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Topic: Riemann curvature tensor

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	Curvature and Riemann Tensor (Site not responding. Last check: 2007-11-01)
		In a nutshell, the tensor G Eddington mentioned (sometimes called "Einstein tensor") is a sort of average of the Riemann curvature over all directions.
		Thus Riemann curvature is the basic notion for expressing gravitational fields; and although the expression of Riemann curvature tensor is different depending on our choice of a coordinate system, this curvature is an invariant quantity.
		A sphere has a definite (positive) curvature, and it is the same whatever coordinate system you may choose, and likewise an Euclidean plane is flat (zero curvature), independent of any coordinate system.
	www.bun.kyoto-u.ac.jp /~suchii/Einstein/riemann.curv.html (228 words)

	Curvature tensor
		In differential geometry, the curvature tensor is one of the most important notions; it generalizes Gauss curvature to higher dimensions.
		In two dimensions, the curvature tensor is determined by the scalar curvature - which is the full trace of the curvature tensor.
		In three dimensions, the curvature tensor is specified by the Ricci curvature - which is a partial trace of the curvature tensor.
	www.guajara.com /wiki/en/wikipedia/c/cu/curvature_tensor.html (728 words)

	Curvature tensor -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-01)
		The curvature tensor is given in terms of a (additional info and facts about Levi-Civita connection) Levi-Civita connection (more generally, an (additional info and facts about affine connection) affine connection) (or (additional info and facts about covariant differentiation) covariant differentiation) by the following formula:
		Some authors define the curvature tensor with the opposite sign.
		the curvature tensor measures anticommutativity of the covariant derivative.
	www.absoluteastronomy.com /encyclopedia/c/cu/curvature_tensor.htm (234 words)

	Scalar curvature - Wikipedia, the free encyclopedia
		It is defined as the trace of the Ricci curvature tensor with respect to the metric:
		The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first contract with the metric to obtain a (1,1)-valent tensor in order take the trace (see musical isomorphisms).
		The three are distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices.
	en.wikipedia.org /wiki/Scalar_curvature (239 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)

	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).
		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).
	www.physlink.com /Education/AskExperts/ae98.cfm (689 words)

	Weyl curvature - TheBestLinks.com - Weyl tensor, Tensor, Metric tensor, Ricci curvature, ... (Site not responding. Last check: 2007-11-01)
		Weyl curvature - TheBestLinks.com - Weyl tensor, Tensor, Metric tensor, Ricci curvature,...
		Weyl tensor, Weyl curvature, Tensor, Metric tensor, Ricci curvature, Scalar...
		In other words, it a tensor that has the same symmetries as the Riemann curvature tensor with the extra condition that its Ricci curvature must vanish.
	www.thebestlinks.com /Weyl_tensor.html (318 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.
	www.fh-lueneburg.de /u1/gym03/englpage/chronik/riemann/riemann.htm (1706 words)

	Ricci curvature (Site not responding. Last check: 2007-11-01)
		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).
		Ricci curvature is also used in Ricci flow, where a metric is deformed in the direction of the Ricci curvature.
	www.worldhistory.com /wiki/R/Ricci-curvature.htm (690 words)

	PlanetMath: connection
		This property is the origin of the term ``covariant derivative'' -- the covariant derivative maps tensor fields into quantities which transform in the same manner.
		Note that the lack of tensoriality in the second argument means that a connection is not a tensor field.
		Both the torsion and the curvature are, quite evidently, anti-symmetric in their first two arguments.
	planetmath.org /encyclopedia/Curvature.html (2994 words)

	Geodesics and Curvature (Site not responding. Last check: 2007-11-01)
		One representation of curvature is the Metric Tensor,
		and is the four-dimensional equivalent of the curvature of a two-dimensional surface.
		The Einstein Tensor, therefore, is only affected by local considerations, and is unaffected by gravitational waves from distant sources (which result in a change to the Riemann Tensor).
	www.phy.syr.edu /courses/PHY312.03Spring/keish-walter/project.htm (233 words)

	ipedia.com: Scalar field Article (Site not responding. Last check: 2007-11-01)
		Tensor fields, which associate a tensor to every point in space.
		In general relativity, gravity is associated with a tensor field.
		In Kaluza-Klein_theory spacetime is extended to five dimensions and its riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".
	www.ipedia.com /scalar_field.html (253 words)

	A Comprehensive Introduction to Differential Geometry, Volume 2, 3rd Edition, Publish or Perish, Michael Spivak
		Riemann's curvature tensor is built up from an intuitive study of the second-order terms in the Taylor series expansion of the Riemannian metric.
		Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7.
		Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established.
	allentech.net /techstore/item_0914098713.html (769 words)

	The Riemann Curvature Tensor
		The fact that it is a tensor follows from the homework.
		A necessary and sufficient condition that parallel transport be independent of the path is that the curvature tensor vanishes.
		The coordinates of the covariant curvature tensor are given by
	people.hofstra.edu /faculty/Stefan_Waner/diff_geom/Sec10.html (1765 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.
		Hence when evaluating the derivatives of the Riemann tensor, even at the origin or Riemann normal coordinates, we must consider all four of the metric tensor derivatives in the above expression.
		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 (1687 words)

	Re: Conjecture on Curvature and Torsion
		Thus the Yang-Mills tensor F is really the curvature R in a general setting, in particular the Faraday's tensor for EandM and the Riemann curvature tensor R for tangental bundle of a Riemann space.
		For Einstein's theory all four indexes in the curvature tensor range from 1 to 4 so we can do "many" contractions among them, however there is only one nontrivial contraction and this is Ricci contraction R^a_{bad} which gives us the Ricci tensor (usual GR theory).
		Notice that for usual GR this is zero because of the symmetries of the Riemann tensor, and notice also that when Dim(V)=/=Dim(M) we can't do Ricci contraction.
	www.lns.cornell.edu /spr/1999-11/msg0019508.html (704 words)

	Tensor (Site not responding. Last check: 2007-11-01)
		Tensor field visualization is a challenging task due in part to the Glyphs convey tensor variables by mapping the tensor eigenvectors and
		Tensor arrays, or tensors for short 1are multidimensional arrays with two In this entry we give the definition of a tensor array and establish some
		tensor fascia lata is an excellent choice for free tissue transfer since it is Functional differentiation within the tensor fasciae latae.
	web-server.allworldsites.com /q/web-server-tensor.htm (1033 words)

	Riemann-Tensor demos (Site not responding. Last check: 2007-11-01)
		The input sequences are shown on the left and the six components of the curvature tensor in the right panel.
		Note that the middle top component (the sectional curvature in x,t) is selective to discontinuously moving straight edges with vertical orientation and to continuous motion (of the corners) along that vertical direction.
		Finally, the probability for a window to be placed is higher, if the curvature (as measured by the Riemann tensor) is higher - the scene can now be recognize (at least by some people).
	www.inb.uni-luebeck.de /~eebarth/demos/R (288 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 (1998 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		The fact that the underlying classical work is revealed to be a natural and complete description of the algebraic structure of curvature tensors shows both the computational expertise of earlier geometers and their understan ding of the underlying invariants.
		Those conditions, using the Lorenzian metric of general relativity which is not positive-definite, specify the form of the stress-energy tensor, formally the Ricci tensor of the curvature tensor of the pseudo-Riemannian metric.
		Thus, the Riemann curvature tensor, in (1,3)-form, will also be unchanged, and the (0,4) form of that tensor is only altered by the homothetic factor.
	www.lehigh.edu /dlj0/Desktop/dlj0/yesterday/courses/424sp97-5.lyx (4015 words)

	(Lorentzian) Orthonormal Frame Methods / Riemann Curvature Tensor Interpretation (Site not responding. Last check: 2007-11-01)
		NB: reviews the (orthonormal) tetrad formalism (physical compos of tensors) within GR and relates it to measurements of ``observers''; emphasises in particular the physical interpretation of the Riemann curvature tensor.
		NB: Characterises gravitational radiation in terms of Riemann tensor; assumes propagation with fundamental velocity; gravitational wave fronts appear as discontinuities in Riemann tensor across null 3-surfaces; possibility of propagation of gravitational disturbances with subluminal velocities not excluded.
		NB: Strongly emphasises the role of the Riemann curvature tensor in physical properties of (relativistic) gravitational fields.
	www.mth.uct.ac.za /~webpages/henk/ref_dir/chonf.html (398 words)

	Lecture Notes for Math497C, Fall 2004 (Site not responding. Last check: 2007-11-01)
		Gaussian curvature, Gauss map, shape operator, coefficients of the first and second fundamental forms, curvature of graphs.
		Self adjointness of the shape operator, Riemann curvature tensor of surfaces, Gauss and Codazzi Mainardi equations, and Theorema Egregium revisited.
		The definition of geodesic curvature, and the proof that it is intrinsic.
	www.math.gatech.edu /~ghomi/OldClasses/Math497C/LectureNotes.html (181 words)

	The Mathematical Details
		To see why equation (2) is equivalent to the usual formulation of Einstein's equation, we need a bit of tensor calculus.
		We can use this tensor to compute the relative acceleration of nearby particles in free fall if they are initially at rest relative to one another.
		The Ricci tensor only captures some of the information in the Riemann curvature tensor.
	math.ucr.edu /home/baez/einstein/node10.html (544 words)

	Riemann curvature tensor derivation - Physics Help and Math Help - Physics Forums
		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 (852 words)

	Properties of the Riemann curvature tensor (Site not responding. Last check: 2007-11-01)
		Since these last two equations are valid tensor equations, although they were derived in a local inertial frame, they are valid in all coordinate systems.
		An important use of the curvature tensor comes when we examine the consequences of taking two covariant derivatives of a vector field
		This is closely related to our original derivation of the Riemann tensor from parallel transport around loops, because the parallel transport problem can be thought of as computing, first the change of
	vishnu.mth.uct.ac.za /omei/gr/chap6/node10.html (301 words)

	Geometric Properties of Natural Operators Defined by the Riemann Curvature Tensor (Site not responding. Last check: 2007-11-01)
		Book Description: A central problem in differential geometry is to relate algebraic properties of the Riemann curvature tensor to the underlying geometry of the manifold.
		The full curvature tensor is in general quite difficult to deal with.
		This book presents results about the geometric consequences that follow if various natural operators defined in terms of the Riemann curvature tensor (the Jacobi operator, the skew-symmetric curvature operator, the Szabo operator, and higher order generalizations) are assumed to have constant eigenvalues or constant Jordan normal form in the appropriate domains of definition.
	isbn.nu /9810247524 (398 words)

	P.P. Cook's Tangent Space
		Penrose suggested his Weyl curvature hypothesis in 1979 as a physical origin of the increasing entropy of the universe with time.
		The Weyl curvature tensor is the traceless part of the Riemann curvature, i.e.
		The Weyl curvature hypothesis is that the Weyl curvature is zero at the big bang but rises gradually as the universe ages.
	ppcook.blogspot.com (5821 words)

	The Riemann Tensor
		This is a tensor of mixed tensor of type (1,1).
		Since the left-hand side of (5.8) is a tensor, it follows that
		Theorem: A necessary and sufficient condition for a manifold to be affine flat is that the Riemann tensor vanishes.
	scholar.uwinnipeg.ca /courses/38/4500.6-001/Cosmology/RiemannTensor.htm (332 words)

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