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 RPn1\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).

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