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# Topic: Closed manifold

 Manifold - Wikipedia, the free encyclopedia Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces. Manifolds need not be connected (all in "one piece"): a pair of separate circles is also a topological manifold. A finite cylinder is a manifold with boundary. en.wikipedia.org /wiki/Manifold   (4997 words)

 Closed set - Wikipedia, the free encyclopedia The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. The unit interval [0,1] is closed in the real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the real numbers. The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces. en.wikipedia.org /wiki/Closed_set   (575 words)

 Closed manifold - Wikipedia, the free encyclopedia In mathematics, a closed manifold, or compact manifold, is a manifold that is compact as a topological space. In contexts where manifold includes manifolds with boundary, a closed manifold is defined a compact manifold without boundary (whereas a compact manifold may have a boundary). By the basic properties of compactness, a closed manifold is the disjoint union of a finite number of connected closed manifolds. en.wikipedia.org /wiki/Closed_manifold   (170 words)

 [No title] Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold. Orbifolds are manifolds with singularities such as reflection surfaces, where they resemble manifolds with boundary, and cone lines, where they are modelled (in the direction perpendicular to the cone line) by a cone with an angle of 360/n degrees for some n. PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons. www.ornl.gov /ortep/topology/defs.txt   (5717 words)

 [No title] For such a manifold, the set of closed geodesics is dense in the set of all geodesics. The class of Anosov manifolds is opposite in some sense to the class of homogeneous Riemannian manifolds because the isometry group of an Anosov manifold is discrete \cite{Es}. In the case of an $n$-dimensional Anosov manifold $M$, the stable and unstable distributions are defined on the manifold $\Omega M$ of unit tangent vectors. www.ma.utexas.edu /mp_arc/html/papers/98-412   (4479 words)

 Manifolds Manifolds form a collection of familiar of geometric objects encountered in the study of calculus such as, curves, surfaces, 3-dimensional regions, etc. Roughly speaking, a manifold is a subset which locally looks like Euclidian space, or Euclidian half-space. An example of a compact, disconnected one dimensional manifold is a union of two disjoint circles. Manifolds obtained by the Proposition 1.46, are familiar constructions. www.math.uiowa.edu /~roseman/tom/tom/node3.html   (4066 words)

 [No title] As F-G is continuous, E is closed, and thus C is contained in E and is closed as well. Well, then, C is a non-empty, open and closed subset of a connected space M, hence has to be all of M. This forces E=M too, so that F(x)=G(x) for all x. Of course you are rigth that the statement is false for noncompact manifolds. www.math.niu.edu /~rusin/known-math/94/analytic.mfl   (1109 words)

 [No title] Calabi-Yau manifolds, as well as their moduli spaces, have many interesting properties, such as the symmetries in the numbers forming the Hodge diamond of a compact Calabi-Yau manifold. A p-dimensional closed manifold may or may not be a boundary of a (p + 1)-dimensional closed manifold. The Hodge decomposition theorem states that for a compact manifold, there is a one-to-one correspondence between a closed form and a harmonic form, which indicates the existence of a zero mode. www.geocities.com /jefferywinkler/beyondstandardmodel9.html   (3355 words)

 Ken Richardson's Publications and Preprints The Euler characteristic of a smooth, closed manifold may be defined as the alternating sum of the dimensions of the cohomology groups; we define the basic Euler characteristic of a smooth foliation to be the alternating sum of the dimensions of the basic cohomology groups. I show that given a compact Riemannian manifold on which a compact Lie group acts by isometries, there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. In contrast to the asymptotic expansion of the ordinary heat kernel of a Riemannian manifold, the nature of the expansion at x in M may depend on x, and the coefficients of the powers of t are not necessarily integrable. faculty.tcu.edu /richardson/pubs.htm   (2094 words)

 [No title] Introduction Throughout the paper the term "symplectic manifold" means a closed symplectic manifold (M, !) such that the cohomology class [!] 2 H2(M; R) lies in the integral lattice H2(M)=tors. By the definition, a symplectically aspherical manifold is a symplectic manifold whose symplectic form is symplectically aspherical. Examples of such manifolds were given in [G2 ] as some 4-dimensional closed manifolds obtained as branched coverings. hopf.math.purdue.edu /Ibanez-Rudyak-Tralle/aspherical.txt   (3988 words)

 A Layman Looks into the Closed $3$-manifold.   (Site not responding. Last check: 2007-10-07) The conjecture is this: a closed simply-connected $3$-manifold is homeomorphic to $S^3$. Boundary of a topological space is the set of all points that lie at extreme ends (that is not a standard term, though). Torus is not simply-connected; a loop through the finger-hole of a coffee-mug cannot be shrunk to a point without breaking either the cup or the loop. www.geocities.com /kummini/maths/poincare.html   (623 words)

 [No title]   (Site not responding. Last check: 2007-10-07) For example, one can use dynamical methods to exhibit closed geodesics on a Riemannian manifold, since they naturally correspond to the closed orbits of the manifold's geodesic flow. This field started with Mostow's rigidity theorem: the homotopy type of a closed manifold of constant negative curvature determines its isometry type (except in dimension 2). Continuing fruitful interactions between these fields then led to significant advances in the study of locally symmetric spaces, manifolds of nonpositive curvature, and dynamical systems, in particular the study of actions of "large" groups. www.math.lsa.umich.edu /~spatzier/research.html   (323 words)

 Boris Kruglikov (University of Tromsoe)   (Site not responding. Last check: 2007-10-07) be a Riemannian metric on a closed manifold As a corollary we conclude that a rationally hyperbolic closed manifold does not admit two geodesically equivalent Riemannian metrics, which are strictly non-proportional somewhere. Morover, in the non-simply connected case a manifold admitting such a pair of metrics can be covered by the product of a rationally-elliptic manifold and a torus. www.math.psu.edu /dynsys/DW2004/abstracts2004/node14.html   (120 words)

 Hyperset theory   (Site not responding. Last check: 2007-10-07) An n-set containing itself is considered to be closed in n+1 dimensional space. Definition of N-Dimensional Hyperset: An entity, corresponding to an N-Dimensional Manifold, whose surface elements are (N-1)-Dimensional Hypersets. This is a closed N-manifold in N+1 dimensional space. sweb.uky.edu /~jcscov0/hyperset.htm   (534 words)

 Medicine Bottle -- Recommendations and Resources   (Site not responding. Last check: 2007-10-07) A device used to close the mouth of a bottle is called a bottle cap. A make-shift mail method after stranding on a deserted island is a message in a bottle: current may bring it to a shore where the message is read so that a rescue operation can be started. It is closely related to the Möbius strip and embeddings of the real projective plane such as Boy's surface. www.becomingapediatrician.com /health/96/medicine-bottle.html   (1480 words)

 PbRetFAQs I have read that using the overflows are bad practice for closed loops because of limited real estate for holes in the bottom - especially on the AGA tanks --- and depending on how they are plumbed, they can exceed the capacity of the of the overflow. Closed loops as stated before are tapped into the tank. And the bottom of the manifold would just be straight pipe with 30 - 1/8” holes in it to aerate the substrate and behind the live rock. www.wetwebmedia.com /pbretfaqs.htm   (10906 words)

 Citebase - On the geometry of closed G2-structure Citebase - On the geometry of closed G2-structure We extend it in another direction proving that a compact G2-manifold with closed fundamental form and divergence-free Weyl tensor is a G2-manifold with parallel fundamental form. This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=G/L for which the fundamental 2-form is d d-bar closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and expli... www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0306362   (1250 words)

 Reidemeister torsion in circle-valued Morse theory   (Site not responding. Last check: 2007-10-07) We show that by suitably counting closed orbits and flow lines between critical points of the gradient of a circle-valued Morse function on a manifold, one recovers a form of topological Reidemeister torsion. On a three-manifold with positive first Betti number, we conjecture that a finer version of the Morse theory invariant is equal to the Seiberg-Witten invariant, by analogy with Taubes' Seiberg-Witten = Gromov'' theorem in four dimensions. In the three dimensional case, combining this result with the conjecture in [1], we obtain a formula for the full Seiberg-Witten invariant, which was conjectured by Turaev. math.berkeley.edu /~hutching/pub/rt.html   (529 words)

 Citebase - Loop homology algebra of a closed manifold Citebase - Loop homology algebra of a closed manifold The loop homology of a closed orientable manifold M of dimension d is the ordinary homology of the free loop space M At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0203137   (909 words)

 Plan of Research   (Site not responding. Last check: 2007-10-07) Two pseudo-Riemannian metrics on one manifold are called geodesically equivalent, if their geodesics coincide as unparameterised curves. My ultimate goal is to understand completely the geometry and topology of closed manifolds admitting geodesically equivalent metrics. 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)

 [No title] It is known that closed symplectic manif* *olds can violate all these properties (in contrast with the case of Kaehler manifo* *lds). Recall that a symplectically aspherical manifold is a sy* *m- plectic manifold (M, !) such that !ß2(M) = 0, i.e. We say that a closed smooth manifold M is flexible, if M pos- sesses a continuous family of symplectic forms !t, t 2 [a, b], such that hk(M, * *!a) 6= hk(M, !b) for some k. hopf.math.purdue.edu /Ibanez-Rudyak-Tralle-Ugarte/HomotopySymplecticKahler.txt   (4328 words)

 PbRetFAQ2 Tunze recirculation pump from sump (Master Recirculation Pump (1073.030), this has a 1" (25mm) output so I was planning a 1" pipe (hard) from this to a bulkhead in the base of the tank (inside weir area) sized for the 1" to pass through. Only problem is that closing off certain outlets does not make an awful difference to my manifold flow, it's strong at the pump end and weak at the other. This is a "closed" as in not open (to the air) recirculation system for moving water around with a motive force (pump) located outside an aquatic system. www.wetwebmedia.com /pbretfaq2.htm   (14850 words)

 STS-67 At 1:05pm EST the port Payload Bay Door was closed with the remaining door closed and latched by 1:08pm EST. Flight controllers worked with the crew to close the manifold that supplies oxidizer and fuel to that jet, which effectively stopped the leak. The doors were briefly closed again while residual propellant downstream of the closed manifold dissipated, but are now open and all scheduled operations have resumed. www.astronautix.com /flights/sts67.htm   (5459 words)

 [No title] Basic course in symplectic geometry: A symplectic manifold $(M,\omega)$ is a manifold $M$ along with a closed 2-form $\omega$ which is everywhere nondegenerate (i.e., $\omega_x:T_x M \times T_x M \to \R$ is an everywhere nondegenerate skew-symmetric pairing); thus $\omega$ induces fiberwise isomorphism between $T_x M$ and $T_x^* M$. It's covariantly constant because $g$ and $J$ are (by definition and assumption, respectively); all covariantly constant things are closed by basic differential geometry. Then the tangent bundle is a symplectic vector bundle (i.e., a vector bundle with a smoothly varying symplectic structure on each fiber). www.math.uchicago.edu /~msmukler/sep30   (1231 words)

 Symmetries, isometries and length spectra of closed hyperbolic three-manifolds, Craig D. Hodgson, Jeffrey R. Weeks Previously known algorithms to compute the symmetry group of a cusped hyperbolic three-manifold and to test whether two cusped hyperbolic three-manifolds are isometric do not apply directly to closed manifolds. But by drilling out geodesics from closed manifolds one may compute their symmetry groups and test for isometries using the cusped manifold techniques. To do so, one must know precisely how many geodesics of a given length the closed manifold has. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1048515809   (122 words)

 James Tauber : Closed Manifolds I said previously that we were ready to state the Poincaré Conjecture, but there's one more bit of terminology I want to get out of the way and that is closed manifold. A closed manifold is a compact manifold without a boundary. All connected one-dimensional closed manifolds are homeomorphic to the circle (or 1-sphere). jtauber.com /blog/2005/08/20/closed_manifolds   (419 words)

 UM Mathematics: Faculty-Detail I am especially intrigued by problems which mix ideas and techniques from these areas. Continuing fruitful interactions between these fields then led to significant advances in the study of locally symmetric spaces, manifolds of nonpositive curvature, and dynamical systems, in particular the study of actions of "large" groups such as SL(n,R) or SL(n,Z) for n>2. The study of homogeneous dynamical systems has been very prominent in the last two decades, and has opened up applications to other areas of mathematics, for example number theory and spectral theory. www.math.lsa.umich.edu /people/facultyDetail.php?id=174   (270 words)

 AMCA: Special Manifolds and Shape Fibrator Properties by Violeta Vasilevska   (Site not responding. Last check: 2007-10-07) B, from any closed, orientable PL (n+k)-manifold M to a simplicial triangulated manifold B, such that each point inverse has the same homotopy type as N, are approximate fibrations. Also we introduce a particular type of manifold called special manifold - closed manifold with a non-trivial fundamental group for which all self maps with non-trivial normal images on 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/p/c/76.htm   (190 words)

 Citebase - A Lower Bound of The First Eigenvalue of a Closed Manifold with Positive Ricci Curvature A Lower Bound of The First Eigenvalue of a Closed Manifold with Positive Ricci Curvature We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature. [11] D. Yang, Lower bound estimates on the first eigenvalue for compact manifolds with positive Ricci curvature, Pacic Journal of Mathematics, 190(1999), 383-398. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0406437   (417 words)

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