# Quotient Manifolds

### Quotient Space

* Let M be a manifold equipped with an **equivalence relation** ∼, i.e., a relation that is 
  * 1\. reflexive: x ∼ x for all x ∈ M, 
  * 2\. symmetric: x ∼ y if and only if y ∼ x for all x, y ∈ M, 
  * 3\. transitive: if x ∼ y and y ∼ z then x ∼ z for all x, y, z ∈ M.
* The set \[x] := {y ∈ M : y ∼ x} of all elements that are equivalent to a point x is called the **equivalence class** containing x.
* The set M/∼ := {\[x] : x ∈ M} of all equivalence classes of ∼ in M is called the **quotient of M** by ∼. Notice that the points of M/∼ are subsets of M.
* At this point we haven't talked about manifold structure yet; the quotient of M is a **quotient space**. The set M is called the **total space** of the quotient M/∼.

#### How to understand a quotient space?

* The quotient space can refer to quotients of vector spaces in linear algebra, or quotients of topological spaces in topology.
* In algebra, intuitively, the quotient space is a certain kind of "**collapsed**" version of an original space. Specifically in group theory (or ring/field theory), a quotient structure is formed by partitioning a given structure into a set of equivalence classes and then performing the operations "**modulus**" these equivalence classes.
  * For example, if we take the integers and use the equivalence relation `x ~ y` if `x - y` is a multiple of, say, 5, we have the quotient structure Z/5Z, also known as the integers mod 5. This structure has five equivalence classes, \[0], \[1], \[2], \[3], \[4], and adding or multiplying any two of them "wraps around" once you hit 5.
  * The fundamental group of SO(3) is Z/2Z, which is also a quotient space in this form.
* In topology, a quotient space is a topological space formed from another one by **identifying** or **"gluing together"** points which are equivalent under a certain equivalence relation.
  * For example, it allows us to turn a line (more specifically, the real line R) into a circle (S^1). We can define an equivalence relation on R where two points x and y are considered equivalent if their difference is an integer, i.e., `x ~ y` if and only if `x - y` is an integer. The equivalence classes under this relation are sets of the form `{x + n | n in Z}`, and each of these sets represents a distinct point in the circle. The circle S^1 is thus the quotient space of R under this equivalence relation.
  * Similarly, the 2-dimensional torus can be obtained as the quotient space of a square, by first gluing the left and right edges (now it is a cylinder) and then gluing the top and bottom edges.

### Quotient Manifolds

* The mapping π : M → M/∼ defined by x → \[x] is called the **natural projection** or **canonical projection**. 
* Let (M, A+) be a manifold with an equivalence relation ∼ and let B+ be a manifold structure on the set M/∼. The manifold (M/∼, B+) is called a **quotient manifold** of (M, A+) if the natural projection π is a submersion.
* Proposition: Let M be a manifold and let M/∼ be a quotient of M. Then M/∼ admits at most one manifold structure that makes it a quotient manifold of M.

### Real Projective Space $$\mathbb{RP}^{n-1}$$

* The real projective space RP^n−1 is the set of all directions in R^n, i.e., the set of all straight lines passing through the origin of R^n. 
* It can be obtained as the quotient space of R^n - {0} under the equivalence relation that two points x and y are considered equivalent if and only if there exists a non-zero real number λ such that x = λy. It consists of the set of all equivalence classes \[x], where x is a point in R^n - {0}. Each equivalence class represents a distinct line through the origin in R^(n+1).
* Recall that R^n - {0} is the p = 1 particularization of the noncompact Stiefel manifold R^n×p\_∗.
* RP^n−1 can also be formed by identifying antipodal points of the unit sphere S^n−1 in R^n. This is another way to obtain RP^n−1 as the quotient space under an equivalence relation.

### Grassmann Manifold

* Let n be a positive integer and let p be a positive integer not greater than n. Let **Grass(p, n)** denote the set of all **p-dimensional subspaces** of R^n.
* We can first show that there exists a one-to-one correspondence between Grass(p, n) and a quotient set R^n×p\_∗/GL\_p, and then show that this quotient set admits a (unique) structure of quotient manifold. This can endow Grass(p, n) with a matrix manifold structure.
  * Recall that the noncompact Stiefel manifold R^n×p\_∗ is the set of all n × p matrices with full column rank.
  * Let ∼ denote the equivalence relation on R^n×p\_∗ defined by X ∼ Y ⇔ span(X) = span(Y), where span(X) denotes the subspace {Xα : α ∈ R^p} spanned by the columns of X ∈ R^n×p\_∗.
  * This equivalence relation identifies all n × p matrices if they are column equivalent, meaning that they have the same column space (which is the space they spanned), and that one can be obtained from the other by a sequence of elementary column operations.
  * Since each elementary column operation corresponds to exactly a matrix in the general linear group GL\_p, the quotient set R^n×p\_∗/∼ becomes R^n×p\_∗/GL\_p.
  * What follows is a proposition to show that this quotient set admits a structure of quotient manifold. Omitted here for simplicity.
* Endowed with its quotient manifold structure, the set R^n×p\_∗/GL\_p is called the **Grassmann manifold** of p-planes in R^n and denoted by Grass(p, n).
* The particular case Grass(1, n) = RP^n is the real projective space discussed previously.

#### Notes, Matrix Representation, and Comparison to the Stiefel manifold

* If a matrix X and a subspace P satisfy P = span(X), we say that P is the **span** of X, that X spans P, or that X is a **matrix representation** of P.
* The Stiefel manifold is the set of all **ordered orthonormal** (or just linearly independent for noncompact Stiefel manifold) **p-tuples of vectors in R^n**.
  * 1\) linear independence required for p-tuples of n-vectors, in order to form a basis.
  * 2\) the order (of n-vectors) matters; this is reflected in the ordering of columns in its matrix representation.
  * 3\) can be represented by a matrix uniquely; there is a one-to-one correspondence between each point on the Stiefel manifold and an orthonormal (or linearly independent) n × p matrix representation.
* The Grassmann manifold is a space that parametrizes all **p-dimensional subspaces** in an n-dimensional vector space.
  * 1\) linear independence required for p-tuples of n-vectors, in order to form a basis.
  * 2\) unordered; p-tuples in different orderings span the same subspace. 
  * 3\) no unique matrix representation; each point on the Grassmann manifold represents a subspace, and corresponds to multiple n × p matrices that can span this subspace. Essentially, what a manifold captures is the geometric object independently of any particular representation of it.
* The Stiefel manifold can be seen as a "covering" of the Grassmann manifold.
  * More formally, there is a natural projection map from the Stiefel manifold St(p, n) to the Grassmann manifold Grass(p, n), which takes an ordered orthonormal basis for a subspace and forgets about the order and orientation to give just the subspace itself.
  * This map is a fiber bundle, with the fibers being the set of all ordered orthonormal bases for a given p-dimensional subspace (which is isomorphic to the orthogonal group O(p)).
  * For every point in the Grassmann manifold (i.e., every p-dimensional subspace in the n-dimensional space), there are multiple corresponding points in the Stiefel manifold (i.e., multiple ordered bases for that subspace).