Learn About Grassmann Manifolds: Intro, Charts, Atlas

In summary, the conversation is about GRASSMANN MANIFOLDS and the following subjects were discussed: introduction, charts and atlas, as well as other related topics such as Milnor and differentiable manifolds. The book mentioned was "Characteristic Classes" and an example was provided to illustrate the concept of lines through the origin in 3D space. The set of all lines in G is covered by three sets each isomorphic to a plane, making G a 2-dimensional manifold with three coordinate charts. It was also mentioned that there is a 2:1 surjection from the unit sphere onto G. The conversation then moves on to discussing planes through (0,0,0) in space and how they are related to
  • #1
bigli
16
0
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
 
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  • #2
read milnor, differentiable manifolds
 
  • #3
the book name

what is your book name that introduce to me ??
 
  • #4
these are leture notes on differential topology, widely available for several decades. or maybe now in the book characteristic classes.
 
  • #5
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.
 
  • #6
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)?
 

1. What are Grassmann manifolds?

Grassmann manifolds, also known as Grassmannians, are spaces that represent all possible subspaces of a given vector space. They are named after Hermann Grassmann, who first studied them in the 19th century.

2. How are Grassmann manifolds different from other manifolds?

Grassmann manifolds are unique in that they are non-compact and infinite-dimensional. This means that they do not have a finite number of dimensions and they do not have boundaries. They also have a highly symmetrical structure, making them useful in many areas of mathematics and physics.

3. What are the applications of Grassmann manifolds?

Grassmann manifolds have a wide range of applications in fields such as differential geometry, algebraic geometry, and physics. They are used to study symmetry, quantum mechanics, and geometric optimization, among other things.

4. What are the charts and atlas of a Grassmann manifold?

Charts and atlas are tools used to describe and visualize manifolds, including Grassmann manifolds. A chart is a local coordinate system that maps points on the manifold to points in a coordinate space, while an atlas is a collection of charts that cover the entire manifold.

5. How can one learn about Grassmann manifolds?

One can learn about Grassmann manifolds through various resources such as textbooks, online lectures, and workshops. It is recommended to have a strong understanding of linear algebra and multivariable calculus before delving into the study of Grassmann manifolds.

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