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# Topic: Diffeomorphism

 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. For a connected manifold M the diffeomorphisms act transitively on M: this is true locally because it is true in Euclidean space and then a topological argument shows that given any p and q there is a diffeomorphism taking p to q. en.wikipedia.org /wiki/Diffeomorphism   (718 words)

 Anosov diffeomorphism - Wikipedia, the free encyclopedia If a flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponentially contracting, and one that is exponentially expanding, and a third, non-expanding, non-contracting one-dimensional sub-bundle, then the flow is called an Anosov flow. It was proved that any other Anosov diffeomorphism on a torus is topologically conjugate to one of this kind. The problem of classifying manifolds that admit Anosov diffeomorphisms turned out to be very difficult, and still as of 2005 has no answer. en.wikipedia.org /wiki/Anosov_diffeomorphism   (981 words)

 5.3 Physical meaning of diffeomorphism invariance, and its implementation in the quantum theory   (Site not responding. Last check: 2007-11-07) Conventional field theories are not invariant under a diffeomorphism acting on the dynamical fields. Thus, diffeomorphism invariance is not a feature of just the gravitational field: It is a feature of physics, once the existence of relativistic gravity is taken into account. Diffeomorphism invariance is the technical implementation of a physical idea, due to Einstein. relativity.livingreviews.org /Articles/lrr-1998-1/node11.html   (538 words)

 Qgravity.org: Technical Summary of Loop Quantum Gravity   (Site not responding. Last check: 2007-11-07) When applied to spatially diffeomorphism invariant functions of the phase space it yields finite operators on the space of diffeomoprhism invariant states, when applied to scalar functions it gives operators on the kinematical state space that trasform under the action of the unitary representation of the spatial diffeomorphsim group. First, at the level of spatially diffeomorphism invariant observables, a sufficient set has been constructed and diagonalized, in closed form, to label a complete basis of states in terms of their eigenvalues, for each of a large set of theories. Diffeomorphism invariant observables are then promoted to physical observables, defined on spacelike slices picked out by the gauge conditions. www.qgravity.org /loop   (7364 words)

 [No title]   (Site not responding. Last check: 2007-11-07) We show that given any sufficiently smooth diffeomorphism can be written as the composition of a finite number of diffeomorphisms which are symmetric under reflection, essentially one-dimensional and about as differentiable as the given one. In our case, the groups considered will be groups of differentiable diffeomorphisms of the sphere and the subgroups in which we factor will be groups of diffeomorphisms that commute with reflections across a plane and which are essentially one dimensional. Given a diffeomorphism $f \in \Diff^l(M,N)$ and a $C^l$ vector field $v$, the map ${\rm Exp}(f,v):M \to N$ defined by ${\rm Exp}(f,v)(x) = \exp_{f(x)}v(f(x))$ -- where $\exp$ denotes the Riemannian geometry exponential associated to a $C^\infty$ metric -- is a $C^l$ mapping. www.ma.utexas.edu /mp_arc/papers/98-10   (3333 words)

 [No title]   (Site not responding. Last check: 2007-11-07) Newsgroups: sci.math.research From: palais@binah.cc.brandeis.edu Subject: Re: Diffeomorphism question Date: Sun, 20 Sep 1992 13:58:37 GMT John Baez asks: >> Let's call the "standard" embedding of S^2 in S^3 that given >> by w = 0,where S^3 is given by w^2 + x^2 + y^2 + z^2 = 1 >> in R^4. It follows that your question is equivalent to asking whether any smooth embedding of the 2-sphere in the 3-sphere is the restriction of an embedding of a 2-disk--- i.e., the Schoenflies Problem---which is well known has a positive answer. Subject: Re: Diffeomorphism question Date: Mon, 21 Sep 1992 03:58:42 GMT Just to let everyone know, my question about two-spheres in three-spheres was answered by Richard Palais and Dan Asimov. www.math.niu.edu /~rusin/known-math/93_back/knots   (593 words)

 [No title] Observe that being a diffeomorphism is an open property in the Whitney topology. I vaguely remember that this statement (in general C^1 case) *was* in Hirsh, but it may be a glitch of my memory in the years that passed... The compatibility condition is that, really, the real-analytic manifold structure is smoothly diffeomorphic with the original smooth manifold. www.math.niu.edu /~rusin/known-math/97/cinfty   (1433 words)

 6.8 Diffeomorphism invariance   (Site not responding. Last check: 2007-11-07) The next step in the construction of the theory is to factor away diffeomorphism invariance. This is the reason for which the loop representation, as defined here, is of great value in diffeomorphism invariant theories only. There are two distinct possibilities for factoring away the diffeomorphisms in the quantum theory, yielding two distinct version of the theory. relativity.livingreviews.org /Articles/lrr-1998-1/node21.html   (455 words)

 Diffeomorphism -   (Site not responding. Last check: 2007-11-07) Formally, given two manifolds M and N, a bijective map $f$ from M to N is called a diffeomorphism if both $f:M\to N$ and its inverse $f^\left\{-1\right\}:N\to M$ are differentiable (if these functions are r times continuously differentiable, f is called a $C^r$-diffeomorphism). Condition 2 excludes diffeomorphisms going from dimension $n$ to a different dimension $k$ (the matrix of $df$ would not be square hence certainly not invertible). $f\left(x\right)=x^3$ is not a diffeomorphism from $\mathbb\left\{R\right\}$ to itself because its derivative vanishes at 0. psychcentral.com /psypsych/Diffeomorphism   (722 words)

 Vasylkevych, Sergiy (2004-04-09) Poisson structures for PDEs associated with diffeomorphism groups. ... We study Poisson and Lie-Poisson structures on the diffeomorphism groups with a smooth metric spray in connection with dynamics of nonlinear PDEs. The key difficulty in finding a suitable context for that arises from the fact that the integral curves of Euler equations are not differentiable on the Lie algebra of divergence free vector fields of Sobolev class Hs. This technique is generalized to an arbitrary diffeomorphism group possessing a smooth spray. etd.caltech.edu /etd/available/etd-05212004-052003   (262 words)

 [No title] Since a map with the preceding two properties is* * automatically an equivariant diffeomorphism, one more application of the equivariant isotopy * *extension theorem implies that h is equivariantly isotopic to a map that is an equivariant diffeo* *morphism near the boundary, the 0-dimensional G-components, and the set MHi. Therefore a relativ* *e version of the uniqueness of smoothings of 3-manifolds implies that k is isotopic to a dif* *feomorphism by an isotopy that is fixed off a compact set (the relative smoothing result is impli* *cit in [Moi]; the results of [KiS2, Essay V] for 3-manifolds contain explicit theorems on relative smooth* *ings). At this stage we have* * an equivariant homeomorphism that is an equivariant diffeomorphism near the singular set, and * *one can deform this map to be an equivariant diffeomorphism everywhere exactly as in the orien* *tation-preserving case. hopf.math.purdue.edu /Kwasik-Schultz/ir3.txt   (8198 words)

 New Physics Reviews 1, 1 The quantum states of matter and curved space are functions of a countable number of "gauge" diffeomorphism equivalence classes of loops and N open surfaces with boundaries. A diffeomorphic Kronecker delta is defined between different sets of open surfaces and loops. The classical matter fields that describe the classical clock cannot be the same as those that describe the quantum reference frame (p.15) Smolin introduces a new classical massless scalar matter "clock field" T that defines the physical time. www.qedcorp.com /pcr/pcr/NPR1.htm   (3395 words)

 [No title]   (Site not responding. Last check: 2007-11-07) One of the simplest examples of an Anosov diffeomorphism is Arnold's cat map: the diffeomorphism of the torus R Even more generally, a hyperbolic automorphism of a simply connected nilpotent Lie group N that fixes a lattice G in N gives an Anosov diffeomorphism f of the compact homogeneous space N/G. The map f is called an Anosov automorphism, and the space N/G is called a nilmanifold. Every known example of an Anosov diffeomorphism is, up to dynamical equivalence, an Anosov automorphism of a nilmanifold, or is finitely covered by one. www.math.utah.edu /~margalit/maxdehn/aut05/101305.html   (221 words)

 Randrianarivony, Maharavo; Brunnett, Guido; Schneider, Reinhold : Constructing a diffeomorphism between a trimmed ... This document has two objectives: decomposition of a given trimmed surface into several four-sided subregions and creation of a diffeomorphism from the unit square onto each subregion. We aim at having a diffeomorphism which is easy and fast to evaluate. Some theoretical criterion pertaining to the control points of the internal curves is proposed and proved so as to ensure that the resulting Gordon patch is a diffeomorphism. www.mathematik.tu-chemnitz.de /preprint/2003/SFB393_20.html   (387 words)

 Michor, Peter, Description of Research   (Site not responding. Last check: 2007-11-07) 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. The resulting theory is easy to understand and to apply and leads to the desired result that the group of real analytic diffeomorphisms of a compact manifold is indeed a real analytic Lie group, and opens the way to apply the real analytic theory in the general theory of manifolds of mappings. www.mat.univie.ac.at /~michor/self-est.html   (3906 words)

 Archived Weblog Entry - 02/15/2004: "Diffeomorphism- Today's Topology / Manifolds subject" According to MathWorld(1) "A diffeomorphism is a map between manifolds which is differentiable and has a differentiable inverse." -- Believe it or not, we think that's how Buckaroo Banzai goes from one dimension to another.... Diffeomorphisms and Noncommutative Analytic Torsion (Memoirs of the American Mathematical Society, 673) by John Lott Dynamical Properties of Diffeomorphisms of the Annulus and of the Torus (Smf/Ams Texts and Monographs, V. by Patrice Le Calvez www.llpoh.org /archives/00000923.html   (291 words)

 Amazon.com: The Structure of Classical Diffeomorphism Groups (Mathematics and Its Applications): Books: A. Banyaga   (Site not responding. Last check: 2007-11-07) The method of the proof is generalized to symplectic and volume-preserving diffeomorphisms. Various cohomology classes connected with the flux are defined on the group of diffeomorphisms. The main results on the structure of diffeomorphism groups are applied to showing that classical structures are determined by their automorphism groups, a contribution to the Erlanger Program of Klein. www.amazon.com /exec/obidos/tg/detail/-/0792344758?v=glance   (523 words)

 A. Norton: Diffeomorphism of Space   (Site not responding. Last check: 2007-11-07) These stills are part of a short animation exploring diffeomorphism in three dimensions. This animation is part of an effort to explore possible connections between the spin of fundamental particles, and the geometry of space-time. The demonstration by a visual model of what is partially involved in diffeomorphism was valuable to me, an amateur who is trying to learn some of the applications of quantum concepts to models of mind. oldsite.vislab.usyd.edu.au /gallery/mathematics/diffeo/diffeo.html   (157 words)

 Up To Diffeomorphism We mentioned in the previous section that diffeomorphically equivalent sets can be assigned the same topology, but from the standpoint of a physical theory it isn't self-evident which diffeomorphism is the right one (assuming there is one) for a particular set of physical entities, such as the events of spacetime. A mapping of this kind is called a diffeomorphism, and two sets are said to be equivalent up to diffeomorphism if there is such a mapping from one to the other. However, as noted above, there are infinitely many systems of coordinates that are not diffeomorphically equivalent, so the limitation to equivalent systems up to diffeomorphism can also be seen as quite restrictive. www.mathpages.com /rr/s9-02/9-02.htm   (965 words)

 Roland Steinbauer's Homepage Applying algebras of generalized functions in a genuinely geometrical context demands diffeomorphism invariance of the respective construction. This property, however, is clearly indispensable if one wants to construct analoga of the local theory in intrinsic terms on differentiable manifolds. As a necessary ingredient they had to (re-)introduce infinite dimensional calculus into their theory which they claimed to be diffeomorphism invariant. www.mat.univie.ac.at /~stein/research/research_ndg.php   (960 words)

 Computations in Riemann Geometry - Curvature Invariants We will define a diffeomorphism as a coordinate transformation which is differentiable to all orders and whose inverse exists and is likewise differentiable to all orders. This "passive" viewpoint is most convenient when doing computations on computers, and underscores the notion that diffeomorphism invariance in a physical theory such as General Relativity is equivalent to saying that physics should not depend on the coordinate system we use to describe it. This is actually a pair of diffeomorphisms, each covering one half of the x-y plane for any value of z (and t), excluding the origin. www.rwc.uc.edu /koehler/crg/invariants.html   (2367 words)

 string theory, general relativity, diffeomorphism invariance The diffeomorphism times Weyl\ninvariance of the 2 dimensional worldsheet theory is exact by\nrequirement of consistency.\n\nYou do expect (and get) additional corrections to Einstein\'s equation\nat order \\$\\alpha\'^2\\$ and beyond. diffeomorphism and Weyl invariance of the 2 dimensional worldsheet These\nnew symmetries - and the corresponding new physical phenomena - become\nrelevant especially at the very short distances comparable to l_{string}\nand shorter.\n\nandgt; if it does break diffeomorphism invariance, then is it correct to say\nandgt; string theory merges general relativity with quantum field theory?\n\nNo, it does not break diffeomorphism invariance in any physical way. www.physicsforums.com /showthread.php?t=39976   (6700 words)

 The geometry and analysis of the averaged Euler equations and a new diffeomorphism group   (Site not responding. Last check: 2007-11-07) The geometry and analysis of the averaged Euler equations a new diffeomorphism group on two subgroups of the volume preserving diffeomorphism group of a smooth n-dimensional compact subset We are then able to prove that the limit of zero viscosity for the corresponding viscous equations is a regular limit. www.math.ucdavis.edu /~shkoller/normal/normal.html   (155 words)

 Gerard 't Hooft at Kitpi25 - Page 3 In other words, I reckon the Larsson thread is based on a misconception at the very start, besides which it seems more concerned with the work of string theorists (like Helling, Policastro) and string-related writing than with the main LQG canon. In the subsequent sub-sections, we constructed a natural, diffeomorphism covariant representation of the quantum analog of this holonomy flux algebra. The basic idea of the procedure used to solve the diffeomorphism constraint is rather simple: One averages the kinematical states with the action of the diffeomorphism group to obtain physical states. www.physicsforums.com /showthread.php?t=46806&page=3   (3040 words)

 n-TRANSITIVITY OF CERTAIN DIFFEOMORPHISM GROUPS (ResearchIndex)   (Site not responding. Last check: 2007-11-07) If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies. It is shown that some groups of diffeomorphisms of a manifold act n- transitively for each finite n. Let M be a connected smooth manifold of dimension dim M 2. citeseer.ist.psu.edu /22452.html   (299 words)

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