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	NationMaster - Encyclopedia: Ricci tensor
		Furthermore, the Ricci tensor on a Riemannian manifold is symmetric in its arguments
		Ricci curvature is also used in Ricci flow, where a metric is deformed in the direction of the Ricci curvature.
		scalar curvature, Einstein tensor, ricci scalar, Weyl tensor, Weyl curvature tensor
	www.nationmaster.com /encyclopedia/Ricci-tensor (803 words)

	Demystifying Einstein’s Field Equations
		The LHS of this equation describes the space-time geometry and the RHS describes the associated mass-energy responsible for that curvature.
		Ricci tensor defines this amount of deviation in terms of volume in a curved space from that of flat space.
		Ricci Scalar is just a number that defines the curvature of space-time.
	www.hitxp.com /phy/rel/gr/210906.htm (2858 words)

	Real Algebraic and Analytic Geometry
		Curvature bounds which play the role of Ricci, scalar curvature and Einstein tensor bounds are introduced for subanalytic topological manifolds.
		It is shown, using metric properties of subanalytic sets, that an upper (lower) bound on the sectional curvature in the sense of Alexandrov implies an upper (lower) bound on the Ricci curvature and on the Einstein tensor.
		In the same way, an upper (lower) bound on the Ricci curvature or on the Einstein tensor implies an upper (lower) bound on the scalar curvature.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG/preprints/0072.html (101 words)

	PlanetMath: Ricci tensor
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		scalar curvature, Einstein tensor, ricci scalar, Weyl tensor, Weyl curvature tensor
		This is version 6 of Ricci tensor, born on 2005-02-16, modified 2006-09-07.
	planetmath.org /encyclopedia/RicciScalar.html (319 words)

	PlanetMath: Ricci tensor
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		scalar curvature, Einstein tensor, ricci scalar, Weyl tensor, Weyl curvature tensor
		This is version 6 of Ricci tensor, born on 2005-02-16, modified 2006-09-07.
	www.planetmath.org /encyclopedia/WeylCurvatureTensor.html (319 words)

	General relativity Info - Bored Net - Boredom (Site not responding. Last check: )
		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 is the Ricci curvature tensor, is the Ricci curvature scalar, is the metric tensor, is the cosmological constant, is the stress-energy tensor, is pi, is the speed of light and is the gravitational constant which also occurs in Newton's law of gravity.
	www.borednet.com /e/n/encyclopedia/g/ge/general_relativity.html (1861 words)

	Ricci curvature Information
		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.
		Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
	www.bookrags.com /wiki/Ricci_curvature (669 words)

	Quantum Conformal Gravity, Higgs, and Spin Networks
		Sardanashvily's Clifford algebra structure should allow treatment of the interaction between spinors and gravity to be represented by curvature in a Clifford Manifold as done by William Pezzaglia in gr-qc/9710027 in his derivation of the Papapetrou Equations.
		the "extrinsic curvature" of this 3-geometry relative to the 4-geometry of the enveloping spacetime...
		"curvature of space" must (1) be a single number (a scalar) that (2) depends on the inclination...
	valdostamuseum.org /hamsmith/cnfGrHg.html (8522 words)

	Reference.com/Encyclopedia/Riemannian geometry
		If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
		The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space.
		If M is a complete Riemannian manifold with negative sectional curvature then any abelian subgroup of the fundamental group of M is isomorphic to Z.
	www.reference.com /browse/wiki/Riemannian_geometry (870 words)

	Notes on Differential Geometry by B. Csikós
		Vector fields along hypersurfaces, tangential vector fields, derivations of vector fields with respect to a tangent direction, the Weingarten map, bilinear forms, the first and second fundamental forms of a hypersurface, principal directions and principal curvatures, mean curvature and the Gaussian curvature, Euler's formula.
		Umbilical, spherical and planar points, surfaces consisting of umbilics, surfaces of revolution, Beltrami's pseudosphere, lines of curvature, parameterizations for which coordinate lines are lines of curvature, Dupin's theorem, confocal second order surfaces; ruled and developable surfaces: equivalent definitions, basic examples, relations to surfaces with K=0, structure theorem.
		Curvature operator, curvature tensor, Bianchi identities, Riemann-Christoffel tensor, symmetry properties of the Riemann-Christoffel tensor, sectional curvature, Schur's Theorem, space forms, Ricci tensor, Ricci curvature, scalar curvature, curvature tensor of a hypersurface.
	www.cs.elte.hu /geometry/csikos/dif/dif.html (588 words)

	Electronic Research Announcements
		As an application, we show that the well-known noncompact Yamabe problem (of prescribing constant positive scalar curvature) on a manifold with nonnegative Ricci curvature cannot be solved if the existing scalar curvature decays ``too fast'' and the volume of geodesic balls does not increase ``fast enough''.
		We also find some complete manifolds with positive scalar curvature, which are conformal to complete manifolds with positive constant and with zero scalar curvatures.
		R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff.
	www.ams.org /era/1997-03-06/S1079-6762-97-00022-X/home.html (639 words)

	Bulletin of the American Mathematical Society
		J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128.
		L. Gao and S.-T. Yau, The existence of negatively Ricci curved metrics on three-manifolds, Invent.
		I. Nikolaev, Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. Aleksandrov, Sibirsk.
	www.ams.org /bull/2001-38-03/S0273-0979-01-00904-1/home.html (1043 words)

	CMS/CAIMS Summer 2004 Meeting
		The spectral geometry of the Riemann curvature tensor
		Negatively curved spaces have a remarkably rich and diverse structure and are interesting from both a mathematical and a physical perspective.
		A curvature invariant of order n is a scalar obtained by contraction from a polynomial in the Riemann tensor and its covariant derivatives up to the order n.
	www.cms.math.ca /Events/summer04/abs/ait.html (1338 words)

	The Field Equations
		It strikes many people as ironic that Einstein found the principle of general covariance to be so compelling, because, strictly speaking, it's possible to express almost any physical law, including Newton's laws, in generally covariant form (i.e., as tensor equations).
		We are able to single out a more or less unique contraction of the curvature tensor only because of that tensor’s symmetries (described in Section 5.7), which imply that of the six contractions of R
		When Hubble and other astronomers began to find evidence that in fact the large-scale universe is expanding, and Einstein realized his ingenious introduction of the cosmological constant had led him away from making such a fantastic prediction, he called it "the biggest blunder of my life”.
	www.mathpages.com /rr/s5-08/5-08.htm (1713 words)

	Henk van Elst -- Abstracts
		In this thesis extensions of the $1+3$ decomposition formalism are developed, partially in fully covariant form, and partially on the basis of choice of an arbitrary Minkowskian orthonormal reference frame, the timelike direction of which is aligned with ${\bf u}/c$.
		In particular, the Bianchi identities for the Weyl curvature tensor occur in fully expanded form, as they are given a central role in the extended formalism.
		In this paper the quantum cosmological consequences of introducing a term cubic in the Ricci curvature scalar $R$ into the Einstein--Hilbert action are investigated.
	www.mth.uct.ac.za /~webpages/henk/abstr.html (2838 words)

	Comparison Geometry - Cambridge University Press
		The simple idea of comparing the geometry of an arbitrary Riemannian manifold with the geometries of constant curvature spaces has seen a tremendous evolution recently.
		A genealogy of noncompact manifolds of nonnegative curvature: history and logic R. Greene; 5.
		Construction of manifolds of positive Ricci curvature with big volume and large Betti numbers G. Perelman; 9.
	www.cambridge.org /us/catalogue/catalogue.asp?isbn=0521592224 (313 words)

	outline1.html
		The RIEMANN CURVATURE TENSOR is a tensor of rank (1,3) at each point of spacetime.
		There is a lot of information about spacetime curvature encoded in the rate at which this ball changes shape and size.
		This process, which turned one subscript on the Ricci tensor into a superscript, is called RAISING AN INDEX.
	www.math.ucr.edu /home/baez/gr/outline2.html (3217 words)

	Application to the scalar-tensor theories. (Site not responding. Last check: )
		It follows from (89) that contrary to general relativity, the scalar-tensor theories (defined by (74)) predict the existence of a first-order geometrical scintillation effect produced by gravitational waves.
		This effect is proportional to the amplitude of the scalar perturbation.
		This formula shows that the contribution of the scalar wave to the scintillation cannot be zero, whatever be the direction of observation of the distant light source.
	www.obs-hp.fr /www/preprints/pp129/node5.html (545 words)

	General_relativity
		It unifies special relativity and Isaac Newton's law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time, this curvature being produced by the mass-energy and momentum content of the spacetime.
		The relationship between stress-energy and the curvature of spacetime is governed by the Einstein field equations.
		Spacetime curvature is created by stress-energy within the spacetime: This is described in general relativity by the Einstein field equations.
	www.brainyencyclopedia.com /encyclopedia/g/ge/general_relativity.html (5098 words)

	General relativity (Site not responding. Last check: )
		Curvature can be measured entirely within a surface, and similarly within a higher-dimensional manifold such as space or spacetime.
		The special theory of relativity (1905) modified the equations used in comparing the measurements made by differently moving bodies, in view of the constant value of the speed of light, i.e.
		The Ricci tensor and scalar curvature are themselves derivable from the metric, which describes the metric of the manifold and is a symmetric 4 x 4 tensor, so it has 10 independent components.
	www.freedownloadsoft.com /info/general-relativity.html (2708 words)

	Michael Anderson - Home Page
		Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, I,
		Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds, II,
		Scalar curvature and the existence of geometric structures on 3-manifolds, I,
	www.math.sunysb.edu /~anderson/papers.html (410 words)

	Springer Online Reference Works
		Manifolds of constant Ricci curvature are called Einstein spaces.
		The Ricci tensor of an Einstein space is of the form
		The Ricci curvature can be defined by similar formulas also on pseudo-Riemannian manifolds; in this case the vector is assumed to be anisotropic.
	eom.springer.de /R/r081780.htm (211 words)

	The Citizen Scientist - Society for Amateur Scientists
		The curvature of space-time is the underlying reason that gravitational waves are possible.
		The basic idea behind the field equation is that the right hand side of the equation is the presence of matter and energy, on the left hand side there is the curvature of spacetime.
		The equation tells us that the presence of matter and energy causes the curvature of spacetime, while the curvature of spacetime causes the density of matter and energy to behave in a special way.
	www.sas.org /tcs/weeklyIssues_2005/2005-01-07/feature3 (889 words)

	Transactions of the American Mathematical Society
		In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive.
		We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.
		J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff.
	www.ams.org /tran/2003-355-05/S0002-9947-02-03242-7/home.html (822 words)

	Is The Universe Closed?
		The unboundedness of space has a greater empirical certainty than any experience of the external world, but its infinitude does not in any way follow from this; quite the contrary.
		Space would necessarily be finite if one assumed independence of bodies from position, and thus ascribed to it a constant curvature, as long as this curvature had ever so small a positive value.
		If the universe is quasi-Euclidean, and its radius of curvature therefore infinite, then r would vanish. But it is improbable that the mean density of matter in the universe is actually zero; this is our third argument against the assumption that the universe is quasi-Euclidean.
	www.mathpages.com /rr/s7-01/7-01.htm (866 words)

	Transactions of the American Mathematical Society
		M. Anderson, Scalar curvature, metric degeneration and the static vacuum Einstein equations on 3-manifolds, I.
		D. Yang, Convergence of Riemannian manifolds with integral bounds on curvature I, Ann.
		Keywords: Integral curvature bounds, maximum principle, gradient estimate, excess estimate, volume and Gromov-Hausdorff convergence.
	www.ams.org /tran/2001-353-02/S0002-9947-00-02621-0/home.html (365 words)

	[No title]
		Next: what could I say about the homotopy groups of a manifold of negative sectional curvature (by Hadamard's theorem the universal cover is R^n, so the higher homotopy groups vanish and pi_1 must be infinite).
		Dave then asked what I could say about the sectional curvature of a surface minimally embedded in a three-manifold.
		Then they asked me how many different kinds of curvature I could define (the curvature tensor, sectional curvature, Ricci curvature, scalar curvature...) There followed some discussions amongst the examiners over different definitions of Ricci and scalar curvature that appear in the literature.
	www.math.princeton.edu /graduate/generals/milley_peter (776 words)

	Course Outline for Math 240 ABC
		We intend to discuss some of the beginnings of this theory, which provides a beautiful example of topology applied to geometry through the theory of ordinary differential equations.
		In higher dimensions, many types of curvature have been studied: sectional curvature, Ricci curvature, scalar curvature.
		After discussing geodesics and curvature, the remainder of the course will depend upon student interests.
	www.math.ucsb.edu /~moore/240.html (865 words)

	Transactions of the American Mathematical Society
		Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor.
		A. Chang, M. Gursky and P. Yang, An equation of Monge-Ampère type in conformal geometry, and four manifolds of positive Ricci curvature, to appear in Ann.
		B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm.
	www.ams.org /tran/2003-355-03/S0002-9947-02-03132-X/home.html (413 words)

	Centre Emile Borel - Ricci curvature and Ricci flow
		Analytic aspects of Ricci flow and related flows
		Introduction to the Ricci flow: the work of Hamilton in dimensions 3 and 4 by Z. Djadli
		Find here the poster of the first announcement.
	www-fourier.ujf-grenoble.fr /~besson/Borel (199 words)

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