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Topic: Einstein manifold

Related Topics

Ricci curvature tensor

Sectional curvature

Curvature of Riemannian manifolds

Metric tensor

Tangent bundle

Ricci flow

Tensor

Vector space

Campeche state

Local coordinates

Hyperbolic space

Endomorphism

Torsion

Chaos theory

Hyperbolic geometry

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	Einstein manifold - Wikipedia, the free encyclopedia
		An Einstein manifold is a Riemannian manifold (M,g) whose Ricci tensor is proportional to the metric tensor:
		Einstein manifolds with k = 0 are also called Ricci-flat manifolds.
		In general relativity, these manifolds (in the pseudo-Riemannian case) can be thought of as vacuum solutions of Einstein's equations with a cosmological constant proportional to k.
	en.wikipedia.org /wiki/Einsteinian_manifold (127 words)

	Archive of Astronomy Questions and Answers
		Einstein's minimalist adoption of "g-mu-nu" as the embodiment of the gravitational field was significant and has far-reaching ramifications.
		Einstein's appropriation of the metric tensor so that it also represented the gravitational field led to an inevitable, logical conclusion: If you took away the gravitational field, this meant that "g-mu-nu" would be everywhere and for all time equal to zero, but so too would the metric for spacetime.
		Einstein's own interpretation of the reality of the points in the spacetime manifold is best expressed in his own book Relativity: The Special and the general theory written in 1952 a few years before his death.
	einstein.stanford.edu /content/relativity/q2442.html (2386 words)

	The Hole Argument
		Appealing as manifold substantivalism once was, after one sees what trouble it causes, it becomes tempting to insist that the manifold of events lacks properties essential to spacetime.
		For the manifold substantivalist, this must be a matter of objective physical fact: either the galaxy passes through E or not.
		Einstein's quest came to a happy close in late November 1915 with the completion of his theory in generally covariant form.
	plato.stanford.edu /entries/spacetime-holearg (5340 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Einstein Algebras and Interpretations of General Relativity In this section, I would like to consider what possible interpretations of Einstein algebras could be like, and how they might differ, in particular, from interpretations of GR in the tensor formalism.
		The fundamental object of the Heller/Sasin theory is a non-commutative Einstein C-algebra E. Restricting E to an appropriate commutative subalgebra produces a (Geroch) Einstein* algebra isomorphic to a TADM model of general relativity.
		The diffeomorphism redundancy of (Geroch) Einstein algebras is seen to be a property of the commutative restriction of E, and not a property of E itself.
	philsci-archive.pitt.edu /archive/00001052/00/Bain.doc (4252 words)

	[No title]
		We extend it to variations of the metric in a Riemannian Einstein manifold with boundary, and apply it to Einstein cone-manifolds, to isometric deformations of Euclidean hypersurfaces, and to the rigidity of Ricci-flat manifolds with umbilic boundaries.
		Throughout this paper, $M$ is an Einstein manifold of dimension $m+1\geq 3$, and $D$ is its Levi-Civita connection.
		If $g$ is an Einstein metric, we say that a 2-tensor $h$ is an ``Einstein variation'' of $g$ if the associated variation of the metric induces a variation of the Ricci tensor which is proportional to $h$, so that $g+\epsilon h$ remains, to the first order, an Einstein manifold with constant scalar curvature.
	www.math.psu.edu /era-mirror/1999-01-003/1999-01-003.tex.html (2523 words)

	RESEARCH in DIFFERENTIAL GEOMETRY
		Kähler manifolds (a class of complex manifolds which includes algebraic manifolds) possess a differential 2-form, w, which is non-degenerate (i.e., wÙw is never 0), and closed (i.e., dw = 0).
		The Einstein condition is fundamental to general relativity.
		), is a differentiable invariant of the manifold.
	facpub.stjohns.edu /~watsonw/diffgeom.htm (1548 words)

	Steve Carlip - Einstein manifolds, spacetime foam, and the cosmological constant
		Einstein manifolds, spacetime foam, and the cosmological constant
		Contrary to this expectation, I show that the number of hyperbolic manifolds grows so fast with volume that the sum over topologies is dominated by arbitrarily large manifolds with arbitrarily complicated topologies.
		This phenomenon indicates an instability in the sum over topologies that may be relevant to the ``cosmological constant problem,'' the observation that the vacuum energy density of the universe is at least 120 orders of magnitude smaller than one would expect from quantum fluctuations at the Planck scale.
	camel.math.ca /CMS/Events/summer98/s98-abs/node83.html (205 words)

	Programa
		I shall revew the basic definitions of a Riemannian geometry on the space of paths of a manifold, which is endowed with the probability measure of the Brownian motion of the underlying manifold.
		We give a reduction procedure to determine (locally) the surfaces with constant Gauss curvature in a three-dimensional manifold which are invariant under the action of a one parameter subgroup of the isometry group of the ambient space.
		On the first jet manifold of a fibration with total space a Poisson manifold and base space the real line, we can define an affine Jacobi structure and a Lie affgebroid structure which are compatible in a certain sense.
	www.um.es /wgp2004/es/programa_es.html (4763 words)

	Global Interpretations of Local Experience
		Indeed, Einstein even hoped that the exclusion of singularities might (somehow) lead to an understanding of atomistic and quantum phenomena within the context of a continuum theory, although he acknowledged that he couldn't say how this might come about.
		Therefore, a great deal of classical general relativity and its treatment of fl holes, etc., is based on the acceptance of singularities in the manifold, although this is often accompanied with a caveat to the effect that in the vicinity of a singularity the classical field equations may give way to quantum effects.
		It is therefore not surprising that Einstein and his successors have regarded the effects of a gravitational field as producing a change in the geometry of space and time.
	www.mathpages.com /rr/s7-08/7-08.htm (4897 words)

	The Spectrum Of An Asymptotically Hyperbolic Einstein Manifold - Lee (ResearchIndex)
		This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its "ideal boundary" at infinity.
		J.M. Lee, The spectrum of an asymptotically hyperbolic Einstein manifold.
		3 Einstein metrics with prescribed conformal infinity on the b..
	citeseer.ist.psu.edu /lee95spectrum.html (452 words)

	Citebase - The mass of asymptotically hyperbolic Riemannian manifolds
		In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space.
		A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin.
		In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and with nonempty boundary.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:math/0110035 (1657 words)

	Analysis-Geometry Seminar
		We restrict our attention to the case of Riemannian solvmanifolds, since every known example of a homogeneous Einstein manifold of negative Ricci curvature is a solvmanifold, that is, a solvable Lie group with a left-invariant metric.
		Complementing Heber's existence results, we give an explicit description of a continuous family of Einstein manifolds with a positive dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature.
		We construct two hyperbolic manifolds which are isoscattering, and which have the same (conformally equivalent) boundaries at infinity, but which are nonetheless not isometric.
	www.math.neu.edu /~mcowen/AGSeminar99-00.html (2096 words)

	Nearly-Kaehler (Site not responding. Last check: 2007-10-08)
		These manifolds are also of interests in theoretical physics, as they are carrying a natural Hermitian metric connection with totally skew-symetric, parallel torsion.
		It should be noted that in six dimensions, nearly Kähler manifolds toghether with their canonical Hermitian connection give simple solutions to the string equations with constant dilation function.
		It describes a holonomic condition ensuring that a strict nearly Kähler manifold be a twistor space under a quaternionicKähler manifold of positive scalar curvature.
	www.mathematik.hu-berlin.de /~nagy/nkresz.html (663 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Affine manifolds, connection coefficients, covariant derivatives of vector, 1-form and tensor fields, parallel propagation along a differentiable curve, intrinsic derivatives, Geodesics.
		Physical foundations of G.R., space-time as a Riemannian manifold, Einstein's field equations.
		The linearized Einstein field equations, Newton's theory as a first approximation, further (exact) analogies between Newton's and Einstein's theories.
	www.maths.tcd.ie /pub/official/Courses/442in9899.html (263 words)

	On A Harmonic Property Of The Einstein Manifold Curvature (ResearchIndex)
		On A Harmonic Property Of The Einstein Manifold Curvature (1995)
		Abstract: The harmonicity condition of the curvature 2-form of a pseudoRiemannian manifold is formulated on the basis of annulment of this form by the de Rham-Lichnerowicz Laplacian.
		It is well known [1] that on the metric space-time manifold M the Maxwell equations can be expressed as dF = 0 ; ffiF = 0 ; (1) where F is electromagnatic field strength 2-form and d, ffi are the operations of exterior differentiation...
	citeseer.ist.psu.edu /babourova95harmonic.html (241 words)

	Ian's home page
		Thus, to try to prove geometrization, one could flow the metric on the manifold proportional to the Ricci curvature, and hope that the flow converges to a fixed point (up to scaling), which would be an Einstein metric.
		an Einstein manifold, but it collapses along a characteristic submanifold, and the uncollapsed part is hyperbolic, while the collapsed part has one of Thurston's other geometries.
		Then he could analyze the singularities, and when a neck pinch occured, he could cut it off and cap off the spheres by balls, continuing the flow on the new manifold, which was similar again to what he did in the case of 4-manifolds with positive isotropic curvature.
	www2.math.uic.edu /~agol/blog/030226.html (1292 words)

	Albert Einstein: Religion and Science
		Everything that the human race has done and thought is concerned with the satisfaction of deeply felt needs and the assuagement of pain.
		It is in this striving after the rational unification of the manifold that it encounters its greatest successes, even though it is precisely this attempt which causes it to run the greatest risk of falling a prey to illusions.
		But whoever has undergone the intense experience of successful advances made in this domain is moved by profound reverence for the rationality made manifest in existence.
	www.sacred-texts.com /aor/einstein/einsci.htm (5273 words)

	Course Title:
		This semester we will continue with the construction of harmonic maps and minimal surfaces in Riemannian manifolds for the first part of the semester.
		We will then study the conformally compact manifolds, covering some work of Graham-Fefferman, Anderson and Qing.
		The question of existence and uniqueness of the conformally compact Einstein manifold given the conformal structure of the boundary will be discussed.
	www.math.princeton.edu /graduate/Spring03courses/MAT552.html (116 words)

	AMCA: On Compact 7-dimensional Einstein Homogeneous Manifolds by Yu.G. Nikonorov (Site not responding. Last check: 2007-10-08)
		The complete classification of 5-dimensional compact homogeneous spaces with invariant Einstein metric was obtained by D.Alekseevsky, Isabel Dotti and C. Ferraris [2].
		The classification of compact homogeneous Einstein manifold of dimension 6 was found by Eu.Rodionov and Yu.Nikonorov [3].
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/d/w/07.htm (187 words)

	Riemannian Geometry (Site not responding. Last check: 2007-10-08)
		Riemannian geometry is designed to describe the universe of creatures who live on a curved surface and who are unaware of space outside and can only measure distances and areas on the surface.
		250B is about manifolds with a notion of a distance, how this can be used to define curvatures and how the curvatures can characterize a manifold.
		Einstein realized that this theory could be used to describe how space curves under the influence of gravity which lead to the general theory of relativity.
	math.ucsd.edu /~lindblad/250b/250b.html (290 words)

	CPT Lunchtime Seminars (Site not responding. Last check: 2007-10-08)
		Generalised fl holes have a horizon given by an arbitrary Einstein manifold.
		In Anti-de Sitter space, these fl holes are dual to gauge theory on a curved background given by the same Einstein manifold.
		I will argue that the dual thermal field theory effect is a novel phase transition induced by inhomogeneous Casimir pressures and characterised by a "condensation of pressure".
	fourier.dur.ac.uk /php/seminars.php3?series=MCPT&date=324&year=2004&identifier=2989 (132 words)

	Re: Kaluza-Klein theories revisited
		This is nice because the ultimate goal of Kaluza-Klein is to get Everything from Einstein's equations, and we want the curled up space to be Einstein so we can get it from a self-consistant cosmology.
		I've tried for the last few days now to write down what the metric actually is for this space, and confirm that it is Einstein, but I've failed.
		I know the mathematicians are going to want to tell me all about Hopf fibrations, but, geeze, all I want is to be able to write down the metric explicitly for S^3 x S^5 / S^1, and know how to get it.
	www.lns.cornell.edu /spr/1999-09/msg0018051.html (716 words)

	No Title
		be isometries of a complete, connected Riemannian manifold M.
		#2 (Theorem 4.6, Boothby) Let M be a compact manifold.
		Use partition of unit to show that M can be embedded into some Euclidean space.
	www.math.ucsb.edu /~wei/teach/240/hwb (362 words)

	Geometry Topology Seminar
		Einstein metrics on odd dimensional spheres, including exotic spheres.
		Abstract: A Hermitian surface is a two-dimensional complex manifold endowed
		Kahler manifolds with constant eigenvalues of the Ricci tensor
	www.fiu.edu /~lenesst/GTop/GTop.htm (961 words)

	Theoretical Elementary Particle Physics - Chicago
		The theoretical underpinnings of the subject are finally taking shape, due to the recent discovery of nonperturbative dualities between different descriptions of the theory.
		The spacetime manifold on which particles and waves propagate is being replaced by a more "stringy" notion of geometry at short distances and/or in strong fields.
		One issue of interest to me is how string theory incorporates fl holes as quantum states.
	physics.uchicago.edu /t_part.html (3097 words)

	Topics: E
		Effective Action of a Group on a Manifold > see group action.
		Einstein Manifold / Metric / Space > see types of spacetimes.
		, constructed from the riemann tensor, appearing in the lhs of the einstein equation.
	www.phy.olemiss.edu /~luca/Topics/e.html (1171 words)

	Transactions of the American Mathematical Society (Site not responding. Last check: 2007-10-08)
		-curvatures are the curvatures of generalized Einstein tensors, and for
		Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension
		, and it is nonpositive for a conformally flat manifold with zero scalar curvature.
	80-www.ams.org.library.uor.edu /tran/2005-357-10/S0002-9947-05-04001-8/home.html (270 words)

	abstract math/0009235 (Site not responding. Last check: 2007-10-08)
		We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant.
		We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial.
		Our proof is based on a Bochner type argument on harmonic maps.
	www.math.umn.edu /~leung/Papers/HarmonicMapConfCpt/HarmonicMapConfCpt.html (48 words)

	Citebase - Einstein manifolds and conformal field theories
		Authors: Gubser, Steven S. In light of the AdS/CFT correspondence, it is natural to try to define a conformal field theory in a large N, strong coupling limit via a supergravity compactification on the product of an Einstein manifold and anti-de Sitter space.
		The central charge and a part of the chiral spectrum are calculated, respectively, from the volume of T
		admits any supersymmetry: it is this manifold which characterizes the supergravity solution corresponding to a large number of D3-branes at a conifold singularity, discussed recently in hep-th/9807080.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-th/9807164 (1648 words)

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