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Math Help - Manifolds

  1. #1
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    Exclamation Manifolds

    I was asked to consider the set X of all straight lines in R^2, not necessarily through the origin.

    a) I had to introduce a nature manifold structure into X, and write down all the charts explicitly, showing their domains and codomains.

    For this I know,

    A line in R^2 has an equation of the form ax + by + c = 0, where a and b must not both be 0.

    If a ≠ 0 then we can divide by a and write the equation as x + b'y + c' = 0. The map taking this line to the point (b',c') in R^2 is one chart.

    If b ≠ 0 then we can divide by b and write the equation as a'x + y + c' = 0. The map taking this line to the point (a',c') in R^2 is also a chart.

    Those two charts together specify the manifold structure.

    b) What is the dimension of the manifold X?

    I know that its a 2-dimensional manifold, as the charts go to the space R^2.

    But am struggling with the following question?

    c) I need to define a natural surjection p:X -> RP^1. I need to write it explicitly using the local coordinates on X which i introducted in a) and the natural local coordinates on RP^1. And also need to show that p is a smooth.

    Any help is appreciated.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Here's a hint. The space RP^1 can be identified with the set of all lines through the origin in R^2. So the natural surjection would correspond to the map that sends a line in R^2 to the line through the origin that has the same gradient. Now you have to describe that map explicitly in terms of what it does on charts.
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  3. #3
    Junior Member
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    How to show a smooth map

    Thanks for your hint, its been helpful.

    I let U0 = {y=mx+c} and U1 = {nx+d}

    I defined the natural surjection p: X -> RP^1 by send c, d terms to zero:

    U0 = {y=mx+c} -> {y=mx} = [1:m]
    and
    U1 = {x=ny+d} -> {x=ny} = [n:1]

    How can i show that p is a smooth map.

    Any help is appreciated.
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