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Topic: Lorentzian manifold

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Lorentzian metric

complex manifold

differentiable manifolds

pseudo Riemannian manifold

smooth manifolds

symplectic manifolds

Calabi Yau manifolds

Variable Length Intake Manifold

topological manifolds

Flag manifold

Curvature of Riemannian manifolds

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	Pseudo-Riemannian manifold - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Lorentzian_manifold (460 words)

	Lorentzian Manifold With A Group Of Conformal Transformations That Has A Normal One-Parameter Homothety Subgroup - ... (Site not responding. Last check: 2007-08-09)
		We declined the assumptions of homogeneity for (M; g) and of connectedness for H and proved that (M; g) is a plain...
		Lorentzian Manifold With A Group Of Conformal Transformations..
		Podoksenov M. N., Lorentzian manifold with a group of conformal transformations, that has a normal subgroup of homotheties.
	citeseer.ist.psu.edu /podoksenov96lorentzian.html (304 words)

	dfggeo.uni-muenster.de - Projekte
		On compact manifolds we study questions of spectral rigidity versus isospectral defomability, relations to the length spectrum, and properties of metrics and potentials which are not spectrally determined.
		Since in general relativity, space is described by a spacelike hypersurface of a Lorentzian manifold, the physical question under consideration is how and to which extent the energy of a physical system controls the gravitational field.
		The manifolds which are of interest here are those which admit a group action of low cohomogeneity, in particular those of cohomogeneity at most two.
	dfggeo.uni-muenster.de /abstracts.php (3932 words)

	AMCA: Twistor and Killing spinors on Lorentzian manifolds and their relations to CR and Kaehler geometry by Helga Baum
		is a Lorentzian metric on M, whose conformal class is an invariant of the CR structur.
		In particular, each Lorentzian manifold with real Killing spinors is an Einstein space of positive scalar curvature, and in case of dimension 4 of constant sectional curvature.
		Further classes of Lorentzian manifolds with imaginary Killing spinors can be obtained by the study of indecomposable but non-irreducible holonomy representations for metrics of index 1 or 2.
	at.yorku.ca /c/a/d/q/76.htm (671 words)

	General relativity - ExampleProblems.com
		In this theory, spacetime is treated as a 4-dimensional Lorentzian manifold which is curved by the presence of mass, energy, and momentum (or stress-energy) within it.
		Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian.
		Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.
	www.exampleproblems.com /wiki/index.php/General_relativity (4245 words)

	Dirac operators on Lorentzian manifolds and their quantization
		In analogy to the axiomatic approach of Gromov and Lawson for Riemannian manifolds it is planed to develop the axiomatics of generalized Dirac operators on Lorentzian manifolds.
		This means that we want to characterize a class of first-order differential operators acting on sections in a vector bundle over the Lorentzian manifold M which are closely related to the geometry of M. In particular, the square of the principal symbol should be scalar and coincide with the Lorentz metric.
		Consider a Ricci-flat 4-dimensional Lorentzian manifold which is foliated by compact spacelike 3-dimensional manifolds of constant mean curvature.
	users.math.uni-potsdam.de /~baer/dfg/projekt1-dfg-SPP-1154.html (628 words)

	Dirac operators on Lorentzian manifolds
		The spinors (and the Dirac equation) on a curved manifold is defined
		manifold admits spinors at all, globally, which means that it admits a spin
		Lorentzian manifold in precisely the same way as you do on a Riemannian
	www.physicsforums.com /showthread.php?t=72323 (3762 words)

	Uri Bader's talk
		The group of conformal transformations of the sphere is non-compact, while the conformal group of any other compact Riemannian manifold is compact (Ferrand, Obata ~70).
		A Lorentzian manifold is a manifold modelled over the space-time (in the same manner that a Riemannian manifold is a manifold modelled over the space).
		We will introduce a certain Lorentzian manifold, sometimes called "the boundary of the anti-de-Sitter space", which resembles the symmetric properties of the sphere.
	people.brandeis.edu /~kleinboc/EP0304/bader.html (129 words)

	Curled-Up Dimensions
		We can imagine a universe cylindrical in all directions, temporal as well as spatial, by embedding the entire four-dimensional spacetime in a manifold of eight dimensions, two of which are purely imaginary, as follows:
		This leads again to the locally Lorentzian four-dimensional metric (1), but now all four of the dimensions X,Y,Z,T are periodic.
		So here we have an everywhere-locally-Lorentzian manifold that is closed and unbounded in every spatial and temporal direction. Obviously this manifold contains closed time-like worldlines, although they circumnavigate the entire universe. Whether such a universe would appear (locally) to possess a directional causal structure is unclear.
	www.mathpages.com /rr/s7-04/7-04.htm (820 words)

	CJM - Curvature Estimates in Asymptotically Flat Lorentzian Manifolds
		We consider an asymptotically flat Lorentzian manifold of dimension (1,3).
		An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form.
		The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero.
	journals.cms.math.ca /cgi-bin/vault/view/finster3363 (74 words)

	Springer Online Reference Works
		Photons correspond to null geodesics and freely falling particles to time-like geodesics (cf.
		The interpretation of a singularity as a physical singularity in the sense that space-time or gravity diverges in some sense may be misleading.
		There are theorems which state that under "physically reasonable" conditions there are singularities in the Lorentzian manifolds representing space-times [a2], [a1] (but see also [a4] and [a3]).
	eom.springer.de /N/n120010.htm (335 words)

	Amazon.com: "covering manifold": Key Phrase page (Site not responding. Last check: 2007-08-09)
		Show that the covering transformations form a group and that if x, y E M, a covering manifold of M, then there is at most one covering transformation taking x to y.
		Let M = G/H = G/H, where G is the universal covering of a connected Lie group G. Then any covering manifold M1 of M is a homo- geneous space of the group 0 with group model G/H1i where H C Ht...
		Let M = G/H = 0/ft, where G is the universal covering of a connected Lie group G. Then any covering manifold M1 of M is a homo- geneous space of the group 0 with group model G/H1 i where H C...
	www.amazon.com /phrase/covering-manifold (478 words)

	[No title]
		The smooth function $\mathcal{T}(x,t)=t$ is a {time-function}, that is the Lorentzian gradient $\nabla^L \mathcal{T}$ is a timelike vector field, where $$ \nabla^L\mathcal{T}(x,t) = \big({\bf 0},-\frac{1}{\beta(x)}\big).
		Trajectories joining two given events have been studied in \cite{ba}, \cite{cm} on complete {\em stationary} Lorentzian manifolds, in \cite{ba1}, \cite{cm1} on open subsets of stationary Lorentzian manifolds and in \cite{agm} in a different setting.
		Explicitly, from (\ref{electric}) and the expression of $\nabla$ in a static manifold (see for example \cite[Proposition 7.35]{onei}) \begin{equation} \label{ele} E = -(\beta)^{1/2} \nabla A_2 -\nabla \beta.
	www.ma.hw.ac.uk /EJDE/Volumes/2004/10/bartolo-tex (5104 words)

	Solutions to HW4
		Let us say that a Lorentzian manifold is geodesically complete if every geodesic can be infinitely extended in both directions with respect to the natural parameter s (interval or proper time).
		It is physically natural to consider such a manifold as a space/time without singularities because any observer moving there in a spaceship can do it forever by the ship's internal clock.
		Also if a manifold M can be included as an open set into a geodesically complete manifold, then we will consider M as having no true singularities, because the apparent singularities are caused by the fact that we consider only a part of a true space.
	www.math.neu.edu /~bratus/diffgeom/sol4/sol4.htm (887 words)

	Lessons from Topological Quantum Field Theory
		In quantum field theory on curved spacetime, space and spacetime are not just manifolds: they come with fixed `background metrics' that allow us to measure distances and times.
		: that is, a Lorentzian manifold with boundary whose metric restricts at the boundary to the metrics on
		-dimensional manifold and spacetime is a cobordism between such manifolds.
	math.ucr.edu /home/baez/quantum/node2.html (1456 words)

	2003 Annual Meeting of the Israel Mathematical Union (Site not responding. Last check: 2007-08-09)
		There is a known model to a conformal action of the simple Lie group SO(2,n) on a Lorentzian manifold (know to physicists as the boundry of the Anti-de-Siter space).
		We will consider a conformal action of a simple Lie-group on a compact Lorentzian manifold, different from this model action, which has no fixed points.
		Abstract: We present several recent developments associating topological features of a manifold, with the corresponding geometric inequalities of systolic type, satisfied by an arbitrary metric on the manifold.
	imu.org.il /zichron/geomabs.html (554 words)

	Preprint No. 417 (Site not responding. Last check: 2007-08-09)
		The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors.
		A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function.
		The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor.
	www-sfb288.math.tu-berlin.de /abstractNew/417 (178 words)

	Dirac operators on Lorentzian manifolds
		It is kind of interesting to think of the Dirac operator as the "square root" of the Laplacian.
		So one way to think of it is that on a Riemannian manifold, the Dirac operator is the "square root" of the Laplacian.
		On a Lorentzian manifold, the Dirac operator is the "square root" of the wave equation.
	www.physicsforums.com /showthread.php?t=72272 (500 words)

	Giannoni, Masiello: Geodesics on product Lorentzian manifolds (Site not responding. Last check: 2007-08-09)
		Giannoni, F. Masiello, A. Geodesics on product Lorentzian manifolds.
		, On the geodesic connectedeness of Lorentzian manifolds, Math.
		, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, Jour.
	www.numdam.org /numdam-bin/item?id=AIHPC_1995__12_1_27_0 (146 words)

	Topics: Lorentzian Geometry (Site not responding. Last check: 2007-08-09)
		Lorentzian str: A reduction of the bundle of frames F(M) to the Lorentz group, as a subgroup of GL(n,R).
		Conditions: The nasc for a manifold M is that M be non-compact, or that the Euler number
		Time orientability: If a Lorentzian manifold is not time-orientable, it admits a 2-fold time-orientable covering [@ Markus AM(55)].
	www.phy.olemiss.edu /~luca/Topics/geom/lorentz.html (488 words)

	Time Travel Portal :: View topic - On causality and closed geodesics of compact Lorentzian manifolds and static ... (Site not responding. Last check: 2007-08-09)
		On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes
		As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.
		On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes by Miguel Sánchez.
	timetravelportal.com /viewtopic.php?t=1548 (194 words)

	On almost cosymplectic Lorentzian hypersurfaces immersed in a Lorentzian manifold, Radu Rosca
		On almost cosymplectic Lorentzian hypersurfaces immersed in a Lorentzian manifold, Radu Rosca
		On almost cosymplectic Lorentzian hypersurfaces immersed in a Lorentzian manifold
		[5] OBATA, M., The Gauss map of immersions of Riemanman manifolds in space o constant curvature.J. Diff.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.kmj/1138846725 (154 words)

	Citebase - The mass of a Lorentzian manifold
		Citebase - The mass of a Lorentzian manifold
		We define a physically reasonable mass for an asymptotically Robertson-Walker (ARW) manifold which is uniquely defined in the case of a normalized representation.
		Users are cautioned not to use it for academic evaluation yet.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0403002 (118 words)

	ADM-Split
		In general relativity spacetime is considered to be curved four-dimensional Lorentzian manifold.
		being the trace-reduced Ricci tensor for the four-dimensional manifold and the stress-energy tensor
		This equation treats time and space variables on an equal base.
	www.tat.physik.uni-tuebingen.de /~koellein/bericht-WEB/node19.html (141 words)

	Topics: Spacetime Subsets (Site not responding. Last check: 2007-08-09)
		Achronal set: A subset S of a Lorentzian manifold M is called an achronal set if none of its points is in the chronological future of any other, or there are no p, q
		(S) is always open, if the manifold is everywhere Lorentzian (no singular points).
		Idea: The part of a spacetime manifold that can be connected to an asymptotic region by both future and past-directed timelike curves.
	www.phy.olemiss.edu /~luca/Topics/st/subsets.html (578 words)

	www.myspace.com/darthkazi (Site not responding. Last check: 2007-08-09)
		I'm sure I'll start with porn and move up from there like George Lucas.
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