Dear Ali Al-Fatemi

In general, differential geometry is not smooth in understanding to nonmathematical students. It needs many prerequisite courses in

1) Elementary and advanced calculus, 2) Linear algebra, 3) Topology and 4) Real and Mathematical analysis.

So it is natural to face some difficulty in understanding the topic.

The definition of Grassman manifold is the set of all k - dimensional vector subspaces in Rn (The usual euclidean vector space) denoted by Gk,n (R), or Gk(Rn).

Example: G1(R2) the set of all 1- dimensional vector subspaces of R2.

In other words, any vector space in the plane that has a dimension one is a straight line passing through the origin.

Therefore, G1(R2) is the set of all st.lines that are passing through the origin in the plane. The union of all such vector subspaces is the plane R2.

Similarly, G2(R3) is the set of all planes that are passing through the origin in the space R3.

G1(R3) is the set of all st.lines that is passing through the origin in the space R3.

Best regards

Dear Ali Al-Fatemi

In general, differential geometry is not smooth in understanding to nonmathematical students. It needs many prerequisite courses in

1) Elementary and advanced calculus, 2) Linear algebra, 3) Topology and 4) Real and Mathematical analysis.

So it is natural to face some difficulty in understanding the topic.

The definition of Grassman manifold is the set of all k - dimensional vector subspaces in Rn (The usual euclidean vector space) denoted by Gk,n (R), or Gk(Rn).

Example: G1(R2) the set of all 1- dimensional vector subspaces of R2.

In other words, any vector space in the plane that has a dimension one is a straight line passing through the origin.

Therefore, G1(R2) is the set of all st.lines that are passing through the origin in the plane. The union of all such vector subspaces is the plane R2.

Similarly, G2(R3) is the set of all planes that are passing through the origin in the space R3.

G1(R3) is the set of all st.lines that is passing through the origin in the space R3.

Best regards

dear Ali AI-Fatemi

One motivation may be understanding (covariant) derivative in (higher dimensional) surfaces. For example, consider a surface in R^3, the tangent plane of a smoothly varying point can be thought of as a smooth path in G_2(R^3). If you introduce the canonical fibre bundle structure, then a smooth tangent vecter attached to the point can be thought of as a section of the bundle, then you can do derivative as in Euclidian space intuitively.

Another thing I know is the constructing of the universal bundle, it's something like the universal covering, you can form a bundle map to map every vector bundle on paracompact base space into it. It brings the tool of Stiefle-Witney classes in characteristic class theory.

I am a student too, so above there might be improper understandings.. I hope for better understandings too.

Grassmann manifolds, or "Grassmanians" are fundamental in algebraic geometry, and have very important applications. In simple terms, they are the most immediate natural generalization of projective spaces. The latter are the basic setting for projective geometry. By definition, the elements of a Grassmannian are k-dimensional subspaces of an n-dimensional vector space. Projective spaces are the special case when the dimension k=1. Grassmannians are given the structure of a differential manifold by defining natural coordinate charts on open dense subsets corresponding to all k-element subsets of the integers (1,2, .., n), with natural coordinates given by the elements of the associated k x (n-k) dimensional affine coordinate matrices (relative to a chosen basis). They are "algebraic varieties", which means that they can be embedded into projective spaces as the intersection of the vanishing locus of a finite number of homogeneous polynomials. They are also homogeneous spaces: U(n)/(U(k) x U(n-k)) of the unitary groups U(n) (in the complex case) or O(n)/(O(k) x O(n-k)) of the orthogonal groups O(n) (in the real case). These groups act on the the Grassmannians as transitive groups of isometries with respect to naturally defined Riemannian metrics, and as such, Grassmannians are also the most basic examples of Riemannian symmetric spaces. Grassmannians also play a fundamental role as "universal spaces" into which all algebraic varieties can be embedded, and all compact Riemannian spaces can be embedded isometrically. You can find everything you are likely to want to know about Grassmannians in the textbook "Principles of Algebraic Geometry", by Griffiths and Harris, Chapter 1, section 5.

Tracking objects using Grassmann manifold appearance modeling based on wireless multimedia sensor networks

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Also see

Computing equilibria of GEI by relocalization on a Grassmann manifold

Peter M. Demarzo, B.Curtis EavesJournal:Journal of Mathematical Economics