1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex Projective Space

  1. Mar 5, 2009 #1
    I've been thinking...and am starting to think that I don't understand complex projective space...So, it's defined as ( Cn+1 \{0,0} / C\{0} ). Now, I think this is just the set of planes in 4 space that pass through the origin... and one can consider how they would all intersect a 3 sphere and think of it as S3/U(1) where U(1) is the circle group... and the hopf function will take all these circles and map them to the 2 sphere isomorphically... but the problem I have is... just pick any 3 of the 4 basis vectors in C^2 and span two planes with them...essential you can just look at R^3 for this... and think of the plane spanned by XY and XZ....well they intersect at the whole X axis...which means there are elements that belong to both planes...but in the case of ( Cn+1 \{0,0} / C\{0} ), these planes are supposed to be equivalence classes...meaning it should divide the space into disjoint sets...and thus, you can't have an element in 2 equivalence classes...Unless, both these planes are actually in the same equivalence class, which is just mind blowing since you can find a bunch more planes that will intercet XY and XZ and before you know it, all of R^3 will belong to the same equivalence class...So, clearly something is wrong with this way of thinking of it...Anyone?
  2. jcsd
  3. Mar 6, 2009 #2
    The set of 2-dimensional subspaces in R^4 is by definition the Grassmannian G(2,4). It has real dimension 2(4-2)=4, so it can not be the same as CP1, which has 2 real dimensions.
  4. Mar 6, 2009 #3


    User Avatar
    Science Advisor

    Heck, I've thought that for years!
  5. Mar 6, 2009 #4

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    In four dimensions, two planes can intersect in a single point, not a line. Take the x-y and z-w planes, for example, which intersect only at the origin.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook