marao

as a mathematical term, I want in all languages to use a word Marao.

One of the mathematical methods is to present a complex object as a set of simple objects. This procedure is widely used in different terms: equivalence, factorization, stratification, foliation. One of them is the concept described below - the expression of a linear space by a set of linear subspaces.

The stratification of the spheres is known, which Hopf noticed and published in 1931. Heinz Hopf (1894.11.19 - 1971.06.03). Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen, Berlin: Springer. It is a foliation of a three-dimensional sphere by circles, and this set of circles is dipheomorphic to a sphere of two dimensions. This situation is easy to describe, if we see a three-dimensional sphere as the set of rays of four-dimensional linear space. I must also draw attention to the analogy: a factorial space of linear space is the decomposition of space into a set of affine subspaces, and Marao is the decomposition of space into a set of linear sub-spaces. I must also draw attention to the analogy: a factorial space of linear space is the decomposition of space into a set of affine subspaces, and Marao is the decomposition of linear space into a set of linear subspaces.

Definition
The set of linear subspaces of a linear space E is a marao if
- the intersection of each pair of two subspaces is zero
- the union of all subspaces is equal to the principal space E.

example
the set of subspaces to dimension one of a linear space E is called a projective space. this set meets the requirements of the definition and therefore it is Marao.

The main example
Let W ⊃ V be an extension of the fields. Each W-linear space is also V-linear. The set of W-linear subspace of dimension one is a projective space. This set as V-linear subspace of the space E Remains again a marao, but of larger dimmension.

the dimension of sub-spaces of the Marao does not exceed half of the dimension of the main space. It is possible only when Marao has an element, Marao trivial, the only element which is itself a linear space E. Marao M with only a subspace E ∈ M.

We have the map of E * non-zero vectors of a linear space E: E * → M, x → m, x ∈ m, This aplication is part of the standard grassmanian fibration.

we have two linear fibrations E → E / p and E → M. The fibers of one are orthogonal to the section of the other and vice versa.

Suppose that M is a marao in a linear space E of dimension n and the dimension of the members of marao k. We choose subspace A of dimension n - k and divide M into two subspaces M* and M^, the first subspace M* consists of a member having the intersection with A equal to zero, and a second subspace M^ have the intersection with A not equal to zero. choose an element p of M*. each element of M*, m ∈ M* is a graph of a linear map of p in A. so M* ⊂ Lin (p, A) = lin (p, E / p) .Marao is subspace of grasmanian. hence TpM ⊂ lin (p, E / p). The tangent application of the map E → M is a linear map E → TpM with the kernel p. hence TpM = E/p. Here, the dimension of Marao is n - k.

The vector u from E / p can correspond to the subspace of the marao, to the vector x of u we associate its containing element xm of the marao, the subset of all xm is denoted by um ⊂ M. x∈xm ∈ um ⊂ M.

If a ∈ E / p we can consider it as the tangent vector in the point p, and we can corespond to a the line described as a subset of the subspaces of M if x∈a content vector x + yr a subspace (x + yr)m (y∈p) r → (x + yr)m, x + yr ∈ (x + yr)m ∈ M

There are many structures on the Grassmannian: the natural linear bundle, the fiber on the point p itself p As linear space, and the tangent space TpM is Naturally isomorphic to Lin (p, E/p). we have an application E → M. Its linear tangent space E → TpM⊂ Lin (p, E/p) with the kernel p. The tangent space TpM is therefore isomorphic to the quotient space E/p.

e size of the Marao sub-spaces does not exceed half of the main space. It is possible only when Marao has an element, Marao trivial, the only element which is itself a linear space with E. Marao M with only a subspace E ∈ M. This is only a trivial Marao element E. We have 'alication of E * non-zero vectors of a linear space E: E * → M, x → m, x ∈ m, This alication is part of the standard grassmanian fibration.

the dimension of subspaces of Marao does not exceed half of the main space. only possible to invert when Marao trivial , with only one element that is itself a basic linear space E.

we have two fibration linear E → E / p and E → M. The fibers of one are orthogonal to the section of the other and vice versa.

we have two linear fibration E → E/p and E → M. The fibers of one are orthogonal to the section of the other and vice versa.

Suppose that M is a marao in a linear space E of dimension n and the dimension of the members of marao k. We choose subspace A of dimension n - k and divide M into two subsets M* and M^, the first subspace M* consists of members with intersection with A equal to zero, and a second subspace M^ with intersection with A not equal to zero. If choose an element p of M*. each element of M*, m ∈ M * is a graph of a linear map p to A. We have M * ⊂ Lin (p, A) = lin (p, E / p). Obviously, Marao is a subspace of grasmanian. hence TpM ⊂ lin (p, E / p) The tangent map of the map E → M is a linear map E → TpM with the kernel p. hence TpM = E / p.

The vector u of E / p can correspond to the subset of the marao, to the vector x of u we associate an element xm of marao content x, this subset of xm (x) corresponding to u is denoted um ⊂ M, xm ∈ um ⊂ M. Le vecteur u de E/p peut correspondre au sous-ensemble du marao, au vecteur x de u on associe un élément xm de maraocontenant x, cet sous-ensemble correspondant a u est noté um ⊂ M, xm ∈ um ⊂ M.

Hopf Fibration

take a note: the marao of k-dimensional subspaces in the n-dimensional linear space marks Mnk

be given a linear space E and in E a Marao M (dimension of the subspaces n), as well as in each of its elements Marao (dimension of the subspaces k). We have a total space N of the fibration (the union of the small Maraos. It is is marao in E a with dimension of the subspaces k) and the base Marao M. on each point the fiber equal to Marao N. Such a fibration can be named as Hopf fibration since the famous Hopf fibration are main examples.

Let's say M is a marao in the linear space in E and N is a marao in the linear space F. Consider the multiplication M × N, all of its elements p × q are the same subspace of the E × F, these subspacess do not cross each other and in union is equal to the entire E × F. This set M × N is a marao in the E × F.

8=ℝ16 → S15 → ℝP15 = M161 → ℂP7 = M162 → M164 → M168 = S8
fiber: ray, two opposite directional ray or the sphere S0, M21 or the sphere S1, M42 or the sphere S2, M84 or the sphere S4, from the second to the end the fiber is rays of the 8-dimensional linear space or the sphere S7.

4=ℝ8 → S7 → ℝP7 = M81 → M82= ℂP3 → M84 = S4
fiber: ray, two opposite directional ray or the sphere S0, M21 or the sphere S1, M42 or the sphere S2, from the second to the end the fiber is rays of the 4-dimensional linear space or the sphere S3.

2=ℝ4 → S3 → ℝP3 = M41 → ℂP1 = M42 = S2
fiber: ray, two opposite directional ray or the sphere S0, M21 or the sphere S1, from the second to the end the fiber is rays of the 4-dimensional linear space or the sphere S3.

ℂ=ℝ2 → S1 → ℝP1 = M21= S1

fiber: ray, two opposite directional ray or the sphere S0, two points or sphere S0.