Other uses Manifold disambiguation File Triangles spherical geometry .jpg thumb 300px The sphere surface of a ball mathematics ball is a two dimensional manifold since it can be represented by a collection ... manifold is a subset of Euclidean space which is locally the graph of a smooth perhaps vector valued function. A more general topological manifold can be described as a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold ... on into high dimensional space . More formally, every point of an n dimensional manifold has a neighborhood ... of a manifold may be more complicated. For example, any point on the usual two dimensional surface ... in the language of topology , they are not homeomorphic. The structure of a manifold is encoded ... in terms of the relatively well understood properties of simpler spaces. For example, a manifold ... metric that allows one to measure distance s and angle s. Symplectic manifold s serve as the phase ... dimensional Lorentzian manifold s model space time in general relativity . This seems out of place ... list of manifolds . Motivational examples Circle Main Circle File Circle with overlapping manifold ... manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small ... a transition map . File Circle manifold chart from slope.svg right thumb Figure 2 A circle manifold ... show that the circle is a manifold, but they do not form the only possible atlas. Charts need ... circles. In this example we see that a manifold need not have any well defined notion of distance ... need not be closed manifold closed thus a line segment without its end points is a manifold. And they are never countable, unless the dimension of the manifold is 0. Putting these freedoms together ... the circle is a differentiable manifold . It is also smooth and analytic because the transition ... of more specialized types of manifold. For example, the circle has a notion of distance between ... more details
Unreferenced date June 2008 In mathematics , an analytic manifold is a topological manifold with analytic function analytic transition maps. Every complex manifold is an analytic manifold. Category Structures on manifolds Category Manifolds topology stub zh ... more details
wiktionarypar manifold TOC right A manifold is an abstract mathematical space which, in a close up view, resembles the spaces described by Euclidean geometry. Manifold may also refer to Science and engineering Exhaust manifold , an engine part which collects the exhaust gases from multiple cylinders into one pipe Hydraulic manifold , a component used to regulate fluid flow in a hydraulic system, thus controlling the transfer of power between actuators and pumps Inlet manifold or intake manifold , an engine part which supplies the air or fuel air mixture to the cylinders Manifold scuba , in a scuba set, connects two or more diving cylinders, so that one diving regulator is fed by all the cylinders Manifold chemistry , a piece of apparatus used in chemistry to manipulate gases Vacuum manifoldManifold magazine , magazine of the University of Warwick mathematical community Media Manifold Records , a record label The Manifold Trilogy by science fiction author Stephen Baxter Manifold magazine , magazine of the University of Warwick mathematical community Software Manifold System , a company that develops a geographic information system GIS software package Places River Manifold , a river in England The Manifold Way , a foot and cycle path which follows the old route of the Leek and Manifold Valley Light Railway, Staffordshire, England People Sir Walter Manifold 1849 1928 an Australian grazier and politician John Manifold 1915 1985 an Australian poet and critic. disambig Category Surnames da Manifold de Manifold es Manifold eo Dukto fa ja tr Manifold ... more details
Noref date December 2009 seealso Classification of manifolds Point set In mathematics , a closed manifold is a type of topological space , namely a compact space compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold. The simplest example is a circle , which is a compact one dimensional manifold. Other examples of closed manifolds are the torus and the Klein bottle . As a counterexample, the real line is not a closed manifold because it is not compact. A Disk mathematics disk is a compact two dimensional manifold, but is not a closed manifold because it has a boundary. Compact manifolds are, in an intuitive sense, finite . By the basic properties of compactness, a closed manifold is the disjoint union of a finite number of connected closed manifolds. One of the most basic objectives of geometric topology is to understand what the supply of possible closed manifolds is. All compact topological manifolds can be embedded into math mathbf R n math for some n , by the Whitney embedding theorem . Contrasting terms A compact manifold means a manifold that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary the boundary may be empty . By contrast, a closed manifold is compact without boundary. An open manifold is a manifold without boundary with no compact component. For a connected manifold, open is equivalent to without boundary and non compact , but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and the line is non compact, but is not an open manifold, since one component the circle is compact. The notion of closed manifold is unrelated with that of a closed set . A disk with its boundary is a closed set, but not a closed manifold. Use in physics The notion of a Shape of the Universe closed universe can refer to the universe being a closed manifold but more likely refers to the universe being a manifold ... more details
Unreferenced date December 2007 A hydraulic manifold is a component that regulates fluid flow between pump s and actuator s and other components in a hydraulic system. It is like a Electric switchboard switchboard in an electrical circuit because it lets the operator control how much fluid flows between which components of a hydraulic machinery. For example, in a backhoe loader a manifold turns on or shuts off or diverts flow to the telescopic arms of the front bucket and the back bucket. The manifold is connected to the levers in the operator s cabin which the operator uses to achieve the desired manifold behaviour. A manifold is composed of assorted hydraulic valves connected to each other. It is the various combinations of states of these valves that allow complex control behaviour in a manifold. See also Block and bleed manifold Hydraulics Category Fluid mechanics Category Hydraulics Manifold, Hydraulic tech stub ... more details
Multiple issues confusing May 2009 context May 2009 orphan May 2009 Manifold integration is a combined concept of manifold learning and data integration , or an extension of manifold learning for multiple measurements. Various manifold learning methods have been developed. However, they consider only one dissimilarity matrix corresponding to one kernel matrix, which represents one manifold of the data set . In practice, however, we use multiple sensors at a time, and each sensor generates data set on one manifold. In such a case, manifold integration is a desirable task, combining these dissimilarity matrices into a compromise matrix that faithfully reflects multiple sensory information on one integrated manifold. For more information, see ref name HChoi2008aaai H. Choi, S. Choi and Y. Choe, Manifold Integration with Markov Random Walks, in Proc. 23rd Association for the Advancement of Artificial Intelligence AAAI 08 , Chicago, Illinois, July 13 17, 2008. ref Notes No footnotes date May 2009 references DEFAULTSORT Manifold Integration Category Artificial intelligence ... more details
unreferenced date December 2010 In dynamical systems , a branch of mathematics , an invariant manifold is a topological manifold that is invariant under the action of the dynamical system. An example is the stable manifold . See also Hyperbolic set Invariant subspace topology stub Category Dynamical systems ru ... more details
Expert subject Mathematics date February 2009 In mathematics , the term simplicial manifold commonly refers to either of two different types of objects, which combine attributes of a simplex with those of a manifold . Briefly a simplex is a generalization of the concept of a triangle into forms with more, or fewer, than two dimensions. Accordingly, a 3 simplex is the figure known as a tetrahedron . A manifold is simply a space which appears to be Euclidean space Euclidean following the laws of ordinary geometry, or more generally a flat Pseudo Riemannian manifold Pseudo Riemannian space in a given Neighborhood mathematics local neighborhood , though it can be greatly more complicated overall. The combination of these concepts gives us two useful definitions. A manifold made out of simplices A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold . This can mean simply that a neighborhood mathematics neighborhood of each vertex i.e. the set of simplices that contain that point as a vertex is homeomorphic to a n dimensional ball mathematics ball . A manifold made from simplices can be locally flat, or can approximate a smooth curve, just as a large geodesic dome appears relatively flat over small areas, and approximates a Sphere hemisphere over its full extent. One can generalize this concept to more dimensions and other kinds of curved surfaces which makes it useful in various kinds of Computer simulation simulations . This notion of simplicial manifold is important in Regge calculus and Causal dynamical triangulation s as a way to discretize spacetime by triangulation triangulating it. A simplicial manifold ... manifold is also a simplicial object in the category mathematics category of manifold s. This is a special case of a simplicial space in which, for each n , the space of n simplices is a manifold. For example, if G is a Lie group , then the nerve category theory simplicial nerve of G has the manifold ... more details
green invariant manifolds of saddle node equilibrium point In mathematics, the center manifold of an equilibrium ... is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium points of dynamical systems is to linearize the system ..., which gives rise to the stable manifold. The stable manifold attracts orbits close to it. Similarly, eigenvalues with positive real part yield the unstable manifold, which repels orbits close ... whose real part is zero, then these give rise to the center manifold. The behavior near the center manifold is not determined by the linearization and thus more difficult to study. Center manifolds ... manifold. Definition Let math dot textbf x f textbf x math be a dynamical system with equilibrium .... Corresponding to the linearized system, the nonlinear system has invariant manifold s, consisting of orbits of the nonlinear system. There is an invariant manifold tangent to the stable subspace and with the same dimension this manifold is the stable manifold . Similarly, the unstable manifold is tangent to the unstable subspace, and the center manifold is tangent to the center subspace. ref harvtxt ... are all precisely zero, rather than just real part zero, then a center manifold is often called a slow manifold . The center manifold theorem The center manifold theorem states that if is C sup ... r sup stable manifold, a unique C sup r sup unstable manifold, and a not necessarily unique C sup r &minus 1 sup center manifold. ref harvtxt Guckenheimer Holmes 1997 , Theorem 3.2.1 ref In example ... Systems year 2011 ref . The general theory currently only applies when the center manifold itself ... an infinite dimensional center manifold. ref cite journal author A.J. Roberts journal J. Austral. Math. Soc. series B pages 480 500 title The application of centre manifold theory to the evolution ... Center manifold and the analysis of nonlinear systems As the stability of the equilibrium correlates ... more details
In mathematics , a homology manifold or generalized manifold is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory . Definition A homology G manifold without boundary of dimension n over an abelian group G of coefficients is a locally compact topological space X with finite G cohomological dimension such that for any x &isin X , the homology groups math H p X,X x, G math are trivial unless p n , in which case they are isomorphic to G . Here H is some homology theory, usually singular homology. Homology manifolds are the same as homology Z manifolds. More generally, one can define homology manifolds with boundary, by allowing the local homology groups to vanish at some points, which are of course called the boundary of the homology manifold. The boundary of an n dimensional first countable homology manifold is an n &minus 1 dimensional homology manifold without boundary . Examples Any topological manifold is a homology manifold. An example of a homology manifold that is not a manifold is the suspension of a homology sphere that is not a sphere. If X × Y is a topological manifolds, then X and Y are homology manifolds. References springer id H h047800 title Homology manifold author E. G. Sklyarenko W. J .R. Mitchell, http links.jstor.org sici?sici 0002 9939 28199010 29110 3A2 3C509 3ADTBOAH 3E2.0.CO 3B2 R Defining the boundary of a homology manifold , Proceedings of the American Mathematical Society , Vol. 110, No. 2. Oct., 1990 , pp. 509 513. topology stub Category Algebraic topology Category Generalized manifolds ... more details
for the equipment used to connect two air cylinders in SCUBA diving Manifold scuba In the subject of manifoldmanifold theory in mathematics , if math M math is a manifoldManifold with boundary manifold with boundary , its double is obtained by gluing two copies of math M math together along their common boundary. Precisely, the double is math M times 0,1 sim math where math x,0 sim x,1 math for all math x in partial M math . Although the concept makes sense for any manifold, and even for some non manifold sets such as the Alexander horned sphere , the notion of double tends to be used primarily in the context that math partial M math is non empty and math M math is compact space compact . Doubles bound Given a manifold math M math , the double of math M math is the boundary of math M times 0,1 math . This gives doubles a special role in cobordism . Examples The sphere n sphere is the double of the ball mathematics n ball . In this context, the two balls would be the upper and lower hemi sphere respectively. More generally, if math M math is closed, the double of math M times D k math is math M times S k math . Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of the M bius strip is the Klein bottle . If math M math is a closed, orientability oriented manifold and if math M math is obtained from math M math by removing an open ball, then the connected sum math M mathrel M math is the double of math M math . The double of a Mazur manifold is a homotopy sphere homotopy 4 sphere . topology stub Category Differential topology Category Manifolds ... more details
An algebraic manifold is an algebraic variety which is also a manifold . As such, algebraic manifolds are a generalisation of the concept of smooth curve s and surfaces . An example is the sphere , which can be defined as the zero set of the polynomial nowrap 1 x sup 2 sup y sup 2 sup z sup 2 sup 1, and hence is an algebraic variety. For an algebraic manifold, the ground field will be the real number s or complex numbers in the case of the real numbers, the manifold of real points is sometimes called a Nash manifold . Every sufficiently small local patch of an algebraic manifold is isomorphic to k sup m sup where k is the ground field. Equivalently the variety is Smooth function smooth free from Singular point of an algebraic variety singular points . The Riemann sphere is one example of a complex algebraic manifold, since it is the complex projective line . Examples Elliptic curve s Grassmannian See also Algebraic geometry and analytic geometry References Nash, J. Real algebraic manifolds . 1952 Ann. Math. 56 1952 , 405 421. See also Proc. Internat. Congr. Math., 1950, AMS, 1952 , pp. 516 517. External links http planetmath.org encyclopedia KAlgebraicManifold.html K Algebraic manifold at PlanetMath http mathworld.wolfram.com AlgebraicManifold.html Algebraic manifold at Mathworld http www.mccme.ru ium postscript s99 notes lec 23.ps.gz Lecture notes on algebraic manifolds Category Algebraic varieties Category Manifolds ... more details
Cape Manifold coord 22 41 S 150 50 E ref Gazetteer of Australia name Cape Manifold id 142248 ref is a coastal headland in central Queensland , Australia . It was named by Captain Cook when he saw it from Keppel Bay to the south on 27 May 1770, from the Number of high Hills over it . ref gutenberg no 8106 name Captain Cook s Journal During the First Voyage Round the World ref He spelt it both Manyfold and Manifold in different journal entries, today Manifold is usual. References references Portal Queensland 1stVoyageCookAus Category Headlands of Queensland Manifold Queensland geo stub ... more details
of Y. T. Siu . An important subclass of K hler manifolds are Calabi Yaumanifold s. Properties Harv Deligne Griffiths Morgan Sullivan 1975 showed that all Massey product s vanish on a K hler manifold ...In mathematics , a K hler manifold is a manifold with unitary group unitary structure a G structure U n structure satisfying an integrability condition . In particular, it is a Riemannian manifold , a complex manifold , and a symplectic manifold , with these three structures all mutually compatible. This threefold ... Hermitian manifold . If the Sp structure is integrable but the complex structure need not be , the notion is an almost K hler manifold if the complex structure is integrable but the Sp structure need not be , the notion is a Hermitian manifold . K hler manifolds are named after the mathematician ... geometric generalization of complex Algebraic variety algebraic varieties . Definition A manifold with a Hermitian metric is an almost Hermitian manifold a K hler manifold is a manifold with a Hermitian ... defined as a complex manifold with an additional structure or a symplectic manifold with an additional structure, or a Riemannian manifold with an additional structure . One can summarize the connection ... metric , i is the almost complex manifold almost complex structure , and math omega math is the almost symplectic manifold almost symplectic structure . A K hler metric on a complex manifold ... manifold. The metric on a K hler manifold locally satisfies math g i bar j frac partial 2 K partial z i partial bar z j math for some function K , called the K hler potential. A K hler manifold, the associated ... manifold s for more details. Examples Complex Euclidean space C sup n sup with the standard Hermitian metric is a K hler manifold. A torus C sup n sup a full lattice group lattice inherits ... manifold. Every Riemannian metric on a Riemann surface is K hler, since the condition for to be closed ... submanifold of a K hler manifold is K hler. In particular, any Stein manifold embedded in C sup ... more details
In mathematics , specifically in differential topology , a Kervaire manifold K sup 4 n 2 sup is a piecewise linear manifold of dimension of a manifold dimension 4 n 2 constructed by harvtxt Kervaire 1960 by plumbing together the tangent bundle s of two n sphere 2 n 1 spheres , and then gluing a ball mathematics ball to the result. In 10 dimensions this gives a piecewise linear manifold with no smooth structure . See also Exotic sphere References citation first M. last Kervaire authorlink Michel Kervaire title A manifold which does not admit any differentiable structure journal Comm. Math. Helv. volume 34 year 1960 pages 257 270 url http retro.seals.ch digbib view?did c1 391766&sdid c1 392119 doi 10.1007 BF02565940 mr 0139172 eom id k k055350 first M.A. last Shtan ko title Kervaire invariant eom id d d031010 first M.A. last Shtan ko title Dendritic manifold Category Differential topology Category Manifolds ... more details
Unreferenced stub auto yes date December 2009 Distinguish Manifold vacuum For the type of laboratory glassware Schlenk line In quantum field theory , the vacuum state may be degenerate . Each pure vacuum state generates its own superselection sector . The space of all pure vacuum states often has a manifold structure and is called the vacuum manifold . Vacuum manifolds arise during the process of spontaneous symmetry breaking from a group G to a subgroup H and the corresponding vacuum manifold has to be a realization of G and contain the quotient space G H. In many cases, it would simply be G H, although it could be larger. Not all vacuum manifolds arise due to spontaneous symmetry breaking. Supersymmetric models often contain moduli space s which is another name for the vacuum manifold. In many cases, the vacuum manifold is parameterized by the values of permissible vacuum expectation value s. This is not the case for spontaneous symmetry breaking due to fermion condensate fermion condensation , though. If the vacuum manifold is homotopy homotopically nontrivial, it s possible for there to be topological sector s. DEFAULTSORT Vacuum Manifold Category Quantum field theory Quantum stub ... more details
In mathematics , a complete manifold or geodesically complete manifold is a Pseudo Riemannian manifold pseudo Riemannian manifold for which every maximal inextendible geodesic is defined on math mathbb R math . Examples All compact space compact manifolds and all homogeneous space homogeneous manifolds are geodesically complete. Euclidean space math mathbb R n math , the sphere s math mathbb S n math and the torus tori math mathbb T n math with their usual Riemannian metric s are all complete manifolds. A simple example of a non complete manifold is given by the punctured plane math M mathbb R 2 setminus 0 math with its usual metric . Geodesics going to the origin cannot be defined on the entire real line. Path connectedness, completeness and geodesic completeness It can be shown that a finite dimensional Connected space Path connectedness path connected Riemannian manifold is a complete metric space if and only if it is geodesically complete. This is the Hopf Rinow theorem . This theorem does not hold for infinite dimensional manifolds. The example of a non complete manifold the punctured plane given above fails to be geodesically complete because, although it is path connected, it is not a complete metric space any sequence in the plane converging to the origin is a non converging Cauchy sequence in the punctured plane. References Citation last1 O Neill first1 Barrett title Semi Riemannian Geometry publisher Academic Press isbn 0 12 526740 1 year 1983 . See chapter 3, pp. 68 . DEFAULTSORT Complete Manifold Category Riemannian geometry Category Manifolds ... more details
Unreferenced date December 2009 In topology , a branch of mathematics , a manifold M may be decomposed ... those pieces are and how they are put together to form M . Manifold decomposition works in two directions one can start with the smaller pieces and build up a manifold, or start with a large manifold ..., it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify 3 manifold s and also in proving the higher dimensional Poincar conjecture . The table below is a summary of the various manifold decomposition techniques. The column labeled M indicates what kind of manifold can be decomposed the column labeled How it is decomposed indicates how, starting with a manifold, one can decompose it into smaller pieces the column labeled The pieces ... pieces are combined to make the large manifold. Expand list date August 2008 frame border border ... 3 manifold has a unique triangulation, unique up to common subdivision. In dimension 4, not all ... torus decomposition Irreducible mathematics Irreducible , orientable , Compact space compact 3 manifold ... manifold Prime decomposition Essentially surfaces and 3 manifold . The decomposition is unique when the manifold is orientable. Cut along embedded Sphere Topology spheres then Adjunction space union by the trivial ... manifold s Connected sum Heegaard splitting Closed manifold closed , orientable 3 manifold s Two Handlebody ... Handle decomposition Any compact smooth manifold smooth n manifold and the decomposition is never unique ... hierarchy Any Haken manifold Cut along a sequence of incompressible surfaces Ball mathematics Topology 3 balls Disk decomposition Certain Compact space compact , orientable 3 manifold s Sutured manifold Suture the manifold, then cut along special surfaces condition on boundary curves and sutures... Ball mathematics Topology 3 balls Open book decomposition Any Closed manifold closed orientable 3 manifold a Link knot theory link and a family of 2 manifold s with Manifold boundary that link Trigenus ... more details
Infobox River river name Manifold image name Manifold valley from Thors Cave .jpg image size 250 caption Manifold Valley from Thor s cave origin South of Buxton near Axe Edge mouth Confluence with the River Dove, Derbyshire Dove br coord 53 3 0 N 1 47 5 W type river region GB display inline,title progression River Dove, Derbyshire Dove River Trent Trent Humber North Sea basin countries length convert 12 mi km elevation mouth elevation discharge watershed Image Dry river Manifold.jpg right thumb A bridge over a dry River Manifold, near Grindon, Staffordshire Grindon . The River Manifold is a river in Staffordshire , England. It is a tributary of the River Dove, Derbyshire River Dove which also flows through the Peak District , forming the boundary between Derbyshire and Staffordshire . The The Manifold rises just south of Buxton near Axe Edge , at the northern edge of the White Peak , known for its limestone beds. It continues for convert 12 mi before it joins the Dove. For part of its course, it runs underground except when in spate , from Wetton Mill to Ilam, Staffordshire Ilam . During this section it is joined by its major tributary, the River Hamps . Villages on the river include ... English Anglo Saxon manig fald many folds , referring to its meander s. Manifold Way See main article Manifold Way The Manifold Way is an convert 8 mi adj on long distance footpath from Hulme End to Waterhouses, Staffordshire Waterhouses , along the former route of the narrow gauge 2 6 Leek and Manifold ... handed over the trackbed to Staffordshire County Council, it is tarmacked throughout. The Manifold ... which overlooks the confluence with the River Hamps. Mining in the Manifold valley The Manifold valley ... that minerals were mined around the Manifold valley thousands of years ago. Nowadays there is little ... engine house still remain. See also Leek and Manifold Valley Light Railway Rivers of the United ... Staffordshire Past Track project historical photos DEFAULTSORT Manifold Category Rivers of Derbyshire ... more details
unreferenced date November 2011 In mathematics , a toric manifold is a topological analogue of toric variety in algebraic geometry . It is an even dimensional manifold with an effective smooth action of n dim compact torus which is locally standard with the orbit space a simple convex polytope . The aim is to do combinatorics on the quotient polytope and obtain information on the manifold above. For example the Euler characteristic , cohomology ring of the manifold can be described in terms of the polytope. The Michael Atiyah Atiyah and Victor Guillemin Guillemin Shlomo Sternberg Sternberg theorem This theorem states that the image of the moment map of a Hamiltonian toric action is the convex hull of the set of moments of the points fixed by the action. In particular, this image is a convex polygon Category Structures on manifolds Category Manifolds Category Topology geometry stub ... more details
In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes . Concepts Parallel move is used to extend an i cell to i 1 cell. In other words, if A and B are two i cells and A is a parallel move of B, then A,B is an i 1 cell. Therefore, k cells can be defined recursively. Basically, a connected set of grid points M can be viewed as a digital k manifold if 1 any two k cells are k 1 connected, 2 every k 1 cell has only one or two parallel moves, and 3 M does not contain any k 1 cells. See also Digital geometry Digital topology Topological data analysis Topology Discrete mathematics References cite paper author Chen, L. and Zhang, J. title Digital manifolds an intuitive definition and some properties publisher Proceedings on the second ACM symposium on Solid modeling and applications, Montreal, Quebec, Canada, year 1993 pages 459 460 Category Digital topology Category Digital geometry ... more details
In mathematics , a Hadamard manifold , named after Jacques Hadamard &mdash sometimes called a Cartan Hadamard manifold , after lie Cartan &mdash is a Riemannian manifold M , g that is complete space complete and simply connected space simply connected , and has everywhere non positive sectional curvature . Examples The real line R with its usual metric is a Hadamard manifold with constant sectional curvature equal to 0. Standard n dimensional hyperbolic space H sup n sup is a Hadamard manifold with constant sectional curvature equal to &minus 1. See also Cartan Hadamard theorem Hadamard space References cite arxiv last Mourougane first Christophe title Interpolation in non positively curved K hler manifolds date 7 Mar 2001 eprint math 0103045 class math.CV Category Riemannian manifolds topology stub fr Vari t de Hadamard ... more details
In mathematics , a topological manifold is a topological space can even be a Hausdorff space separated ... an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifold s, for example, are topological manifolds equipped with a differential structure . Every manifold has an underlying topological manifold, obtained simply by forgetting the additional structure. An overview of the manifold concept is given in that article. This article ..., to some connected open subset of E sup n sup . A topological manifold is a locally Euclidean ... conditions, are discussed below. In the remainder of this article a manifold will mean a topological manifold. An n manifold will mean a topological manifold such that every point has a neighborhood ... manifold X there is a unique integer n such that X is an n manifold. This integer is called the dimension of X. Examples Euclidean space R sup n sup is the prototypical n manifold. Any discrete space is a 0 dimensional manifold. A circle is a 1 manifold. A torus is a 2 manifold or surface as is the Klein bottle . The n dimensional n sphere sphere S sup n sup is a compact space compact n manifold. The n dimensional n torus torus T sup n sup the product of n circle s is a compact n manifold. Projective ... manifolds. Real projective space RP sup n sup is a n dimensional manifold. Complex projective space CP sup n sup is a 2 n dimensional manifold. Quaternionic projective space HP sup n sup is a 4 n dimensional manifold. Manifolds related to projective space include Grassmannian s, flag manifold s, and Stiefel manifold s. Lens space s are a class of manifolds that are quotient space quotient s of odd ... open subset of an n manifold is a n manifold with the subspace topology . If M is an m manifold and N is an n manifold, the product topology product M × N is a m n manifold. The disjoint union ... more details
In complex geometry , a Hopf manifold harv Hopf 1948 is obtained as a quotient of the complex vector space with zero deleted math Bbb C n backslash 0 math by a Group action free action of the group math Gamma cong Bbb Z math of integer s, with the generator math gamma math of math Gamma math acting by holomorphic Contraction mapping contractions . Here, a holomorphic contraction is a map math gamma Bbb C n mapsto Bbb C n math such that a sufficiently big iteration math gamma N math puts any given compact subset math Bbb C n math onto an arbitrarily small neighbourhood of 0. Two dimensional Hopf manifolds are called Hopf surface s. Examples In a typical situation, math Gamma math is generated by a linear contraction, usually a diagonal matrix math q cdot Id math , with math q in Bbb C math a complex number, math 0 q 1 math . Such manifold is called a classical Hopf manifold . Properties A Hopf manifold math H Bbb C n backslash 0 Bbb Z math is diffeomorphic to math S 2n 1 times S 1 math . It is non K hler manifold K hler . Indeed, the first cohomology group of H is odd dimensional. By Hodge decomposition , odd cohomology of a compact K hler manifold are always even dimensional. Hypercomplex structure Even dimensional Hopf manifolds admit Hypercomplex manifold hypercomplex structure . The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperk hler manifold hyperk hler . References Citation last1 Hopf first1 Heinz author1 link Heinz Hopf title Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 publisher Interscience Publishers, Inc., New York id MathSciNet id 0023054 year 1948 chapter Zur Topologie der komplexen Mannigfaltigkeiten pages 167 185 eom id H h110270 first L. last Ornea Category Complex manifolds ... more details
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