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Grassmann manifolds

  1. Jul 7, 2007 #1
    I need to following subjects about GRASSMANN MANIFOLDS,what do I?

    1)introduction(together with details)

    2)charts,atlas(together with details)

    3)depended subjects
     
  2. jcsd
  3. Jul 7, 2007 #2

    mathwonk

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    read milnor, differentiable manifolds
     
  4. Jul 8, 2007 #3
    the book name

    what is your book name that introduce to me ??
     
  5. Jul 9, 2007 #4

    mathwonk

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    these are leture notes on differential topology, widely available for several decades. or maybe now in the book characteristic classes.
     
  6. Jul 9, 2007 #5

    mathwonk

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    heres an example: the set G of all lines through the origin of (x,y,z) space. since each such line is determined by any other point, consider the three planes x=1, y=1, z=1.

    each line through the origin contains a point with at least one non zero coordinate, hence with some coordinate equal to 1, so each such line meets at least one of those planes in a unique point.

    thus the set of all lines in G is covered by three sets each isomorphic to a plane. hence G is a 2 dimensional manifold with three coordinate charts. moreover, there is a 2:1 surjection from the unit sphere onto G, since each point of the sphere determines one line through (0,0,0), and each such line meets the sphere twice.
     
  7. Jul 9, 2007 #6

    mathwonk

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    now consider planes through (0,0,0) in space. can you see why this set is isomorphic to the previous set of lines through (0,0,0)?
     
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