Smooth covering map and smooth embedding

  • Thread starter huyichen
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Now F:S^2->R^4 is a map of the following form:
F(x,y)=(x^2-y^2,xy,xz,yz)
now using the smooth covering map p:S^2->RP^2, p is the composition of inclusion map i:S^2->R^3 and the quotient map q:R^3\{0}->RP^2. show that F descends to a smooth embedding of RP^2 into R^4.

Is the problem asked to show that F。p^(-1) is a smooth embedding? I am confused, and if it is the case, then how should we compute the Jacobian matrix for F。p^(-1)?
 

quasar987

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Essentially, yes, but notice that p^-1 is ill defined. Instead, you must show that there exists a map f: RP^2-->R^4 such that f o p = F (and that it is a smooth embedding). Observe that this only means checking that F is constant on the fibers of p.
 

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