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Manifold

Definition

The abstract definition of a manifold relies on the concepts of charts and atlases.

Chart

  • Let M be a set. A bijection (one-to-one correspondence) ϕ of a subset U of M onto an open subset of R^d is called a d-dimensional chart of the set M, denoted by (U, ϕ).
  • When there is no risk of confusion, we will simply write ϕ for (U, ϕ).
  • The elements of ϕ(x) ∈ R^d are called the coordinates of x in the chart (U, ϕ)

Atlas

  • A (C^∞) atlas of M into R^d is a collection of charts (Uα, ϕα) of the set M such that
    • M = the union of Uα over α,
    • the change of coordinates from one chart to another is smooth. Alternatively, we say that the elements of an atlas overlap smoothly. Rigorous math expressions omitted.
  • Two atlases A1 and A2 are equivalent if A1 ∪ A2 is an atlas. (The union operation applies to the charts of each atlas.)
  • The maximal atlas (or complete atlas) A+ is the set of all charts (U, ϕ) such that A ∪ {(U, ϕ)} is also an atlas.
  • A maximal atlas of a set M is also called a differentiable structure on M.

Manifold

  • A (d-dimensional) manifold is a couple (M, A+), where M is a set and A+ is a maximal atlas of M into R^d, such that the topology induced by A+ is Hausdorff and second-countable.
    • Hausdorff: allows for the uniqueness of limits of sequences. A convergent sequence on a non-Hausdorff topological space may have several distinct limit points.
    • A topological space is second-countable if there is a countable collection B of open sets such that every open set is the union of some subcollection of B. This is to guarantee the space is separable (has a countable dense subset) and metrizable (there exists a metric inducing the topology), and to help resolve the existence of a Riemannian metric and an affine connection.
  • When (M, A+) is a manifold, we simply say “the manifold M” when the differentiable structure is clear from the context, and we say “the set M” to refer to M as a plain set without a particular differentiable structure.
  • Given a chart ϕ on M, the inverse mapping ϕ−1 is called a local parameterization of M.

Types of Manifold

How to recognize a manifold? Check whether a given set X admits an atlas.

Vector Space

  • Every vector space is a linear manifold in a natural way. Charts built in this way are compatible.

Matrix Manifold
Rn×p\mathbb{R}^{n\times p}

  • The set R^n×p is a vector space with the usual sum and multiplication by a scalar.
  • A chart of this manifold is given by ϕ : R^n×p → R^np : X → vec(X), where vec(X) denotes the vector obtained by stacking the columns of X below one another.
  • We will refer to the set R^n×p with its linear manifold structure as the manifold R^n×p. Its dimension is np.

Matrix Manifold
Rn×p\mathbb{R}^{n\times p}_*
(noncompact Stiefel manifold)

  • Let R^n×p_∗ (p ≤ n) denote the set of all n × p matrices whose columns are linearly independent. This set is an open subset of R^n×p, since its complement (linearly dependent n×p matrices with determinant = 0) is closed. (In general, rank or linear independence is not preserved under matrix multiplication.)
  • It admits a structure of an open submanifold of R^n×p. The chart is similar to that of R^n×p.
  • It is referred to as the manifold R^n×p_∗, or the noncompact Stiefel manifold of full-rank n × p matrices. In general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.
  • When p = 1, the noncompact Stiefel manifold reduces to the Euclidean space R^n with the origin removed. (In this case, except for the origin, any single vector by itself can form a linearly independent basis.)
  • When p = n, the noncompact Stiefel manifold becomes the general linear group GL_n, i.e., the set of all invertible n × n matrices. (The columns of such matrices are linearly independent and can serve as a basis.)