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A question about charts

  1. Jan 25, 2016 #1
    I am studying differential geometry and I stumbled on something that I don't understand. When we have a m- dim differential manifold, with U_i and U_j open subsets of M with their corresponding coordinate 20160125_093821.jpg
    function phi. As can be seen in the figure.

    If I understand it correctly phi_j of a point gives us a coordinate of that point. So what does the inverse of a coordinate function do? And how come the product of phi_i * phi_j = psi takes us between subsets?
     
  2. jcsd
  3. Jan 25, 2016 #2

    fresh_42

    Staff: Mentor

    The idea behind is that charts are a locally Euclidean map (coordinates) of the (curved) manifold. Just as street maps in an atlas are. Earth isn't flat, but locally we can pretend it is. This concept requires that overlapping charts are identical on their intersection.
    The image in the textbook says ##φ_i|_{U_i ∩ U_j} = Ψ_i \cdot φ_j|_{U_i ∩ U_j}##.
     
  4. Jan 26, 2016 #3

    Erland

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    Science Advisor

    It is ##\phi_i\phi_j^{-1}## which maps a subset of ##U'_j## to a subset of ##U'_i##, as is written on the page you scanned near the right margin on the upper half.
     
  5. Jan 27, 2016 #4
    So that [tex]$\Psi$[\tex] maps between different subsets comes from a definition?
     
  6. Jan 27, 2016 #5

    Samy_A

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    Homework Helper

    Yes.
    Set ##U_{ij}=U_i \cap U_j##, and assume that ##U_{ij}\neq \varnothing##
    ##\Psi_{ij}## by definition maps ##\phi_j(U_{ij})## to ##\phi_i(U_{ij})##.
    ##\Psi_{ij}## being infinitely differentiable is the condition imposed in (iv) for M to be an m-dimensional differentiable manifold.

    As said above you can view ##\Psi_{ij}## as a coordinate transformation. Point (iv) in the definition requires these transformations to be infinitely differentiable.

    The book explains it better than I could, actually:
    nakahara.jpg
     
    Last edited: Jan 27, 2016
  7. Jan 27, 2016 #6
    Okej. Thank you very much for all the help.
     
  8. Jan 27, 2016 #7

    lavinia

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    The inverse of a coordinate chart is called a "parameterization". It assigns points on the manifold to parameters in a domain in Euclidean space.

    Example:

    ## φ(u,v) = (sin(u)cos(v),sin(u)sin(v),cos(u))## parameterizes a region of the sphere. (u and v need to be restricted so that the map is 1 to 1).

    Another Example:

    ##φ(u,v) = (cos(u),sin(u),cos(v),sin(v))## parameterizes a torus in four dimensional Euclidean space.

    A parameterization generalizes the idea of a parameterized curve,##c(t)##, to more than one parameter.

    - A coordinate chart goes in the other direction. It assigns a domain in Euclidean space to points on a manifold.

    Example: Project the northern hemisphere (minus the equator) of the unit sphere onto the open unit disk in the xy-plane by dropping the z-coordinate.

    - A parameterization followed by a coordinate chart maps a domain in Euclidean space into another domain in Euclidean space. If ##ψ## and ##φ## are two coordinate charts then ##φψ^{-1}## is a parameterization followed by a coordinate chart.

    For smooth manifolds,##φψ^{-1}## is required to be a smooth map.
     
    Last edited: Jan 27, 2016
  9. Feb 7, 2016 #8
    Here's a bit of intuition about charts. An n-manifold M is a space that is locally like Euclidean space of dimension n. If M is a smooth n-manifold — the nicest kind — then it makes sense to talk about smooth functions

    f: M →

    on it, smooth curves

    α: [a,b] → M​

    on it, smooth vector fields

    V: M → T(M)​

    on it, etc. (Here T(M) is the tangent space of M, a concept that your course will soon introduce, which consists of all tangent vectors at all points of M.)

    In order to achieve this, we can always construct M by taking pieces (open sets) of Euclidean space n and "gluing them together smoothly". The idea of charts with their smooth transition functions is the right way to make this idea of smooth glueing precise, and allows for definition of smooth functions, curves, vector fields, etc. associated with M.
     
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