Manifold Definition and 335 Threads

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures (texture) to coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
The study of manifolds requires working knowledge of calculus and topology.

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  1. S

    Does a compact manifold always have bounded sectional curvature?

    Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...
  2. mnb96

    How to find the manifold associated with a Lie Group?

    Hello, I have troubles formulating this question properly. So I will explain it through one example. If we consider the Lie group R=SO(2) of rotations on the plane, we know that we can find a manifold on which the group SO(2) acts regularly: this manifold is the unit circle in ℝ2. In fact...
  3. mnb96

    Example of differentiable manifold of class C^1

    Hello, I read from several sources the statement that the set of points M\inℝ2 given by (t, \, |t|^2) is an example of differentiable manifold of class C1 but not C2. Is that true? To be honest, that statement does not convince me completely, because in order for M to be a manifold, we should...
  4. B

    Gravitation as varying density of a space-propertime manifold?

    Originally I asked on another thread whether the GR effects on light can be described as light passing through a medium of varying density- so exerting its effects by refraction. A.T. kindly posted the following links: http://www.newscientist.com/article/dn24289#.UlsOR-B3Zmh...
  5. Philosophaie

    Formulating x^n Coordinate System for Non-Rectangular/Spherical Riemann Manifold

    I want to be able to formulate x^{n} coordinate system. x^{n} =(x^{1}, x^{2}, x^{3}, x^{4}) How do you do this when the Riemann Manifold is not rectangular or spherical? Also how do you differentiate with respect to "s" in that case. \frac{dx^n}{ds}
  6. TrickyDicky

    Are singularities part of the manifold?

    Mod note: Posts split off from https://www.physicsforums.com/showthread.php?p=4468795 Hi, WN, might the OP be referring to GR instead of SR, more specifically to the expanding FRW universe in which it is impossible to even consider the notion of exansion without agreeing about an "everywhere...
  7. F

    Is General Relativity Dependent on the Existence of a Manifold?

    I understand that SR and GR should not be dependent on the coordinate system. But does GR depend on the existence of a manifold? As I recall, GR is formulated with a metric tensor. And tensors are only defined on manifold. Or is there a manifold independent GR?
  8. Mandelbroth

    Plausibility of a Discrete Point Manifold

    Consider a discrete set of ##k## points. First, is it a manifold? I know that a manifold is a topological space that contains a neighborhood homeomorphic to Euclidean space for each point. Can we just consider each point's neighborhood to be a set containing only that point? Second, would the...
  9. F

    Are second derivative symmetric in a Riemannian manifold?

    Hi all! I was wondering if \partial_1\partial_2f=\partial_2\partial_1f in a Riemannian manifold (Schwartz's - or Clairaut's - theorem). Example: consider a metric given by the line element ds^2=-dt^2+\ell_1^2dx^2+\ell_2^2dy^2+\ell_3^2dz^2 can we assume that...
  10. atyy

    Is a Vector Space a Manifold? Exploring the Relationship in Special Relativity

    I'm asking because I think of Minkowski space as a manifold with a Riemannian metric. However, I've also seen treatments in which an event in spacetime is chosen as origin, and special relativity treated as a vector space given the choice of origin. Does this mean that a vector space is a...
  11. M

    Copper or Brass exhaust manifold.

    Hello All, I am drawing up a design for a water-cooled exhaust manifold for a small engine. It would be a simple design of an exhaust exit pipe mounted to a flange, with a tubular copper heat exchanger liquid filled coil wrapped around the exit pipe right near the cylinder. The copper would...
  12. M

    Why the figure of eight is not a manifold?

    Why the figure of eight is not a manifold? I have read somewhere that if we remove the crossing point than the the figure of eight becomes disconnected, but by removing one point in \mathbb{R}^2 it's still connected. Is there any other proof without removing the crossing point?
  13. F

    Exploring Manifolds: Understanding Locally Euclidean Spaces

    I have a few questions about manifolds. According to Wikipedia.org, A topological manifold is a locally Euclidean Hausdorff space. First question, does locally Euclidean mean that there are a continuous set of points in order that they can be mapped to an infinite set of coordinates in the...
  14. U

    Understanding Manifolds and Riemann Manifolds: Applications and Explanation

    Can anyone explain the concept of manifold and Reimanni manifold in plain language ?? And what are its applications?? Thanks
  15. TrickyDicky

    Is It More Precise to Define Manifolds as Topological Vector Spaces?

    In general terms a manifold can be defined simply as a topological space locally resembling Euclidean space with the resemblance meaning homeomorphic to Euclidean space, plus a couple of point set axioms that avoid certain "patological" manifolds and that some authors reserve for the definition...
  16. M

    How Can the Laplacian Relate to Einstein Manifolds?

    I am researching a hypothesis and looking for anyone who is familiar with differential topology (specifically Einstein manifolds). I have access to the Besse book Einstein Manifolds but am also looking for any open questions in differential topology that I am not aware of. I am attempting to...
  17. S

    Integrating a differential form on a manifold without parametric equations

    I am sure that this can be done, but I haven't been able to figure it out, Is there a way to integrate a differential form on a manifold without using the parametric equations of the manifold? So that you can just use the manifold's charts instead of parametric equations? If you a function...
  18. T

    Power Series for Volume of Balls in Riemannian Manifold

    I'm trying to work out the following problem: Find the first two terms of the power series expansion for the volume of a ball of radius r centered at p in a Riemannian Manifold, M with dimension n. We are given that Vol(B_r(p)) = \int_S \int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t...
  19. W

    Mapping Torus of a Manifold is a Manifold.

    Hi, All: I'm trying to show that the Mapping torus of a manifold X is a manifold, and I'm trying to see what happens when X has a non-empty boundary B. Remember that the mapping torus M(h) of a space X by the map h is constructed like this: We start with a homeomorphism h:X-->X (we...
  20. P

    Calculus on Mobius Band: Let L_{θ} Be the Line

    Let L_{\theta} be the line passing through the point z(\theta)=(\cos\theta,\sin\theta) on the unit circle at angle \theta and with slope \frac{1}{2}\theta. The mobius band is M={(z,v):z\in S^{1},v\in L_{\theta}} my question is , why M is a mobius band?
  21. S

    Confused as to what constitutes a manifold?

    I am having trouble getting a set definition of what constitutes a manifold for example , I have the real plane R^2, and the sphere s = {(x,y,z)|(x,y,z)£R^2, x^2+y^2+z^2=1} Note £, is meant to be "element of". And I have a continuous function f mapping the real plane onto s such that...
  22. M

    How does changing the metric on a manifold affect the shape of the manifold?

    Hi all, I am trying to understand geometric flows, and in particular the Ricci flow. I understand how to calculate the metric tensor from the parametrization of a surface, but I am facing a problems in the concept phase. A metric tensor's purpose is to provide a coordinate invariant...
  23. wolram

    When talking cosmology what exactly is a manifold, is there any

    When talking cosmology what exactly is a manifold, is there any observational evidence that we live in one ?
  24. P

    Need some help on air pressure through a manifold and optimal leak/pressure rate

    I have a small pump system for an aquarium, and have decided to build a small manifold. The total volume of the manifold and the tubing is 2.9828 cu inches. Here is my pump specs: Inflation time < 8s (from 0-200 mmHg in a 100cc tank) Air flow > 70mL/min Max pressure > 360mmHg Leakage max...
  25. Markus Hanke

    Metric of Manifold with Curled up Dimensions

    Would someone here be able to write down for me an example of a metric on a manifold with both macroscopic dimensions, and microscopic "curled up" dimensions with some radius R ? Number of dimensions and coordinates used don't matter. Not going anywhere with this, I am just curious as to how...
  26. Chris L T521

    MHB Manifold Theory: Computing Curvature and Verifying Geometries

    I've posted a bunch of analysis questions as of late. I'm going to change things up a little bit and ask something that involves manifold theory. Here's this week's problem: ----- Problem: (i) Let $\omega$ be a 1-form. Use the structure equations \[\begin{aligned}d\theta^1 &=...
  27. M

    I am not sure - a manifold is locally connected and has countable basis?

    I am not sure -- a manifold is locally connected and has countable basis? There is an Exercise in a book as following : Given a Manifold M , if N is a sub-manifold , an V is open set then V \cap N is a countable collection of connected open sets . I am asking why he put this exercise...
  28. L

    Integer Cohomology of Real Infinite-Dimensional Grassmann Manifold

    I can't seem to find on the web a site that gives the Z cohomology of the infinite dimensional Grassmann manifold of real unoriented k planes in Euclidean space. I am interested in computing the Bockstein exact sequence for the coefficient sequence, 0 -> Z ->Z ->Z/2Z -> 0 to see which...
  29. S

    How do I get the 1st fundamental form on Grassmann Manifold

    Consider G(n,m), the set of all n-dimensional subsapce in ℝ^n+m. We define the principal angles between two subspaces recusively by the usual formula. When I see "Differential Geometry of Grassmann Manifolds by Wong", http://www.ncbi.nlm.nih.gov/pmc/articles/PMC335549/pdf/pnas00676-0108.pdf I...
  30. K

    Manifold Explained in Simple Terms

    Hi there, I find that the term 'manifold' appears in many book of statistical physics or classical mechanics while talking about phase space. I try to search the explanation on online but it is quite abstract and hard to understand what's manifold really refers to. Can anyway explain this a...
  31. S

    What is a Conifold and How is it Related to Orbifold Singularities?

    We know that a Clifford torus is parameterized in 4D euclidean space by: (x1,x2,x3,x4) = (Sin(theta1), Cos(theta1), Sin(theta2), Cos(theta2)) {0<=theta1 and theta2<2pi} Consider that a clifford torus is the immediate result of Circle * Circle Now, have you encountered a similar manifold...
  32. R

    Bifurcations and Center Manifold

    Homework Statement If β=0 the neurone model is \dot{u}= -u \dot{v}= v2 + v - u + \delta If \delta = 1/4 it has critical point (0,-1/2) Transform the system so that the critical point is at the origin so let \bar{v} = v +1/2 and find the equations of motion for (u,\bar{v}) Homework...
  33. D

    Custom Intake Manifold Design for Go-Kart Engine

    I'm new here saw some information i liked here at one point or another and i decided it would be a nice place to join. I am a hgh school senior and next year i plan to go into mechanical engineering. For one of my senior projects I am planning on designing and making a custom intake manifold...
  34. G

    Does the Mystery Manifold theorem exist?

    The theorem supposedly states: A space cannot be bent unless it is a manifold which is embedded in a space of at least one higher dimension. Does anyone know if this theorem actually exists? If it does, I would appreciate a reference to its name and proof.
  35. TrickyDicky

    Is our universe a geodesically complete manifold?

    It's always been my understanding that given the existence of BH singularities and the initial BB singularity our universe couldn't be geodesically complete. But then one of the premises of our cosmology models is that the universe is homogeneous, and all homogeneous manifolds are geodesically...
  36. D

    Quaternion Kaehlerian manifold, definition

    Hello, I am reading the paper of S. Ishihara, Quaternion Kaehlerian manifolds, I need it for understanding of totally complex submanifolds in quaternion Kaehlerian manifolds. I am afraid that I don't understand well the definition of quaternion Kaehler manifold, that is my question is the...
  37. S

    What is the Definition of a Manifold and How Does it Relate to Topology?

    i see the definition of differential manifolds in some book for example, NAKAHARA but what is the definition of manifold in general! and what the definition of topological manifold.
  38. ShayanJ

    Metric: A Property of Manifolds or Coordinate Systems?

    From the things I've read on manifold geometry,metric is a property of the manifold.Maybe you can call it intrinsic. But consider e.g. a euclidean manifold and two coordinate systems on it.say,cartesian and spherical.As you know,the metric for this two coordinate systems is different. So what?Is...
  39. A

    Difference between hilbert space,vector space and manifold?

    Difference between hilbert space,vector space and manifold?? Physically what do they mean? I m really confused imagining them..Explanation with example would help me to understand there application ..THanks in advance
  40. S

    Prove that topological manifold homeomorphic to Euclidean subspace

    Homework Statement Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space. Homework Equations A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood...
  41. W

    What do points on the manifold correspond to in reality?

    In SR the points in Minkowski space correspond to events. I recently read in a GR lecture note that the points on the manifold do NOT correspond to events like in SR (the author even says the points don't have a direct physical meaning). So what do they represent then? And if I continuously...
  42. ShayanJ

    What is the concept of a natural coordinate in manifold geometry?

    I'm just learning manifold geometry and tensor analysis.From the things I've understood till now,an idea came into my mind but I can find it or its negation no where.So I came to ask it here. I can't explain how I deduced this but I think there should be sth like a natural coordinate for a...
  43. M

    What Is the Boundary of a Product Manifold?

    Is it true, that if A and B are oriented manifolds with boundary, having dimensions n and m respectivelly, then the boundary of A\times B is \partial(A\times B)=\partial A\times B + (-1)^n A\times \partial B? If not, then what can we say about the boundary of product manifolds? Could someone...
  44. R

    Topological dimension of the image of a smooth curve in a manifold

    Here is the situation I am concerned with - Consider a smooth curve g:[0,1] \to M where M is a topological manifold (I'd be happy to assume M smooth/finite dimensional if that helps). Let Im(g) be the image of [0,1] under the map g . Give Im(g) the subspace topology induced by...
  45. B

    Is R mod 2pi a Compact Manifold?

    Hi, Why R mod 2pi is a Compact Manifold? Isn't this like a real line which is not compact? How should we prove it using a finite sub-cover for this manifold? bah
  46. V

    Automotive Intake Manifold Turbo flat 4

    Just a introduction to my project, I have designed and building a 69 Volkwagen Bug with a Version 8 subaru Sti Engine, The pan is built with tube braces and coil over suspension, I built a tall back bone frame for strength and for the rest of the frame to be lower for ride height and suspension...
  47. R

    Why Does a=dB Imply ∫a=0 on Compact Manifolds?

    I'm looking at prop 19.5 of Taylor's PDE book. The theorem is: If M is a compact, connected, oriented manifold of dimension n, and a is an n-form, then a=dB where B is an n-1 form iff the ∫a over M is 0. I'm trying to understand why a=dB implies ∫a = 0. If M has no boundary, than this...
  48. L

    Curvature as the source of a field on a two dimensional Riemannian manifold

    I am looking for some intuition into a way of looking at the Gauss curvature on a surface that describes it as the divergence of a potential function - at least locally. I am not sure exactly what the intuition is - but this way of looking at things seems suggestive. Any insight is welcome. In...
  49. M

    Air flow in inlet manifold affecting engine performance

    i want to model air flow in the inlet manifold using CFD ,how do i predict how flow variations and other parameters affect engine performance ? i know how it affects but how do i show it virtually
  50. M

    Is the Levi-Civita connection unique for a given manifold?

    Hi all, To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric. Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics...
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