P.S. This was supposed to go into the homework forum. I had windows open for both forums and I ended up using the wrong one.(adsbygoogle = window.adsbygoogle || []).push({});

Let F be a map from S^{2}in R^{3}into R^{4}, 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 R^{4}such that points on the manifold land in a copy of R^{2}inside R^{4}

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.

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2) Didn't seem to work because I always ended up squaring too many terms to get to a constant. The Jacobian matrix ended up degenerate in too many places to do those by hand, and I also ended up including points in the level set that are not part of the manifold.

Now 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.

For example, if I start with a point p=F(x,y,z) where z^{2}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 z^{2}< .5 and a minus if z^{2}> .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 R^{4}that does not accidentally contain a point on the manifold that is "on a different part" of the manifold than p.

Any suggestions? How do I find and verify my neighborhoods?

P.S. 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.

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# RP2 into R4 embedding

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