# Quotient Manifolds

### Quotient Space

• Let M be a manifold equipped with an equivalence relation ā¼, i.e., a relation that is

• 1. reļ¬exive: x ā¼ x for all x ā M,

• 2. symmetric: x ā¼ y if and only if y ā¼ x for all x, y ā M,

• 3. transitive: if x ā¼ y and y ā¼ z then x ā¼ z for all x, y, z ā M.

• The set [x] := {y ā M : y ā¼ x} of all elements that are equivalent to a point x is called the equivalence class containing x.

• The set M/ā¼ := {[x] : x ā M} of all equivalence classes of ā¼ in M is called the quotient of M by ā¼. Notice that the points of M/ā¼ 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/ā¼.

#### 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 Ļ : M ā M/ā¼ deļ¬ned by x ā [x] is called the natural projection or canonical projection.

• Let (M, A+) be a manifold with an equivalence relation ā¼ and let B+ be a manifold structure on the set M/ā¼. The manifold (M/ā¼, B+) is called a quotient manifold of (M, A+) if the natural projection Ļ is a submersion.

• Proposition: Let M be a manifold and let M/ā¼ be a quotient of M. Then M/ā¼ admits at most one manifold structure that makes it a quotient manifold of M.

• The real projective space RP^nā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 Ī» such that x = Ī»y. 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^nĆp_ā.

• RP^nā1 can also be formed by identifying antipodal points of the unit sphere S^nā1 in R^n. This is another way to obtain RP^nā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^nĆp_ā/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^nĆp_ā is the set of all n Ć p matrices with full column rank.

• Let ā¼ denote the equivalence relation on R^nĆp_ā defined by X ā¼ Y ā span(X) = span(Y), where span(X) denotes the subspace {XĪ± : Ī± ā R^p} spanned by the columns of X ā R^nĆp_ā.

• This equivalence relation identifies all n Ć 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^nĆp_ā/ā¼ becomes R^nĆp_ā/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^nĆp_ā/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 Ć 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 Ć 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).

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