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Enveloping spaces

  1. Nov 3, 2003 #1
    How do you draw a curve with line element ds2=y2dx2+dy2 in a R2 space? Is it just lots of x=a lines, with a-any real number?

    I don't understand why a mapping of phi(x,y)=(x/y,sqrt(x2+y2)) can be an enveloping space for the above curve?

    Any ideas, anyone?
     
  2. jcsd
  3. Nov 17, 2003 #2
    this doesn t quite make sense. if you draw a curve in R2, it inherits an induced metric from the normal euclidean metric.

    this metric is
    [tex]
    ds = \sqrt{1+\left(\frac{df}{dx}\right)^2}dx
    [/tex]
    from the euclidean metric
    [tex]
    ds^2 = dx^2 + dy^2
    [/tex]

    i cannot get the line element you wrote from a curve in euclidean R2
    what is an enveloping space? i have never heard this term before. can you define it please?
     
  4. Nov 21, 2003 #3
    It's from this paper gr-qc/9405063.

    Can I substitute make x'=arctan(y/x) and y'=sqrt(x2+y2) into ds2=dx2+dy2 so that phi'(x,y)=(x',y') be the covering space for the original Euclidean space with (0,0) removed?
     
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