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Intrinsic curvature.

  1. Jan 25, 2009 #1
    Hello all.

    I understand that a two dimensional surface can have curvature without it being referred to a higher dimension. So that a surface such as that of a sphere does not need to refer to a third dimension to determine its own intrinsic curvature and so on for higher dimensions.

    Can a one dimensional space/surface such as a line have intrinsic curvature.

    Matheinste.
     
  2. jcsd
  3. Jan 25, 2009 #2

    Hurkyl

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    All one-dimensional manifolds are homeomorphic to a line or to the circle*.

    Let's start with the line with a Riemann metric. You can choose a base-point and a direction, parametrize the line by directed arclength. Making the change-of-coordinates to use arclength as a coordinate, the metric is the constant tensor field ds ds. Therefore, all Riemann lines are isometric to a (possibly infinite) open interval in R.

    For the circle, I believe the exact same idea shows that all Riemann circles are isometric to a circle in R².

    In other words, the only diffeomorphism-invariant properties of a one-dimensional Riemann manifold are
    1. Whether it's a line or a circle
    2. Its length
    and if it's an infinite line,
    3. Whether it's infinite on both ends, or just one end




    * There's also the long ray and the long line, depending on your exact technical assumptions, but let's ignore those.
     
    Last edited: Jan 25, 2009
  4. Jan 26, 2009 #3
    Thankyou for your reply.

    I have digested what you said and understand it.

    My non rigorous train of thought was that i believe given a surface/2 dimensional manifold we can decide if it is curved by drawing a circle and measuring its diameter and its circumference, all measurements being taken upon the surface, and comparing the ratio with pi.

    Is there a test of this nature for a line.

    Matheinste.
     
  5. Feb 4, 2009 #4
    curvature is a two dimensional concept - curvature for curves is an extrinsic idea and may not be derived from the metric. There is no idea of intrinsic curvature for a curve
     
  6. Feb 4, 2009 #5
    Hello wofsy.

    That's what i thought but was not sure of.

    Thanks.

    Matheinste.
     
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