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RP2 into R4 embedding

  1. Sep 30, 2009 #1
    P.S. I carelessly posted this question in the Topology&Geometry forum first:
    https://www.physicsforums.com/showthread.php?t=341773



    1. The problem statement, all variables and given/known data

    Let F be a map from S2 in R3 into R4, given by

    [tex] F(x,y,z) = (x^2 - y^2, xy, xz, yz) \. [ = (a,b,c,d)] [/tex]

    Eventually I am supposed to show that this is an embedding of the real projective plane, but first I am asked to verify that the image of this map is a manifold at all. And that proved trickier than it looked like.

    I know two ways of verifying that something is indeed a manifold.

    1) Find local diffeomorphisms, taking a neighborhood of the manifold into R4 such that points on the manifold land in a copy of R2 inside R4

    2) Show that the manifold is the level set of some function, where the derivative of that function has full rank at every point inside the level set.

    2. Relevant equations

    I am trying to go with approach (1). I can write down rules for the required functions, but I am not sure how to find neighborhoods that work.

    3. The attempt at a solution

    I have to deal with 6 cases, depending on where in the image my point lies. For example, if I start with a point p=F(x,y,z) where z2 is not 0 or 1/2, I get

    [tex]a = \frac{2c^2 - 2 d^2}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}},
    b = \frac{2cd}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}}[/tex]

    where I use a plus in the denominator if z2 < .5 and a minus if z2 > .5. Once I have a and b in terms of c and d, I could pick a map as follows:

    [tex]g(a,b,c,d) = (a - \frac{2c^2 - 2 d^2}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}}, b - \frac{2cd}{1 \pm \sqrt{1 - 4 (c^2 + d^2)}}, c,d)[/tex]

    This map returns (0,0,c,d) iff (a,b,c,d) was on the manifold in a small enough neighborhood of p. My problem is that I need to pin down an open set in R4 that does not accidentally contain a point on the manifold that is "on a different part" of the manifold than p.

    I also tried working in polar coordinates but I had a very hard time tracking cases and solving for variables in a couple of instances, and it didn't seem to make the search for appropriate neighborhoods any easier.


    Any suggestions? How do I find and verify my neighborhoods?
     
  2. jcsd
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