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Topic: Local diffeomorphism

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	Diffeomorphism - Wikipedia, the free encyclopedia
		In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds.
		A differentiable bijection is not necessarily a diffeomorphism, e.g.
		The diffeomorphism group of a manifold is the group of all its automorphisms (diffeomorphisms to itself).
	en.wikipedia.org /wiki/Diffeomorphism (714 words)

	Local homeomorphism - Wikipedia, the free encyclopedia
		In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces.
		This is a local homeomorphism for all non-zero n, but a homeomorphism only in the cases where it is bijective, i.e.
		A bijective local homeomorphism is therefore a homeomorphism.
	en.wikipedia.org /wiki/Local_homeomorphism (432 words)

	Description of Research - Injectivity of maps
		In [15] we use Pinchuk's counterexample to the real Jacobian conjecture to construct certain local diffeomorphisms which show that there is no higher dimensional version of the spectral theorem of Gutierrez.
		On the positive side, we prove in [15] a sharp injectivity theorem with "nearly spectral" hypothesis, similar in spirit to the one of Gutierrez that is valid in all dimensions.
		Injectivity of local diffeomorphisms from nearly spectral conditions,
	www.nd.edu /~fxavier/Research/injectivity.htm (573 words)

	Diffeology - local differentiability
		V is a plot of Y. Then, f o P is locally a plot of Y, at each point of U, by the axiom of locallity f o P is a plot of Y. Then, f is differentiable.
		Local diffeomorphisms are especially used for defining manifolds, orbifolds and all kind of diffeological spaces, locally modeled on some special category of diffeological spaces.
		Or there exists injective maps which are local diffeomorphism at each point without being surjective, then without being diffeomorphisms.
	www.umpa.ens-lyon.fr /~iglesias/articles/Diffeology/dflg_localdiff.html (768 words)

	[No title]
		\section{Definitions} By a {\em local diffeomorphism} ({\em local vector field}) we mean in what follows either a germ of diffeomorphisms (vector fields) at a point or a representative of this germ defined in a neighborhood of the point.
		Let $\:\frG\:$ be a group of local diffeomorphisms $\:\Phi\::\:(\R^{2d},0)\:\to\:(\R^{2d},0)\:$ and $\:\gs\::\:\frG\:\to\:\R\:$ be a {\em multiplicative character} of $\:\frG,\:$ {\em i.e.}, a homomorphism into the multiplicative group $\:\R^.\:$ \\ {\bf Definition 1.1.} {\em A local* vector field $\:\xi\:$ is said to be $\:(\frG,\gs)$-equivariant if} $$ U_*\xi\:=\:\gs(U)\xi\ \ \ (U\:\in\:\frG).
		A local transformation $\:G\:$ keeps the property of $\:\xi\:$ to be $\:(\frG,\gs)$-equivariant if it commutes with every element $\:U\:\in\:\frG,\:$ {\em i.e.}, $\:UG\:=\:GU.\:$ Such a local diffeomorphism is said to be $\:\frG$-{\em equivariant}.
	www.ma.utexas.edu /mp_arc/html/papers/00-449 (1923 words)

	Luboš Motl's reference frame: Values in physics
		Their structure requires us to consider theories with local gauge symmetries which are themselves "beautiful" and constrain the matter spectrum and the character of the interactions.
		One of the fundamental pillars of general relativity is the equivalence principle that states that in locally inertial frames, the laws of special relativity must be satisfied by all local phenomena.
		The diffeomorphism group is a local symmetry and at the quantum level, all states must be invariant (singlets) under all these local symmetries, much like in the Yang-Mills case.
	motls.blogspot.com /2006/02/values-in-physics.html (4070 words)

	Metanexus Institute (Site not responding. Last check: 2007-11-05)
		To define such local spacetime diffeomorphism invariant observables we must be able to specify a particular event relationally, that is by the physical information available to an observer at that event.
		This is because, when examined in terms of what local observers actually see in a real, generic spacetime, complex enough for them to exist, there is a clear distinction between the past and the future.
		But without these formal constructions, we are left with only the local observables themselves, which are defined in terms of the real information that distinguishes the events from each other, and which are related to each other by the causal structure.
	www.metanexus.net /metanexus_online/show_article2.asp?id=8598 (1684 words)

	[No title]
		By a partial diffeomorphism fl of M we mean a diffeomorphism (analytic if clear from the context) from an open subset D of M onto another open subset of M. The inverse fl-1 of fl is defined on the image of fl.
		We let let Diff(M) denote the set of such partial diffeomorphisms on M. As a general rule, if the Lie bracket of two vector fields, or the composition of two 4 partial diffeomorphisms, appears in a statement, that statement should be taken to be read `if this composition is defined, then...'.
		Examples of locally finitely generated involutive \Phi are finite dimensional Lie algebras of vector fields (just take for the Yi's a basis of this Lie algebra), as well as involutive sets of vector fields for which the distribution D(\Phi) 30 has constant rank (if {Y1(,),.
	www.math.rutgers.edu /~sontag/actions-report.html (14799 words)

	Michor, Peter, Description of Research (Site not responding. Last check: 2007-11-05)
		Paper [9] develops the foundations, [12] treats the diffeomorphism group for a non-compact manifold, [13] treats the principal bundle of embeddings with structure group the diffeomorphism group; these papers culminate in the monography [C].
		[24] states that the (singular) cohomology of a diffeomorphism group is the continuous Lie algebra cohomology (Gelfand-Fuks) of the Lie algebra of all vector fields with values in the module of all smooth functions on the diffeomorphism group.
		We give a description in term of functional analysis by equipping each such algebra with the direct limit locally convex topology with respect to the nuclear Fréechet spaces of smooth functions on $R^n$, and we found a condition for a locally m-convex commutative algebra to be a $C^\infty$-algebra (\v Cebyshev condition).
	www.mat.univie.ac.at /~michor/self-est.html (3906 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In the end, the theory just gave back classical gtr, or rather a local mimic of gtr (unless one does the sensible thing and assumes that the unobservable background is only valid in some chart, and then creates sufficiently many overlapping charts to obtain the full spacetime).
		Diffeomorphism invariance and coordinate independence are absolutely required of physical theories, because of their fundamental nature as models of the world.
		It is a local field theory in the same sense that Maxwell's theory of EM is a local field theory.
	math.ucr.edu /home/baez/PUB/dbwf2 (20256 words)

	Luboš Motl's reference frame: February 2006 (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-05)
		Whenever locality and separation of a system into pieces is a good approximation, the additivity of the action or energy-momentum and/or the multiplicativity of the path integral phase or the wavefunction must be satisfied.
		The only plausible way to prove locality is to show the equivalence of your theory with a theory where the energy and the momentum are additive.
		This is important for approximate locality represented by the additivity of the energy-momentum vector.
	motls.blogspot.com.cob-web.org:8888 /2006_02_01_motls_archive.html (11696 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Subject: Re: local diffeo & RP^n Date: Wed, 09 Aug 2000 15:14:39 -0500 Newsgroups: sci.math Summary: [missing] Ed Hook wrote: > > Actually, since RP^n is compact and hausdorff, the local diffeo is indeed a > > covering map.
		Pick y in Y; this is a regular value because of the local diffeo.
		Now, pick non-intersecting neighborhoods around each of these points such that f restricted to these neighborhoods is a a homeo; this is easily accomplished because of the local diffeo and the "hausdorfness" at hand.
	www.math.niu.edu /~rusin/known-math/00_incoming/covering (972 words)

	CDT \| Musings
		Essentially, you are restricting to metrics for which the geodesic distance between the initial and final slices (for all geodesics between the initial and final slice) is fixed.
		In practical applications, the (infinite) freedom to tinker with the details of the latticization (both the choice of lattice, and the choice of the lattice action) can be used to advantage.
		The restriction on triangulations in CDT is not, strictly, a local one.
	golem.ph.utexas.edu /~distler/blog/archives/000713.html (3481 words)

	[physics/0507133] Are Gravitational Waves Directly Observable?
		The problems of the seemingly "local" observables in classical GR are widely known (see, e.g., the definition of 'gravitational energy' here, and Charles Torre here), but I couldn't figure it out how these problems may be resolved with the ‘pseudo-local’ observables introduced in [Ref.
		NB: If the state plays "a key role" (notice the poetry) in the recovery of the notion of locality itself, but the locality turns out to be "both relative and approximate", then perhaps you're dealing with the potential point(s); more here.
		Nevertheless we describe how, in suitable quantum states and in a suitable limit, the familiar physics of local quantum field theory can be recovered from appropriate such observables, which we term ‘pseudo-local.’ We consider measurement of pseudo-local observables, and describe how such measurements are limited by both quantum effects and gravitational interactions.
	www.god-does-not-play-dice.net /arXiv.html (5177 words)

	Manifold (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-05)
		If the local charts on a Manifold are compatible in a certain sense, one can talk about directions, tangent spaces, and differentiable functions on that manifold.
		A Manifold is said to be homogeneous for its homeomorphism group, or diffeomorphism group, if that group acts transitively on it; this is true for connected manifolds.
		Roughly speaking, it is a space which locally looks like the quotient of Euclidean space by a finite group.
	manifold.iqnaut.net.cob-web.org:8888 (2005 words)

	Diffeology - Manifolds
		Let M be a diffeological space, we call M a manifold if M is locally diffeomorphic to a numerical domain at each point x ∈ M. In other words: M is a manifold if, for each point x ∈ M, there exists a local diffeomorphism F : M ⊃ A → R
		The previous definition is a little bit too general, the dimension of the numerical domain to which the manifold M is supposed to be locally diffeomorphic is, in general, assumed to be constant.
		Then, by definition, the diffeology of M is generated by local diffeomorphisms with R
	www.umpa.ens-lyon.fr /~iglesias/articles/Diffeology/dflg_manifolds.html (1153 words)

	DC MetaData for: Natural operations in differential geometry
		That the exterior derivative $d$ commutes with local diffeomorphisms now means, that $d$ is a natural operator from the functor $\La^kT^$ into functor $\La^{k+1}T^$.
		Hence the classical bundles of geometric objects are now studied in the form of the so called lifting functors or (which is the same) natural bundles on the category $\Mf_m$ of all $m$-dimensional manifolds and their local diffeomorphisms.
		Further, some functors of modern differential geometry are defined on the category of fibered manifolds and their local isomorphisms, the bundle of general connections being the simplest example.
	www.mat.univie.ac.at /~michor/preprint-shadows/kmsbookh.html (1544 words)

	Not Even Wrong » Blog Archive » Schroer’s “Samizdat” (via CobWeb/3.1 ... (Site not responding. Last check: 2007-11-05)
		Rather the local covariance principle (diffeomorphism-independence) reflects itself in an algebraic property which (since states are dual to algebras) in terms of states is the invariance of a whole folium of states, but not that of its individual members (these things are explained in previous contributions of the authors).
		diffeomorphisms and not coordinate transformations) and therefore comparisons with gauge theories (where one was forced to introduce to introduce redundant descriptions) may be misleading (those do not have an active interpretation).
		The first property is the local covariance of Einstein and its very non-trivial recent adaptation to QFT (necessitating a very novel and surprising generalization of being forced to think about all globally hyperbolic manifolds at the same time even if for defining just one QFT in the new sense).
	www.math.columbia.edu.cob-web.org:8888 /~woit/wordpress/?p=471 (15914 words)

	Math 423, Fall, 2002 (Site not responding. Last check: 2007-11-05)
		However, unlike ``tangent'' planes to a surface in space, which intersect, we want to think of these ``planes'' as being separate - linked by the local coordinate structure, but corresponding to the various points ``of tangency''.
		V is the local coordinate map for M. It is clear that this defines a manifold structure on T
		Since that matrix is invertible, and the coordinate transformation on the base part is a diffeomorphism, it is clear that this map is a local C
	www.lehigh.edu /dlj0/yesterday/Desktop/dlj0/courses/423f02-lect5.html (407 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		> > according to lena's original post, the mapping is only a local > diffeomorphism between the real projective space RP^n.
		> Let f : M --> N be a local diffeomorphism of closed (= compact borderless) connected manifolds.
		Then f(M) is both open and closed in N, whence f is surjective.
	www.math.niu.edu /~rusin/known-math/00_incoming/local_diff (135 words)

	Classification of local conformal nets. Case < 1, Yasuyuki Kawahigashi, Roberto Longo
		We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge $c$ less than 1.
		Then, by using the classification of modular invariants for the minimal models by Cappelli-Itzykson-Zuber and the method of $\alpha$-induction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c < 1 and infer our main classification result.
		As an application, we identify in our classification list certain concrete coset nets studied in the literature.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.annm/1111770727 (127 words)

	Dr. Ilka Agricola
		Abstract: One of the problems in differential geometry is to determine as to whether two geometric structures defined on two manifolds are locally isomorphic.
		For example, given two Riemannian structures a question arises how to establish, in a finite number of steps, if there exists a local diffeomorphism that transforms one of the Riemanian structures into the other.
		This problem is particulary known in General Relativity, where a newly obtained exact solution of Einstein's field equations, usually expressed in some local coordinates, should be shown not to be transformable by a coordinate transformation to a solution already known.
	www.mathematik.hu-berlin.de /~agricola/act-old.html (491 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		If \\ (a) The invariant curves ($\gamma_1,$ $\gamma_2$) at $p$ have contact of order $l$ at a degenerate homoclinic crossing $q\in M$, $l\in (1,r]$.\\ (b) There is a $C^r$ local diffeomorphism from a neighborhood of the origin in the plane which locally conjugates $F$ to a linear map.
		We may assume $\phi $ takes the local stable manifold at $o$ to a neighborhood in the $Y$-hyperplane, and the local unstable manifold to a neighborhood in the $X$-hyperplane.
		In $N'$ there are images of $q$, under forward and backward iterates of $F$, on both the local stable and local unstable manifolds of $o$.
	www.ma.utexas.edu /mp_arc/papers/98-208 (2852 words)

	Local diffeomorphism - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-05)
		In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f
		Every local diffeomorphism is also a local homeomorphism and therefore an open map.
		According to the inverse function theorem, a smooth map f : M → N is a local diffeomorphism if and only if the derivative Df
	en.wikipedia.org.cob-web.org:8888 /wiki/Local_diffeomorphism (144 words)

	Citations: Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation ... (Site not responding. Last check: 2007-11-05)
		Herman, "Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation numbers," Bull.
		, since it is the linearization of the conjugacy equation of an analytic circle diffeomorphism to the rotation w 7 w h.
		....: In his work [Ar] on the local linearization problem of analytic diffeomorphisms of the circle, Arnol d discussed in detail this issue ; he complexified the rotation number but he did not prove that the dependence of the conjugacy on it is monogenic.
	citeseer.ist.psu.edu.cob-web.org:8888 /context/2095474/0 (300 words)

	General relativity - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-05)
		The principle of local Lorentz invariance: The laws of special relativity apply locally for all inertial observers.
		Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian.
		The spacetime is static so the theory is not fully relativistic in the sense of general relativity; it is not background independent nor generally covariant under the diffeomorphism group.
	en.wikipedia.org.cob-web.org:8888 /wiki/General_relativity (5651 words)

	Injectivity as a Transversality Phenomenon in Geometries of Negative Curvature (ResearchIndex) (via CobWeb/3.1 ... (Site not responding. Last check: 2007-11-05)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Abstract: this paper we approach the problem of injectivity of two dimensional local diffeomorphisms from the point of view of geometries of negative curvature.
		1 Injectivity of local diffeomorphisms from nearly spectral co..
	citeseer.ist.psu.edu.cob-web.org:8888 /96148.html (276 words)

	AMCA: The fundamental form on a Lie groupoid of diffeomorphisms by Ivan Belko
		(B) of k-jets of local diffeomorphisms of differentiable manifold B is essential in the Ehresmann method.
		(B) consists of k-jets of local diffeomorphism of B. The Lie groupoid \Pi (B) is a Lie groupoid of linear isomorphisms of the tangent bundle TB.
		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/k/m/18.htm (426 words)

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