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Topic: Pseudo Riemannian manifold

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	RIEMANNIAN MANIFOLD at axiss.inepar.edu.mx!
		In Riemannian geometry, a Riemannian manifold (M,g) (with Riemannian metric g) is a real differentiable manifold M in which each tangent space is equipped...
		Pseudo-Riemannian manifolds are crucial in Physics and in particular in General Relativity where space-time is modeled as a 4-pseudo Riemannian manifold...
		Homogeneous Riemannian manifolds whose geodesics are orbits, in andquot;Topics in Geometry:...
	axiss.inepar.edu.mx /index.php?riemannian-manifold (1854 words)

	Reference.com/Encyclopedia/Pseudo-Riemannian manifold
		In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.
		Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime.
		Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space.
	www.reference.com /browse/wiki/Pseudo-Riemannian_manifold (445 words)

	Compact pseudo-Riemannian manifolds with parallel Weyl tensor \| Department of Mathematics
		A pseudo-Riemannian manifold (M,g) of dimension n > 3 which is neither conformally flat nor locally symmetric, such that the Weyl conformal tensor of g is parallel, is called an ECS manifold (short for "essentially conformally symmetric").
		The ECS manifolds shown to exist are nontrivial torus bundles over the circle, and arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.
		Namely, every compact ECS manifold has an infinite fundamental group, its Euler characteristic is zero, and its real Pontryagin classes all vanish; for any compact ECS Lorentzian manifold (M,g), some two-fold covering manifold of M is a bundle over the circle; and, finally, every four-dimensional ECS Lorentzian manifold is noncompact.
	www.math.ohio-state.edu /node/26668 (227 words)

	PlanetMath: Ricci tensor
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		In geometry, a pseudo-Riemannian manifold that satisfies this equation is called Ricci-flat.
		It is possible to prove that a manifold is Ricci flat if and only if locally, the manifold, is conformally equivalent to flat space.
	planetmath.org /encyclopedia/RicciScalar.html (319 words)

	Pseudo-Riemannian manifold - Definition, explanation
		In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric,
		The key difference between a Riemannian metric and a pseudo-Riemannian metric is that a pseudo-Riemannian metric need not be positive-definite, merely nondegenerate.
		The signature of a pseudo-Riemannian manifold is just the signature of the metric (one should insist that the signature is the same on every connected component).
	www.calsky.com /lexikon/en/txt/p/ps/pseudo_riemannian_manifold.php (348 words)

	NationMaster - Encyclopedia: Pseudo Riemannian manifold
		In physics, differentiable manifolds serve as the phase space in classical mechanics and four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity.
		A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one).
		A Kähler manifold is a manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense.
	www.nationmaster.com /encyclopedia/Pseudo-Riemannian-manifold (558 words)

	Pseudo-Riemannian manifold - Encyclopedia, History, Geography and Biography
		Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime.
		Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space.
		A pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric $(0,2)$ tensor which is nondegenerate at each point on the manifold.
	www.arikah.com /encyclopedia/Pseudo-Riemannian_manifold (492 words)

	Riemannian Manifold (Site not responding. Last check: )
		Sub-Riemannian manifold - In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold.
		Riemannian manifold - In Riemannian geometry, a Riemannian manifold (M, g) is a real differentiable manifold M in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point.
		KÃ¤hler manifold - In mathematics, a KÃ¤hler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible.
	http.the-cba.com /Riemannian-Manifold.html (1267 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		An alternate definition of a differentiable manifold is a topological space with a sheaf of functions, which is locally isomorphic to Euclidean space with the sheaf of differentiable functions.
		A Finsler manifold is a generalization of a Riemannian manifold, in which the inner product is replaced with a vector norm; this allows the definition of length, but not angle.
		A symplectic manifold is a manifold equipped with a closed, nondegenerate 2-form.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=smooth_manifold (2882 words)

	physics - Pseudo-Riemannian manifold (Site not responding. Last check: )
		In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, (0,2) tensor which is nondegenerate at each point on the manifold.
		The key difference between a Riemannian metric and a pseudo-Riemannian metric is that a pseudo-Riemannian metric need not be positive-definite, merely nondegenerate.
		The signature of a pseudo-Riemannian manifold is just the signature of the metric (one should insist that the signature is the same on every connected component).
	www.physicsdaily.com /physics/Pseudo-Riemannian_manifold (371 words)

	Differentiable manifold Information
		A differentiable manifold is a topological manifold with a globally defined differentiable structure.
		Any topological manifold can be given a differentiable structure locally by using the homeomorphisms in its atlas, combined with the standard differerentiable structure on the Euclidean space.
		Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude,...
	www.bookrags.com /wiki/Differentiable_manifold (2780 words)

	20th WCP: Mathematical Models of Spacetime in Contemporary Physics and Essential Issues of the Ontology of Spacetime
		The core of this conception is constituted by Weil's definition of the world as a 3+1 dimensional manifold, with a defined metric field (metric field is defined by the components of a geometrical object-metric tensor).
		In the field theory of nature, spacetime considered a manifold with a defined metric field is precisely the substance, the primary substratum of events.
		Theories of spacetime in mathematical physics, while considering continua and metric manifolds, cannot explain the difference between time dimension and space dimensions, they are also unable to explain by means of geometry the unidirection of the passage of time, which can be comprehended only by means of thermodynamics.
	www.bu.edu /wcp/Papers/Math/MathGos.htm (2817 words)

	Research Papers of P. Gilkey
		[8] The spectral geometry of a Riemannian manifold, J.Diff.Geo.
		[60] The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary.
		[127] Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture.
	darkwing.uoregon.edu /~gilkey/respap.html (3638 words)

	Differential Geometry Seminar　(2006） / The 21st Century COE Program /Math of OCU(J)
		Complexification of a pseudo-Riemannian manifold and anti-Kaehler geometry
		It is known that the complexification $M_g^{\bf c}$ of a $C^{\omega}$-pseudo-Riemannian manifold $(M,g)$ is defined as a tubular neighborhod (equipped with the complex structure $J^g$ arising from $g$) of the zero section ($=M$) of the tangent bundle $TM$ of $M$.
		Pluriharmonic maps from an n-dimensional complex manifold into a symmetric space are known to have a loop group formulation, that is they come in families parametererized by a spectral parameter in the unit circle.
	math01.sci.osaka-cu.ac.jp /21COE/Differential_Geometry_Seminar/Differential_Geometry_Seminar_e_06 (1875 words)

	Plan of Research
		Now I more-or-less understand the Riemannian case, and try to understand the case when both metrics are pseudo-Riemannian.
		Two pseudo-Riemannian metrics on one manifold are called geodesically equivalent, if their geodesics coincide as unparameterised curves.
		The Riemannian analog of the Beltrami problem for closed 2-manifolds was understood already 1998 (this is a joint work with P. Topalov).
	www.minet.uni-jena.de /~matveev/Forschung/research_proposal.html (436 words)

	Riemannian Geometry
		An N-dimensional Riemannian manifold is characterized by a second-order metric tensor g
		If there exists a coordinate system at a point on the manifold such that the metric components are constant in the first and second order, then the manifold is said to be totally flat at that point (not just asymptotically flat).
		The “connection” of this manifold is customarily expressed in the form of Christoffel symbols.
	www.mathpages.com /rr/s5-07/5-07.htm (2235 words)

	Springer Online Reference Works
		A connection in a principal fibre bundle over a (pseudo-)Riemannian manifold whose curvature satisfies the harmonicity condition (the Yang–Mills equation).
		Examples of non-Einstein Riemannian connections satisfying the Yang–Mills equation are Riemannian connections of conformally-flat metrics with constant scalar curvature and non-constant sectional curvature.
		Examples of non-Riemannian connections satisfying the Yang–Mills equation are connections in the normal bundle of a totally-geodesic submanifold of a symmetric space, or of a quaternionic submanifold of a quaternionic space, induced by the Riemannian connections of these spaces.
	eom.springer.de /Y/y099030.htm (1005 words)

	TALKS. ABSTRACTS (Provisional).
		We study harmonic-Killing vector fields in Kähler manifolds, obtaining that in the compact case such vector fields coincide with the holomorphic ones.
		Calculation of cohomology of these manifolds is closely related to the Gontcharova theorem in the theory of infinitedimensional Lie algebras, namely Theorem.
		The notion of flat pencils of metrics is introduced by Dubrovin and very important for the theory of Frobenius manifolds, the theory of associativity equations in two-dimensional topological field theory and the theory of integrable systems of hydrodynamic type.
	www.math.ist.utl.pt /~jmourao/om/omix/abstracts.html (4163 words)

	Nicholas C. Petridis
		Geometry, 1974, v 9, No, "I believe that this paper will contribute in answering the long standing conjecture of Willmore, viz., that a compact Ricci flat manifold is flat" writes S.I. Golberg in 1975.
		Distance and Volume Decreasing Theorems for a family of Harmonic Mappings of Riemannian Manifolds, Lectures Notes in Math., Springer-Verlang, 1980.
		On Quasiconformal Mappings of Hermitian Manifolds to be published in the Novosibirsk Research Center volume in honor of the distinguished Geometer-Analyst Yu.
	www.math.uoc.gr /dept/petridis-special.html (604 words)

	CJM - Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four
		A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four.
		Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat.
		If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to {mathbb R} All metrics for the simply connected non-reductive Einstein spaces are given explicitly.
	journals.cms.math.ca /cgi-bin/vault/view/fels4448 (149 words)

	A Linear Solution of the Four-Dimensionality Problem
		Presented here is a rigorous formulation of several implicit assumptions of standard physics which leads to a first-order theory shown to possess a real-world model: if an observer’s logic is Boolean, he is bound to perceive his spacetime as a four-dimensional pseudo-Riemannian manifold of signature 2, with an ideal big bang geometry.
		The connections between the type of an observer’s logic and large-scale structure of the observable universe yield a testable prediction, existence of positive cosmological constant and suggest a non-standard integration-over-spacetime technique.
		To compare, the notion of smooth affine manifold (a starting point for the working physicist) is far more complicated.
	members.tripod.com /~Xperiment/4D.htm (2958 words)

	Transactions of the American Mathematical Society
		Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields
		Abstract: In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space.
		W.H. Paus, Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields, Ph.D. thesis, University of New South Wales, Sydney, 1996.
	www.ams.org /tran/1998-350-10/S0002-9947-98-02016-9/home.html (587 words)

	Speakers CurGeo 03
		Given a smooth manifold M, let V be a submanifold of the tangent bundle TM such that the projection of V to M is a submersion.
		Canonical affinors on the tangent bundle of a symplectic manifold (jointly with W.M. Mikulski).
		Canonical affinors on the tangent bundle of a symplectic manifold (jointly with J. Kurek).
	www.diffiety.ac.ru /conf/curgeo03/participants.htm (1905 words)

	Spacetime information - Search.com
		By combining the two concepts into a single manifold, physicists are able to significantly simplify the form of most physical laws, as well as describe the workings of the universe at both supergalactic and subatomic levels in a more uniform way.
		The concept of geodesics becomes critical in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in space-time, that is, free from any external influences.
		Second, for a manifold, the property of connectedness and path-connectedness are equivalent and one requires the existence of paths (in particular, geodesics) in the spacetime to represent the motion of particles and radiation.
	www.search.com /reference/Spacetime (2539 words)

	Plan of Research
		Now I more-or-less understand the Riemannian case, and try to understand the case when both metrics are pseudo-Riemannian.
		Two pseudo-Riemannian metrics on one manifold are called geodesically equivalent, if their geodesics coincide as unparameterised curves.
		The Riemannian analog of the Beltrami problem for closed 2-manifolds was understood already 1998 (this is a joint work with P. Topalov).
	home.mathematik.uni-freiburg.de /matveev/Forschung/research_proposal.html (436 words)

	Asymptotics Of The D'alembertian With Potential On A Pseudo-Riemannian Manifold (ResearchIndex)
		Let be the Laplace-d'Alembert operator on a pseudo-Riemannian manifold (M; g).
		We derive a series expansion for the fundamental solution G(x; y) of + H, H 2 C 1 (M), which behaves well under various symmetric space dualities.
		0.1: The Dirac operator on Lorentzian spin manifolds and the Huygens..
	citeseer.ist.psu.edu /24174.html (286 words)

	XVIth Annual Geometry Festival: Abstracts of Talks
		These rigidity results have implications for the classification of holomorphic bundles over compact Kähler manifolds that are generated by their global sections but for which one or more of the Schur-Chern classes vanish, and this will be explained in the talk.
		Frobenius manifolds were introduced by the author in the beginning of 90s as the geometric setup of the so-called equations of associativity discovered by physicists E.Witten, R.Dijkgraaf, E. and H. Verlinde.
		For mathematicians it is best known appearance of Frobenius manifolds in the theory of the genus zero Gromov-Witten invariants of compact symplectic manifolds, although Frobenius manifolds appear also in other branches of mathematics.
	www.math.neu.edu /geomfest/abstracts.html (977 words)

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