What is Manifold: Definition and 327 Discussions

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.

View More On Wikipedia.org
Recent contents Top users
  1. P

    Null geodesic in 2 dimensional manifold

    I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).
  2. M

    Metrics on a manifold, gravity waves, gauge freedom

    Suppose I have a manifold. I say that it can support a certain configuration of gravity field described by metric tensor \gamma. I do not write \gamma_{\mu\nu}, because that would immediately imply a reference to a particular chart. A tensor field, however, exists on a manifold unrelated to this...
  3. R

    What is the difference between a manifold and a metric space?

    I always thought one could define a manifold as a collection of points with a distance function or metric tensor. But in a layperson's book by Penrose, he defined a manifold as a collection of points with a rule for telling you if a function defined on the manifold is smooth. He says this is...
  4. M

    Closed Orbit of a Flow on a Manifold

    Homework Statement Let γ be a closed orbit of the flow φ on the manifold M and suppose there exists T>0 and X0 є γ such that φT(X0) = X0. Prove that φT(X) = X for every X0 є γ. Furthermore locate two closed orbits γ1 and γ2 and positive periods T1 and T2 for the flow of r ̇=r(r-1)(r-2); θ...
  5. K

    General relativity, integration over a manifold exercise

    Homework Statement The problem I am facing is 2.9 in Sean Carroll's book on general relativity (Geometry and Spacetime) I should note that I am not studying this formally and so a full solution would not be unwelcome, though I understand that forum policy understandably prohibits it. The...
  6. L

    Differential Topology: 1-dimensional manifold

    Homework Statement Given S1={(x,y) in R2: x2+y2=1}. Show that S1 is a 1-dimensional manifold. Homework Equations The Attempt at a Solution Let f1:(-1,1)->S1 s.t. f1(x)=(x,(1-x2)1/2). This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S1. I was...
  7. Spinnor

    Use 3D lattice and vector field(s) to represent curved manifold?

    I would like to try and map a small piece of a 3 dimensional curved manifold using a flat 3 dimensional space, and a vector field. Will the following work? Take a 3 dimensional cube of size a*a*a that lies in a 6 dimensional space, R^6, with coordinates x1,x2,x3,x4,x5,x6. Let this cube be a...
  8. V

    The Lagrangian/Hamiltonian dynamics of a particle moving in a manifold

    I posted this problem on the Classical Mechanics Subforum last week but have not received many responses - hopefully someone can help here as I've spent hours racking my brain, trying to work this out! Homework Statement There is a particle of mass 'm' moving in a manifold with the following...
  9. V

    The Lagrangian/Hamiltonian dynamics of a particle moving in a manifold

    Hi I am trying to work through the solution to the attached problem (see attachments). Now, I can't understand several things in the solution: The Lagrangian in question is: L={\frac{m}{2}}{g_{ij}(x)}.{\dot{x^{i}}{\dot{x^{j}} 1)is g_{ij} a matrix with diag(-1,1,1,1), ie. the metric tensor...
  10. G

    Manifold: what's the meaning of this name?

    Dear all, I've always wondered where the name "manifold" comes from? Any idea? Thanks, Goldbeetle
  11. Spinnor

    Are there geodesics for Calabi–Yau manifold?

    Say I sit at some point P in a Calabi-Yau manifold. Are there geodesics which start from P and return to P? Are there "geodesics" which start from P and return to P but may make a "side trip first"? Is the number of geodesics which start at P and end at P infinite or finite and does that...
  12. R

    Differentiable manifold not riemannian

    I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric. thanks
  13. J

    Spacetime manifold: initial condition or result of GR?

    I apologize for the poorly worded title. Let me try to explain my question better. A scientific theory must be predictive to be useful. Since we only know what happened in the past, the global topology of spacetime cannot be an input to the theory. Given space-like slices/"chunk" of the...
  14. S

    What is Tubes Assembly and Manifold for engines?

    Dear all, I am so sorry for my stupid questions. Currently, I am looking for documents (lecture notes) on tubes assembly and manifold for aircraft engine. What are those tubes assembly and manifold? What are those characteristic? I did quite a lot google search, however, I only found any good...
  15. Z

    Another manifold definition deficiency?

    Conventional manifold definition refers to the neighbor of every point having a Euclidean space description. http://en.wikipedia.org/wiki/Manifold" But if most manifolds have additional property of some curvature, then won't such manifold definition actually be describing a tangent space i.e...
  16. Y

    Vanishing first betti number of kaehler manifold with global SU(m) holonomy

    Hi, I have the following question. In "Joyce D.D. Compact manifolds with special holonomy" I read on page 125 that a compact Kaehler manifold with global holonomy group equal to SU(m), has vanishing first betti number, or more specifically vanishing Hodge numbers h^(1,0)= h^(0,1) = 0...
  17. Y

    Holonomy of compact Ricci-flat Kaehler manifold

    Hi, I have come across the following apparent contradiction in the literature. In "Joyce D.D., Compact manifolds with special holonomy" I find on page 125 the claim that if M is a compact Ricci-flat Kaehler manifold, then the global holonomy group of M is contained in SU(m) if and only if the...
  18. A

    Prove that x^4+y^4=1 is a manifold

    Can someone help me out whit this I proved for the circle, but I can't prove it for this -Prove that x^4+y^4=1 (the set of points) is a manifold For the circle it was easy, but how do I take on this case? Thanks
  19. F

    Manifold Questions: Particle Interaction vs Element Separation

    Is there a difference between a manifold that is a result of particle interactions and say a system of elements where there is no interactions? E.g. Two particles interact with one another by exchanging force carriers and as a result they create a manifold in the form of a sphere. Isn't this...
  20. S

    The Hawking 4D closed manifold

    Hi, I am struggling to understand Stephen Hawking's view of the universe as a 4D closed manifold. In a recent interview, I believe he had this to say: What I don't understand is how this theory is compatible with the scientific observation that the universe is expanding? I have 2 questions: 1)...
  21. B

    Why is the tangent space of a lie group manifold at the origin the lie algebra?

    Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me. I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this...
  22. B

    Volume of parametrized manifold

    I don't think this is a difficult problem, but I am not sure about what is being asked in the question. I got it from Munkres' Analysis on Manifolds page 193 Q 2. Homework Statement Let A be open in R^k; let f : A-->R be of class C^r; let Y be the graph of f in R^(k+1), parametrized by...
  23. E

    Understanding the Non-Manifold Property of Euclidean Half-Space

    Because of boundary points, I can sort of see intuitively why Euclidean half-space, i.e. {(x_1, ... , x_n) : x_n >= 0} is not a manifold, but is there a simple rigorous argument for why Euclidean half-space is not homeomorphic to an open set of R^n. I do not know too much topology and the...
  24. T

    Arclength on a PseudoRiemann Manifold

    Wikipedia gives a confusing definition of a path's length and I would like some clarity. Let M be a pseudo-Riemann manifold with metric g and let a and b be points in M.If y is a smooth function from R->M where y(0) = a and y(1) = b, then it's length is the integral \int_0^1\sqrt{\pm...
  25. G

    How to say a given space is a manifold?

    How to say a given space is a manifold? The only thing that props in my mind is to check if every open set has a euclidean coordinate chart on it. But, what if the space I am dealing with is not fully understood apriori? As in, how were the spaces of thermodynamic equilibrium states, phase and...
  26. F

    K-dimensional manifold problem

    Homework Statement Suppose X ⊂ R^n is a k-dimensional manifold and Y ⊂ R^p is an l-dimensional manifold. Prove that: X × Y = {[x,y] ∈ R^n × R^p : x ∈ X and y ∈ Y} is a (k+l)-dimensional manifold in R^(n+p). (Hint: Recall that X is locally a graph over a k-dimensional coordinate plane...
  27. quasar987

    A problem about integral curves on a manifold

    I must demonstrate in two ways that if c(t) is an integral curve of a smooth vector field X on a smooth manifold M with c'(t_0)=0 for some t_0, then c is a constant curve. I found one way: If \theta denotes the flow of X, then because X is invariant under its own flow, we have c'(t)=X_{c(t)} =...
  28. S

    Yau's result for the Ricci curvature on Kahler manifold

    Hi, I've been reading through Yau's proof of the Calabi conjecture (1) and I was quite intrigued by the relation R_{i\bar{j}} = - \frac{\partial^2}{\partial z^i \partial \bar{z}^j } [\log \det (g_{s\bar{t}}) ] derived therein. g_{s \bar{t} } is a Kahler metric on a Kahler manifold (I'm...
  29. quasar987

    Connected components of a manifold

    I got this book here that mentions en passant that the connected components of a (topological) manifold are open in the manifold. That's not true in a general topological space, so why does Hausdorff + locally euclidean implies it? I don't see it.
  30. JasonJo

    Tangent bundle of a differentiable manifold M even if M isn't orientable

    This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo. Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a...
  31. snoopies622

    Can a 4D Schwarzschild Manifold be Embedded in 5D Hyperspace?

    Can a 4-dimensional manifold with the Schwarzschild metric be embedded into a flat manifold of 5 (or more if necessary) dimensions? In other words, are there functions of t,r,\theta , \phi and M such that if x_1 = f_1 (t,r,\theta ,\phi ,M) x_2 = f_2 (t,r,\theta ,\phi ,M) . . etc...
  32. W

    Puzzles about the ''groud state manifold''

    in atomic physics, sometimes one would encounter the termilogy ''ground state manifold'' my question is, the ground state of an atom is usually unique How come the ''ground state manifold''? It means several nearly degenerate level? are these level stable?
  33. G

    What are the real-world implications of tangent vectors to a manifold?

    Dear all, I can formally understand one of the many definitions of tangent vectors to a manifold, but what are they in reality? It should depend on the nature of the points of the manifold, for example, if M={set of events of general relativity}, then vectors are velocities. Other examples...
  34. D

    High intake manifold temperature

    I have a intake manifold temperature that is way to high (105 by mid day normally 90) that is killing me.I have replaced the aftercooler and rebuilt the aux water pump that supplys the cooling water to the aftercooler as well as cleaned out the secondary cooling marely tower,all to no avail is...
  35. L

    Intake manifold for turbo applications

    Hello all, this is my first post on this site, so I'll try not to look stupid. I am currently trying to design a new intake manifold for a turbo 1.6L engine. I have never attempted to this before, but I have a few ideas in mind, and would like some feedback. First, I could use a D-shape pipe...
  36. P

    Manifold & Metric: Does it Need a Metric?

    Does a manifold necessarily have a metric? Does a manifold without metric exist? If it exists, what is its name?
  37. P

    Manifold and Metric: Answers to Your Questions

    Does a manifold necessarily have a metric? Does a manifold without metric exist? If it exists, what is its name?
  38. A

    Manifold gradient problem

    Hi all: I have just met a problem. If say there is a triangle ijk on a manifold, D(i), D(j), D(k) are the geodesic distances from a far point to i,j,k respectively. Then g = [D(i) - D(k); D(j) - D(k)], what does g describe? Does is describe the gradient of the vertex k? If u = Vi-Vk, v =...
  39. A

    Is (S^n) X R Parallelizable for All n?

    Hi, I am new to manifold and having a hard time on it. :frown: Could anyone please help me on the following problem. Please write down your thoughts. Thanks alot. Prove that (S^n) X R is parallelizable for all n.
  40. R

    Let M be a three dimensional Riemannian Manifold that is compact. .

    Let M be a three dimensional Riemannian Manifold that is compact and does not have boundary. I believe manifolds that are compact and without boundary are called closed. So, my manifold M is closed. I'm interested in knowing the answers to the following questions. Under what conditions is...
  41. D

    Proving Existence of Vector Field X for 1-Form w on Smooth Manifold M

    Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.
  42. Z

    Do We Already Have Evidence Of Dark Energy - Our Manifold?

    DO WE ALREADY HAVE EVIDENCE OF DARK ENERGY - OUR MANIFOLD? If LIGO I with it’s 10^-21 sensitivity, VIRGO etc. don’t detect gravity waves, might this then be interpreted as indicating that C_R pseudo-Riemanian spacetime continuum (i.e. manifold’s) stiffness is not INsignificant; rather...
  43. E

    Riemannian manifold and general relativity

    Homework Statement My book ("General Relativity" by Hobson on page 32) says that an N-dimensional manifold has 1/2 * N * (N-1) independent metric functions. I am confused about why there is a limit at all to the number of independent metric functions g_{\mu \nu} . It probably has to do with...
  44. quasar987

    A contradiction in Spivak's Calculus on manifold?

    Homework Statement I don't have great expectation that this will get a reply but here goes, because this is bugging me. I will assume that you are familiar with the notation used by Spivak. In the last section of chapter 4, he shows how to integrate a k-form on R^m over a singular k-cube...
  45. M

    Vector Subspace or Linear Manifold.

    Is there any difference between a vector subspace and a linear manifold. Paul Halmos in Finite Dimensional Vector Spaces calls them the same thing. Hamburger and Grimshaw in Linear Trasforms in n Dimensional Vector Space does not use the word subspce at all. Planet Math says a Linear...
  46. S

    Why are generalized momenta cotangent vectors in symplectic manifolds?

    I've been reading about the abstract formulation of dynamics in terms of symplectic manifolds, and it's amazing how naturally everything falls out of it. But one thing I can't see is why the generalized momenta should be cotangent vectors. I can see why generalized velocities are tangent...
  47. B

    Smooth Atlas of Differentiable Manifold M

    can you be given a suitable smooth atlas to the subset M of plane that M to be a differentiable manifold? M={(x,y);y=absolute value of (x)}
  48. honestrosewater

    Why water cracks hot exhaust manifold

    This cute mechanical engineer mentioned in a message that he had designed some manifolds, so that's why I'm asking this (in addition to my general interest in physics, of course). I'm not sure if they were intake or exhaust manifolds, but that's not the problem now. I figured that...
  49. B

    Exploring the Relationship between k-forms and l-forms on m Manifold

    So I was wondering about this... if \omega is a k-form and \eta is a l-form, and m is a k+l+1 manifold in \mathbb{R}^n, what's the relationship between \int_M \omega\wedge d\eta and \int_M d\omega\wedge \eta given the usual niceness of things being defined where they should be, etc. etc. The...
  50. M

    Smooth deformation of a Lorentzian manifold and singularities

    How can a smooth deformation of a Lorentzian manifold possibly create one or more singularities?
Back
Top