# 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 $$\mathbb{R}^{n\times p}$$ * 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 $$\mathbb{R}^{n\times p}\_\*$$ (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.) --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://wiki.hanzheteng.com/math/differential-geometry/manifold.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.