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Topic: Curvature tensor

	[No title] (Site not responding. Last check: 2007-11-07)
		Later, the curvature tensor was understood in terms of connectionss.
		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.informationgenius.com /encyclopedia/c/cu/curvature_tensor.html (669 words)

	Ricci curvature - Wikipedia, the free encyclopedia
		In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor.
		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.
	en.wikipedia.org /wiki/Ricci_tensor (651 words)

	PlanetMath: tensor (Site not responding. Last check: 2007-11-07)
		A tensor is the mathematical idealization of a geometric or physical quantity that may be represented relative to a given frame of reference as an array of numbers
		Some well known examples of tensors in geometry are quadratic forms, and the curvature tensor.
		Examples of physical tensors are the energy-momentum tensor, and the polarization tensor.
	planetmath.org /encyclopedia/Tensor.html (852 words)

	Scalar curvature - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold.
		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).
	en.wikipedia.org /wiki/Scalar_curvature (239 words)

	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 is given in terms of a Levi-Civita connection (more generally, an affine connection)
		the curvature tensor measures anticommutativity of the covariant derivative.
	en.wikipedia.org /wiki/Curvature_tensor (303 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)

	Geodesics and Curvature (Site not responding. Last check: 2007-11-07)
		In General Relativity, gravity is seen as a distortion of spacetime, which is referred to as a curvature (because of its similarities to a curved 2-dimensional surface in 3-dimensional space).
		One representation of curvature is the Metric Tensor,
		and is the four-dimensional equivalent of the curvature of a two-dimensional surface.
	physics.syr.edu /courses/PHY312.03Spring/keish-walter/project.htm (233 words)

	Curvature : Curvature tensor (Site not responding. Last check: 2007-11-07)
		Curvature is the amount by which a curve, surface, or other manifold deviates from a straight line or (hyper)plane.
		For a plane curve C, the curvature at a given point P has magnitude equal to the reciprocal of the radius of an osculating[?] circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
		The magnitude of curvature at points on physical curves can be measured in diopters (alternative spelling: dioptre); a diopter is one per meter.
	www.eurofreehost.com /cu/Curvature_tensor.html (348 words)

	General relativity Article, Generalrelativity Information (Site not responding. Last check: 2007-11-07)
		Curvature can be measured entirely within a surface, andsimilarly within a higher-dimensional manifold such as space or spacetime.
		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 stressenergy tensor at that point; the latter tensor being a measure of the density of matterand energy.
		General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matterand 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.
	www.anoca.org /theory/space/general_relativity.html (2378 words)

	Estimating The Tensor Of Curvature Of A Surface From A Polyhedral Approximation - Taubin (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Estimating The Tensor Of Curvature Of A Surface From A Polyhedral Approximation (1995)
		In this paper we describe a method to estimate the tensor of curvature of a surface at the vertices of a polyhedral approximation.
		Estimating the tensor of curvature of a surface from a polyhedral approximation.
	citeseer.ist.psu.edu /taubin95estimating.html (515 words)

	Ricci curvature (Site not responding. Last check: 2007-11-07)
		In differential geometry, Ricci curvature is 2-valent tensor, obtained as a trace of the full curvature tensor.
		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).
		Bishop-Gromov inequality states that if Ricci curvature of a complete m-dimensional Riemannian manifold is ≥0 then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Eulidean m-space.
	www.sciencedaily.com /encyclopedia/ricci_curvature_1 (561 words)

	Clearing up the market cycle... best Riemann Curvature Tensor (Site not responding. Last check: 2007-11-07)
		The continuous determination of spacetime geometry by the Riemann curvature tensor The continuous determination of spacetime geometry by the Riemann curvature tensor It is shown that generically the Riemann tensor of a Lorentz (or positive...
		The spectral geometry of the Riemann curvature tensor
		The spectral geometry of the Riemann curvature tensor The spectral geometry of the Riemann curvature tensor Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold.
	ascot.pl /th/Fourier5/Riemann-Curvature-Tensor.htm (669 words)

	Science Forums and Debate - Relativity (Site not responding. Last check: 2007-11-07)
		This tensor does not explicitly appear in the GR field equations, although the Ricci tensor is a contraction of it.
		And vectors are first rank tensors, and if two vectors have the same direction, I don't see any problem with dividing one by the other, and so here is another case where one tensor can be divided by another.
		LHS is curvature, RHS is energy density (actually the stress energy tensor whose unit of measure is energy density) divided by a universal constant force F = c
	www.scienceforums.net /forums/printthread.php?t=9322 (3608 words)

	Maxima Manual - Tensor
		Tensor Package - will set to zero, in exp, all occurrences of the tensori that have derivative indices.
		Tensor Package - will set to zero, in exp, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.
		Tensor package) This function first computes the covariant components LR[i,j] of the Ricci tensor (LR is a mnemonic for "lower Ricci").
	www.ma.utexas.edu /maxima/maxima_27.html (1080 words)

	Curvature and Riemann Tensor (Site not responding. Last check: 2007-11-07)
		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 /%7Esuchii/Einstein/riemann.curv.html (228 words)

	The Weyl Tensor
		The Weyl tensor (or conformal tensor) is defined to be the 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.
		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)

	Curvature tensor - InfoSearchPoint.com (Site not responding. Last check: 2007-11-07)
		Later, the curvature tensor was understood in terms of connections.
		Connections, parallel transport and curvature form the so-called golden triangle of Riemannian geometry.
		In the Cartan formalism, the curvature is given as a matrix $\Omega$ of 2-forms.
	www.infosearchpoint.com /display/Riemann_tensor (744 words)

	IBM Research \| Technical Paper Search \| Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation
		Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by iso-surface construction algorithms, has become a basic step in many computer vision algorithms.
		The tensor of curvature of a surface is a function that assigns a quadratic form to each point on the surface, and can be described by a 3 x 3 symmetric matrix for each surface point.
		The principal curvatures and principal directions of the surface at a point are eigenvalues and eigenvectors of the matrix associated with the point.
	domino.research.ibm.com /library/cyberdig.nsf/3addb4b88e7a231f85256b3600727773/e2fbf6f6180e2b43852565930064abc1 (303 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/old/outline2.html (3152 words)

	The curvature tensor and geodesic deviation (Site not responding. Last check: 2007-11-07)
		It is important to distinguish two different kinds of curvature: intrinsic and extrinsic .
		the curvature it has in relation to the flat three- dimensional space it is part of.
		It is clear therefore that when we talk of the curvature of spacetime, we talk of its intrinsic curvature since the worldlines [ geodesics ] of particles are confined to remain in spacetime.
	www.mth.uct.ac.za /omei/gr/chap6/node8.html (442 words)

	Math "Newb" Wants to know what a Tensor is - Page 6 - Physics Help and Math Help - Physics Forums
		Since the curvature is a 2-form, Stokes' theorem equates the integral of curvature over the area of the film to net rotation of the basis as it traverses the loop, regardless of the shape of the film.
		Since the curvature tensor is a (1,1) tensor valued 2 form, it assigns an endomorphism of the tangent space to an oriented pair of orthonormal unit vectors.
		Well since the gaussian curvature is a scalar, and can be any scalar, it cannot be represented by a rotation matrix since a rotation matrix is essentially a point on the circle, while the gaussian curvature is essentially a point on the real line.
	www.physicsforums.com /showthread.php?p=387819 (5192 words)

	[No title]
		A 4 tensor R defined at a point P of Mm is an algebraic curvature tensor if the identities of equation (0.1) hold at P.
		Note that if R is an algebraic curvature tensor, then there exists a metric "ge* xtending the metric on TP Mm so that R is the curvature* tensor of "gat P.
		The non-zero curvatures are R(ea; eb)eb = Cea and R(ea; eb)ea = -Ceb fora
	hopf.math.purdue.edu /Gilkey-Leahy-Sadofsky/GLSeigen.txt (3994 words)

	Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation (Site not responding. Last check: 2007-11-07)
		Estimating principal curvatures and directions of a surface from a polyhedral approximation with a large number of faces has become a basic step in many computer vision algorithms, particularly those targeted at medical applications.
		The author wishes to design an algorithm for accurately and efficiently estimating the principal curvature and direction at each point of an underlying unknown smooth surface from a polyhedral approximation.
		This paper presents a simple new efficient algorithm for estimating the principal curvature and principal direction at each vertex of a polyhedral approximation of a smooth surface.
	www.stanford.edu /~ctj/liteseer/segparmf/taubin94tensor/taubin94tensor.html (219 words)

	General Relativity and Spacetime (Site not responding. Last check: 2007-11-07)
		We can imagine a sphere of a given radius existing on it's own (where flat creatures could walk around), but because we live in 3D space, it is easier to think of a sphere embedded in three dimensions.
		Gauss' curvature K is an invariant geometrical property of any surface, because it does not depend on the choice of coordinate system (as seen with the case with the plane).
		Actually this is the only independent function out of 16 in the curvature tensor.
	www.astro.ku.dk /~cramer/RelViz/text/geom_web/node2.html (900 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		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.
		The Ricci tensor is a classical example of a contraction of the curvature tensor; a sum over an upper and lower index of a tensor of type \begin_inset Formula \((r,s) \) \end_inset invariantly giving a tensor of type \begin_inset Formula \((r-1,s-1).
		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.
	www.lehigh.edu /dlj0/Desktop/dlj0/yesterday/courses/424sp97-5.lyx (4015 words)

	Kerr Black Holes, Ch.5. Petrov Types (Site not responding. Last check: 2007-11-07)
		The type of a spacetime gives considerable information, not merely about the character of its curvature, but (as we shall see) also about other geometric invariants, notably null geodesics.
		In particular, for type D spacetimes such as Kerr's, these congruences are not only geodesic but also "shearfree"--roughly speaking, if a beam of such light initially has circular cross-section, then it keeps this property as it propagates.
		A final perspective on Kerr spacetime is provided by the Goldberg-Sachs theorem, which for arbitrary Ricci-flat spacetimes gives necessary and sufficient curvature conditions for the existence of shear-free geodesic null congruences.
	www.math.ucla.edu /~bon/kerr/intro5.html (336 words)

	[No title]
		Recall that the conformal scalar curvature of a Hermitian surface (M; J; g) is the scalar curvature with respect to g of the canonical Weyl structure associated with the Hermitian structure (g; J) (cf.
		Let r, R, æ and s be the Levi{Civita connection, the curvature tensor, the Ricci tensor, and the scalar curvature of the Riemannian metric g, respectively.
		It is well known that k is the scalar curvature with respect to g of the canonical Weyl structure associated with the Hermitian structure (g; J) (cf.
	www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/test/file147.html (1523 words)

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