvature of an arbitrary surface with 1-st fundamental form {{E,F},{F,G}}."; curvatureOperator::usage="curvatureOperator[metrictensor][xx,yy][zz][point] is the curvature tensor R(X,Y)Z of the Levi-Civita connection of a local Riemannian metric on R^n given by a symmetric matrix 'metrictensor'.
	www.mathematik.hu-berlin.de /~gollek/seminarSS2002/riem.m (893 words)

	Unifying Gravity and EM - Page 3 (Site not responding. Last check: 2007-10-30)
		Consistent with this, in my initial proposal I made a point to say the antisymmetric tensor is represented by a spin 1 field, and the symmetric tensor is represented by a spin 2 field.
		The next question is to ask: "What tensor can be formed out the the Christoffel symbol?" The correct answer provided in GR books is the rank 4 Riemann curvature tensor.
		A problem with the Ricci tensor is that its divergence is not zero, a problem for energy conservation.
	www.physicsforums.com /showthread.php?p=784181 (4455 words)

	Riemannian Geometry & Tensor Calculus -- from Mathematica Information Center
		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)

	Archive of Astronomy Questions and Answers
		The Weyl tensor is that part of the curvature that is NOT determined by the local distribution of matter and energy in space-time.
		This means that the Weyl 'conformal' tensor, the Ricci Tensor and the Metric tensor all contribute to the full specification of these cosmologies with the Weyl Tensor accounting for the so-called 'trace-free' components, and the metric and Ricci tensors accounting for the 'non-zero trace' portion of Riemann curvature.
		This means that although the Weyl Tensor may have had a value close to zero at the Big Bang singularity, the merging of numerous fl holes at the future 'Big Crunch' singularity will almost certainly mean that the Weyl tensor will not vanish at this future time.
	www.astronomycafe.net /qadir/q1474.html (805 words)

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