Learn About Grassmann Manifolds: Intro, Charts, Atlas

[bigli]

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

 

[mathwonk]

read milnor, differentiable manifolds

 

[bigli]

the book name

what is your book name that introduce to me ??

 

[mathwonk]

these are leture notes on differential topology, widely available for several decades. or maybe now in the book characteristic classes.

 

[mathwonk]

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.

 

[mathwonk]

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)?