ManifoldSystem is a geographic information system GIS software package developed by manifold.net that runs only on Microsoft Windows . ManifoldSystem handles both vector and raster data, includes spatial SQL, a built in Internet Map Server IMS , and other general GIS features. ManifoldSystem has an active user community with a mailing list and online forums. History The development team for Manifold ... and development of ManifoldSystem was launched. The company soon changed its name to Manifold Net to match the new product s name. ManifoldSystem releases ManifoldSystem was first sold in January ... Enterprise editions of ManifoldSystem allowing collaboration by teams using shared components. The 5.x series also introduced a new spatial SQL and fuzzy logic using the Decision Support System. Releases ... increases when using NVIDIA CUDA reduce the time required for complex surface calculations in Manifold from minutes to seconds. For its use of NVIDIA CUDA ManifoldSystem won the 2008 Geospatial Innovator ... System. http galvarezhn.cartesianos.com category manifold Spanish site with topics about Manifold ... companies Category GIS software de ManifoldSystem es Manifold software ..., Release 4.50 emphasized general GIS features of broader interest and emerged as Manifold ... Map Service WMS image servers using Manifold as a client and extended IMS support to include OGC WMS when using Manifold as a server. 7.00 further extended IMS to include OGC WFS T and image server ... as well. 7.00 also introduced the Manifold Image Server interface API, allowing users to create modules that enable usage within Manifold of image servers such as Virtual Earth , Google Maps , Yahoo , Ask.com ... 2008 spatial product available in 2007. 8.00 also introduced a Manifold written spatial extender for Microsoft SQL Server 2005 as well as generic spatial DBMS capability from Manifold enabling spatial ... the Manifold.net website. http lists.directionsmag.com discussion list.php?f 29 the Manifold ... more details
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 ... y sup 2 sup 1, where the Cartesian coordinate system y coordinate is positive indicated ... 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 ... 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 ManifoldSystem , 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
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
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
green invariant manifolds of saddle node equilibrium point In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point 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 ... manifold. Definition Let math dot textbf x f textbf x math be a dynamical system with equilibrium point x . The linearization of the system at the equilibrium point is math dot textbf x A textbf x .... 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 ..., 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 ... 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 ... more details
and associated high manifold vacuum, this system is simple and almost foolproof. In fact, if the throttle ...Distinguish Vacuum manifoldManifold vacuum , or engine vacuum in an internal combustion engine is the difference in air pressure between the engine s Inlet manifold intake manifold and Earth s atmosphere . Manifold vacuum is an effect of a piston s movement on the Stroke engine Induction stroke induction stroke and the choked flow through a throttle in the intake manifold of an engine. It is a measure ... in the engine. In some engines, the manifold vacuum is also used as an Automobile ancillary power auxiliary power source to drive engine accessories. Manifold vacuum should not be confused with venturi ... is controlled by the amount of fuel supplied to the cylinder, and so has no throttle as such. Manifold ... of the engine, and the density of the intake stream in the intake manifold. In most applications ... . Restricting the input flow reduces the density and hence pressure in the intake manifold , reducing ... , as the engine must pump material from the low pressure intake manifold into the exhaust manifold ... , ambient air is free to fill the intake manifold, increasing the pressure filling the vacuum . A carburetor or fuel injection system adds fuel to the airflow in the correct proportion, providing energy to the engine. When the throttle is opened all the way, the engine s air induction system ... . Supercharger s and turbocharger s can Turbocharger Pressure increase boost boost manifold pressure ... a MAP sensor which measures the manifold absolute pressure to control fuel flow. Manifold vacuum vs. venturi vacuum Manifold vacuum is caused by a different effect than venturi vacuum , which is present ... mass flow through the engine and hence the total power output . Manifold vacuum may also be ported ... valve opens, the opening is effectively below it, and the opening sees nearly the full manifold vacuum. Ported vacuum is often used for distributors and emissions items. Manifold vacuum in cars ... more details
Infobox Book name Manifold Space image Image Manifold Space UK.jpg 200px Cover to the first edition author Stephen Baxter cover artist country Great Britain language English language English series Manifold Trilogy Manifold genre Science fiction novel publisher Voyager Books Voyager UK & br Del Rey Books US release date 6 October 2000 media type Print Hardcover Hardback & Paperback pages 464 pp First hardback edition size weight isbn ISBN 0 00 225771 8 First hardback edition oclc 43718099 preceded by Manifold Time followed by Manifold Origin Image Manifold Space.jpg thumb right 150px Cover to the US edition Manifold Space is a science fiction book by author Stephen Baxter , first published in the United Kingdom in 2000, then released in the United States in 2001. It is the second book of the Manifold Trilogy Manifold series and examines another possible solution to the Fermi paradox . Although it is in no sense a sequel to the first book it contains a number of the same characters, notably protagonist Reid Malenfant, and similar artifacts. The Manifold series contains four books, Manifold Time , Manifold Space , Manifold Origin , and Phase Space book Manifold Phase Space . Plot summary Extraterrestrial life Alien activity is discovered in a Kirkwood gap the aliens are identified as self ... notes in an earlier speech A target system, we assume, is uninhabited. We can therefore program for massive and destructive exploitation of the system s resources, without restraint, by the Von ... in on the changing solar system over many centuries. Eventually, it is revealed that in this version ... Manifold Books. Madeleine Meacher Other main character, former aviator pilot of the futuristic ... name Publishers Weekly cite journal date January 8, 2001 title MANIFOLD Book Review journal Publishers ... last Cassada first Jackie date January 1, 2001 title Manifold Book Review journal Library Journal volume 126 issue 1 pages 162 ref Awards Manifold Space was named by Library Journal as one of the best ... 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
The set of all possible configurations of a system is modelled as a manifold, and this manifold s cotangent bundle describes the phase space of the system. Any real valued differentiable function, H , on a symplectic manifold can serve as an energy function or Hamiltonian . Associated to any Hamiltonian ...In mathematics , a symplectic manifold is a smooth manifold , M , equipped with a Closed and exact differential ... to Hamilton s equations . The Hamiltonian vector field defines a flow on the symplectic manifold, called ... , Hamiltonian flows preserve the volume form on the phase space. Definition A symplectic form on a manifold ... of , namely d , is identically zero. A symplectic manifold consists a pair M , , of a manifold M and a symplectic form . Assigning a symplectic form to a manifold M is referred to as giving M a symplectic structure . Linear symplectic manifold There is a standard linear model, namely a symplectic ... natural geometric notions of submanifold of a symplectic manifold. symplectic submanifolds potentially ..., i.e. each tangent space is an isotropic subspace of the ambient manifold s tangent space. Similarly ... of maximal dimension, namely half the dimension of the ambient symplectic manifold. Lagrangian submanifolds ... of a symplectomorphism in the product symplectic manifold nowrap 1 M × M , × &minus .... Lagrangian fibration A Lagrangian fibration of a symplectic manifold M is a fibration where all ... of a symplectic manifold K , given by an Immersion mathematics immersion nowrap 1 i L K ... back of sub 2 sub by . Special cases and generalizations A symplectic manifold endowed with a metric tensor metric that is Almost complex manifold Compatible triples compatible with the symplectic form is an almost K hler manifold in the sense that the tangent bundle has an almost complex structure ... of a Poisson manifold . The definition of a symplectic manifold requires that the symplectic form be non degenerate everywhere, but if this condition is violated, the manifold may still be a Poisson ... 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
intake system. main Variable length intake manifold Variable Length Intake Manifold VLIM is an internal ...Refimprove date April 2009 Original research date June 2011 In automotive engineering , an inlet manifold or intake manifold is the part of an engine that supplies the fuel air mixture to the cylinder engine cylinder s. The word manifold comes from the Old English word manigfeald from the Anglo Saxon ..., an exhaust manifold collects the exhaust gas es from multiple cylinders into one pipe. Image ... function of the intake manifold is to evenly distribute the combustion mixture or just air in a direct ... piston engine , a partial vacuum lower than atmospheric pressure exists in the intake manifold. This manifold ... auxiliary systems power assisted brake s, emission control devices, cruise control , ignition system ignition advance, windshield wiper s, power window s, ventilation system valves, etc. This vacuum ... crankcase ventilation system . This way the gases are burned with the fuel air mixture. The intake manifold has historically been manufactured from aluminum or cast iron, but use of composite ... fuel injectors spray fuel droplets into the air in the manifold. Due to electrostatic forces some of the fuel will form into pools along the walls of the manifold, or may converge into larger droplets ... head porting File Manifold comparison.jpg right thumb 340px Comparison of a stock intake manifold ... to a custom built one used in competition bottom . In the custom built manifold, the runners to the intake ... efficiency of the engine s fuel air intake. The design and orientation of the intake manifold ... begins to equalize with lower pressure air in the manifold. Due to the air s inertia, the equalization will tend to oscillate At first the air in the runner will be at a lower pressure than the manifold. The air in the manifold then tries to equalize back into the runner, and the oscillation repeats ... manifold design feature, the exhaust manifold design, as well as the exhaust valve opening time ... more details
wikify date February 2011 In mathematics , the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold . One of the main methods of simplifying dynamical systems , is to reduce the dimension of the system to that of the slow manifold&mdash center manifold theory rigorously justifies the modelling. ref J. Carr, Applications of centre manifold .... Correspondingly, the nonlinear system has invariant manifold s, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces. There is an invariant manifold tangent to the slow subspace and with the same dimension this manifold is the slow manifold . Stochastic slow ... system of five equations in five variables to explore the notion of a slow manifold of quasi geostrophic flow ref E. N. Lorenz, On the existence of a slow manifold, Journal of Atmospheric Science ... system that is exponentially close to the Lorenz system for which there is a good slow manifold. Eliminate ... dynamics on the slow manifold of the atmosphere oceanic dynamics, ref R. Camassa, On the geometry of an atmospheric slow manifold, Physica D , 84 357&ndash 397, 1995. ref and is thus crucial to forecasting with a climate model . Definition Consider the dynamical system math frac d vec x dt vec f ... . Then the linearization of the system at the equilibrium point is math frac d vec x dt A vec x quad ... manifold three of the subspaces are the stable, unstable and center subspaces corresponding ... , 110 559&ndash 588, 1998. ref Examples Simple case with two variables The coupled system in two ... 2 math has the exact slow manifold math y x 2 math on which the evolution is math dx dt x 3 math . Apart from exponentially decaying transients, this slow manifold and its evolution captures all solutions ... pm i math . Hence there exists a three dimensional slow manifold surrounded by fast waves in the math X math and math Z math variables . Lorenz later argued a slow manifold did not exist ref E. Lorenz ... more details
Novel article template owned by Wikipedia WikiProject Novels Infobox book name Manifold Time title orig translator image Image Mainfold time Stephen Baxter , alternate coverart.jpg center 200px cover to the first edition author Stephen Baxter cover artist Tony and Daphine Hallas Science Photo Library country Great Britain language English language English series Manifold Trilogy Manifold genre Science fiction publisher Voyager Books Voyager UK & Del Rey Books USA release date 1999 media type Print Hardcover Hardback & Paperback pages 456 p. UK hardback edition & 456 p. UK paperback edition size weight isbn ISBN 0 00 225768 8 UK paperback edition & ISBN 0 00 651182 1 UK paperback edition oclc 41258602 preceded by followed by Manifold Space Manifold Time is a 1999 science fiction novel by Stephen Baxter . It is the first of Baxter s Manifold trilogy the others being Manifold Space and Manifold Origin , although the books can be read in any order because the series takes place in a Parallel universe fiction multiverse . The book was nominated for the 2000 Arthur C. Clarke Award . ref name WWE 2000 cite web url http www.worldswithoutend.com books year index.asp?year 2000 title 2000 Award Winners & Nominees work Worlds Without End accessdate 2009 08 03 ref Plot summary Time is set on Earth, the inner part of the Solar System and various other universes onwards from the 21st century. The novel covers a wide range of topics, including the Doomsday argument , Fermi paradox , genetic engineering , and humanity s extinction. Image Mainfold time Stephen Baxter , alternate coverart.jpg left thumb 150px cover to the first edition The book begins at the end of space and time, when the last descendants of humanity face an infinite but pointless existence. Due to proton decay the physical universe has collapsed, but some form of intelligence has survived by embedding itself into a lossless ... References reflist External links http www.worldswithoutend.com novel.asp?ID 590 Manifold Time at Worlds ... more details
In mathematics , a Stein manifold in the theory of several complex variables and complex manifold s is a complex submanifold of the vector space of n complex number complex dimensions. The name is for Karl Stein mathematician Karl Stein . Definition A complex manifold math X math of complex dimension math n math is called a Stein manifold if the following conditions hold math X math is holomorphically convex, i.e. for every Compact space compact subset math K subset X math , the so called holomorphic ... theorem of Behnke and Stein 1948 asserts that X is a Stein manifold. Another result, attributed ... is a Stein manifold. It can be shown quite easily that every closed complex submanifold of a Stein manifold is a Stein manifold, too. The embedding theorem for Stein manifolds states the following Every Stein manifold math X math of complex dimension math n math can be embedded into math mathbb C 2 n 1 math by a biholomorphic proper map . These facts imply that a Stein manifold is a closed complex ... Riemann surface is a Stein manifold if and only if it is not compact. This can be proved using a version of the Runge theorem for Riemann surfaces, due to Behnke and Stein. Every Stein manifold math ... holomorphic functions defined on all of math X math which form a local coordinate system when restricted to some open neighborhood of math x math . Being a Stein manifold is equivalent to being a complex strongly pseudoconvex manifold . The latter means that it has a strongly pseudoconvex or plurisubharmonic ... 1911 . The function math psi math invites a generalization of Stein manifold to the idea of a corresponding ... affine varieties . Stein manifolds are in some sense dual to the elliptic manifold s in complex ... that a Stein manifold is elliptic if and only if it is fibrant object fibrant in the sense of so called holomorphic homotopy theory . Relation to smooth manifolds Every compact smooth manifold of dimension ..., Ann. of Math. 148, 1998 619 693. ref Every closed smooth 4 manifold is a union of two Stein 4 ... 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
of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable. A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus . Any manifold can be described ... differentiable chart. Note that a differentiable manifold as it stands does not have any metric ... terms, a differentiable manifold is a topological manifold with a globally defined differential structure . Any topological manifold can be given a differential structure locally by using the homeomorphism .... He motivated the idea of a manifold by an intuitive process of varying a given object in a new ... principle . A modern definition of a 2 dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surface s. ref See H. Weyl 1955 . ref The widely accepted general definition of a manifold ... Definition A presentation of a topological manifold is a second countable Hausdorff space Hausdorff ... of patching together pieces of a space to make a manifold the manifold produced also contains the data of how it has been patched together. However, different atlases patchings may produce the same manifold, and, on the converse, a manifold does not come with a preferred atlas. And, thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines ... include the following. A differentiable manifold is a topological manifold equipped with an equivalence ... class C sup k sub manifold is a topological manifold with an atlas whose transition maps are all k times continuously differentiable. A smooth manifold or C sup sup manifold is a differentiable manifold for which all the transition maps are smooth function smooth . That is, derivatives of all orders exist so it is a C sup k sup manifold for all k . An equivalence class of such atlas atlases is said to be a smooth structure . An analytic manifold , or C sup sup manifold is a smooth ... more details
Image Manifold scuba.jpg thumb right Schematic representation of double diving cylinders connected with isolating manifold. In scuba diving a manifold is used to connect two diving cylinder s tanks with breathing gas , providing a greater amount of gas for longer dive times and greater safety due to redundancy. Diving with two or more cylinders is associated with technical diving . Function Longer and deeper dives require a greater amount of breathing gas, in turn requiring either a larger cylinder or multiple cylinders. A large tank up to 18 Litre L is very heavy and tends to raise the diver s center of mass , making them unbalanced in water. A simple multiple tank configuration called separate doubles consists of two mechanically attached but otherwise unconnected tanks with two complete Diving regulator regulator sets with a total of two Diving regulator The first stage first stage regulators and two Diving regulator Types of last stage second stage regulators . The function of a manifold is to connect the air supplies of two cylinders called doubles or Twins , allowing the diver to breathe simultaneously from both. The manifold is a metal tube usually made of aircraft grade brass http www.omsdive.com valves.html with two cylinder connectors, two first stage regulator connectors and three valves, as shown in the figure above. The left and the right valves allow to disconnect the corresponding first stage regulators, leaving the entire gas supply to be used through the remaining ... Cylinder to manifold connection malfunction, though rare, can result in an extremely violent gas loss ... for the remainder of the dive. Advantages Compared to the alternatives, the manifold offers the following ... in case of a regulator or manifold malfunction a standard procedure can be used to minimize the gas loss. The diver can localize the malfunction and isolate it from the functioning system by closing the necessary valves. Criticism A manifold is a single point of failure for the gas supply, especially ... 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
vectors X 0 . In a system of local coordinates on the manifold M given by n real valued functions ... and the differential geometry of surfaces , a Riemannian manifold or Riemannian space M , g is a real differentiable manifold M in which each tangent space is equipped with an Inner product space ... geometric notions on a Riemannian manifold, such as angle s, lengths of curve s, area s or volume ..., again in a way that was intrinsic to the manifold and not dependent upon its embedding ... of space. Overview The tangent bundle of a smooth manifold M assigns to each fixed point ... product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a Smooth ... one could define Riemannian manifold as a metric space which is Isometry isometric to a smooth submanifold ... as metric spaces Usually a Riemannian manifold is defined as a smooth manifold with a smooth Section ... curve in the Riemannian manifold M , then we define its length L in analogy with the example ... space connected Riemannian manifold M becomes a metric space and even a intrinsic metric length ... the manifold is compact set compact , any two points x and y can be connected with a geodesic whose ... . Riemannian metrics Let M be a differentiable manifold of dimension n . A Riemannian metric on M ... vector s at each point of M . Relative to this coordinate system, the components of the metric ... d x i otimes mathrm d x j. math Endowed with this metric, the differentiable manifold M , g is a Riemannian manifold . Examples With math frac partial partial x i math identified with e sub i sub 0 ... Euclidean metric . Let M , g be a Riemannian manifold and N M be a submanifold of M . Then the restriction .... Let M , g sup M sup be a Riemannian manifold, h M sup n k sup &rarr N sup k sup be a differentiable ... more details
In mathematics , a Cauchy Riemann CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a Real number complex vector space , or more generally modeled on an Edge of the wedge theorem edge of a wedge . Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L , or in other words a subbundle of the complexified tangent bundle C T M T M C such that math L,L subseteq L math L is formally Frobenius ... is called a CR structure on the manifold M . The abbreviation CR stands for Cauchy Riemann or https ... the antiholomorphic functions. In the complex manifold holomorphic coordinates math T 1,0 ... of a system of smooth real valued functions F sub 1 sub 0, F sub 2 sub 0, ..., F sub k sub 0. Suppose that this system has maximal rank, in the sense that the differentials ... stronger than needed to apply the implicit function theorem in particular, M is a manifold of real dimension 2 n &minus k . We say that M is an embedded CR manifold of CR codimension k . In most applications, k 1, in which case the manifold is said to be of hypersurface ... that k 1 so that the CR manifold is of hypersurface type , unless otherwise noted. The Levi form Let M be a CR manifold of hypersurface type with single defining function F 0. The Levi ... n sup if and only if it is strictly pseudoconvex as a CR manifold. See plurisubharmonic function s and Stein manifold . Abstract CR structures An abstract CR structure on a manifold M of dimension ... form and pseudoconvexity Suppose that M is a CR manifold of hypersurface type. The Levi form ... associated to . Generalizations of the Levi form exist when the manifold is not of hypersurface type ... manifold is the real math 2n 1 math sphere as a submanifold of math mathbb C n 1 math . The bundle math ... Cr Manifold Category Smooth manifolds ... 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
Encyclopedia
top
