Topological Manifolds

What is a Manifold?

Throughout our work in topological spaces, metric spaces, and homeomorphisms, we have built a vocabulary for discussing "spaces" abstractly. The category of all topological spaces is, however, too permissive for differential geometry. Many topological spaces lie too far from \(\mathbb{R}^n\) to support a meaningful theory of differentiation, integration, or measurement. We now isolate a class of spaces that retains enough Euclidean character to do calculus, while still capturing the global complexity of interesting geometric objects: manifolds.

A manifold should be thought of as a space that is "locally Euclidean, globally not necessarily so." The Earth's surface is the prototypical example. Although it is globally a sphere, any small region of it looks essentially like a flat plane. Such a region is flat enough that we can draw a paper map of it. The set of rotation matrices \(\mathrm{SO}(3)\) lives inside the \(9\)-dimensional space of \(3 \times 3\) real matrices, but the set itself is only \(3\)-dimensional, and any small neighborhood of a particular rotation can be parametrized by three angles. Three further examples are the space of all smooth probability distributions on a fixed sample space, the configuration space of a robotic arm, and the orbit of a planet under a gravitational constraint. Each is a curved, non-Euclidean space that nevertheless admits a Euclidean description in the small.

The framework we develop here is purely topological. Only the notion of "local Euclidean similarity" is at issue, and derivatives play no role. This is a deliberate intermediate step. The structures used in physics, robotics, optimization, and machine learning are almost always smooth manifolds, manifolds on which differentiation makes sense. The smooth structure is built on top of the topological one as a second, independent layer. We treat the topological layer in isolation first, both because the topological conditions are subtle and because the layered approach clarifies what differentiation actually adds.

The Three Conditions

A manifold of dimension \(n\) is a topological space satisfying three conditions. The third is the geometric content. The first two are technical requirements that exclude certain pathological spaces.

The third condition is the geometric heart. At every point, \(M\) admits a "chart" that represents a neighborhood of that point as a piece of \(\mathbb{R}^n\). We will study charts and atlases in detail in the next section. For now we examine the role of the first two conditions, which may initially appear technical but turn out to be indispensable.

Why Hausdorff?

Without the Hausdorff condition, even a space that is otherwise locally Euclidean can fail to have a unique limit for convergent sequences. Limits, and therefore continuity in any sharpened form, become ambiguous. The standard cautionary example is the line with two origins. We take two copies of \(\mathbb{R}\) and identify them at every nonzero point, leaving the two origins distinct. The resulting space is locally Euclidean of dimension \(1\) at every point, and it is second-countable. However, the sequence \(x_n = 1/n\) converges both to the first origin and to the second origin, since any neighborhood of either origin eventually contains all \(x_n\). Uniqueness of limits is lost. No usable theory of differentiation, or even of continuous functions, can be built on such a space.

Hausdorffness rules out this kind of identification. Two distinct points of a Hausdorff space can always be separated by disjoint open sets, and the points where a convergent sequence "wants to land" cannot multiply.

Why Second-Countable?

Second-countability is a size constraint. The topology of \(M\) must be describable using countably many open sets. This excludes manifolds that are "too large" for the methods of analysis. The classical example is the long line, a one-dimensional locally Euclidean Hausdorff space that is not second-countable. The long line is, in effect, an uncountable concatenation of copies of \([0, 1)\) ordered by the first uncountable ordinal. In that construction the local Euclidean structure is preserved by the specific well-ordering, but the resulting space contains too many "directions" to admit a countable basis.

The long line cannot be covered by countably many charts, and many constructions break down on it. Those constructions include integration, Riemannian metrics, and partitions of unity.

The deeper role of second-countability emerges in the next page, on the topological properties of manifolds. Combined with local compactness (which manifolds enjoy automatically), second-countability implies paracompactness, which is in turn the technical foundation for partitions of unity. That tool glues local constructions on charts into global structures, and it underlies standard global constructions such as building a Riemannian metric on a manifold or integrating a function over it.

Second-countability also forces the number of connected components to be at most countable.

Topological Invariance of Dimension

The third condition asserts the existence of a homeomorphism \(\varphi : U \to \widehat{U} \subseteq \mathbb{R}^n\) for some \(n\) at each point. A natural question is whether \(n\) is well-defined: could the same manifold be locally Euclidean of dimension \(n\) at some points and of dimension \(m \ne n\) at others? For a connected manifold, the answer is no, and this is a deep theorem.

The proof of this theorem requires tools from algebraic topology that lie beyond the scope of the present series. We treat invariance of dimension as a foundational fact. From now on, when we write "an \(n\)-manifold," the value of \(n\) is a genuine topological invariant of the space.

Open Submanifolds at the Topological Level

We record a useful preliminary observation. Open subsets of manifolds are themselves manifolds. This is the first hint of the "locality" of manifold theory. Structures restrict to open subsets without modification.

The geometric content of a manifold is the system of charts that witnesses the locally Euclidean condition. Those charts are also the means by which we will work with a manifold concretely. We turn to charts and the transition maps that relate them in the next section.

Coordinate Charts

The locally Euclidean condition guarantees that every point \(p \in M\) has a neighborhood that "looks like" a piece of \(\mathbb{R}^n\). Making this concrete requires naming the homeomorphism. The basic data is an open set together with its Euclidean representation, and this pair is called a coordinate chart. The metaphor of a paper map is exact. A chart on a manifold plays the same role as a flat map of a region of the Earth.

The role of a chart is to assign Euclidean coordinates to points of \(M\) within the chart domain. Writing the components of \(\varphi\) as \[ \varphi(p) = (x^1(p), x^2(p), \ldots, x^n(p)), \] each function \(x^i : U \to \mathbb{R}\) is called a coordinate function, or, more loosely, "the \(i\)-th coordinate."

The superscript notation \(x^i\) denotes an index and not a power. This notation is standard in differential geometry, and it foreshadows the upper/lower index distinction we will formalize when we introduce smooth structures and the Einstein summation convention.

There are several equivalent notations for a chart, all of which appear freely in the literature:

• \((U, \varphi)\), the chart-name notation, emphasizing the homeomorphism.
• \((U, (x^1, \ldots, x^n))\), the coordinate notation, emphasizing the individual coordinate functions.
• \((U, (x^i))\), the shorthand, when the dimension is understood.

Charts Centered at a Point and Coordinate Balls

Two refinements of the chart concept appear constantly in practice. The first selects a chart based at a specified point.

Centered charts exist whenever charts do. Given any chart \((U, \varphi)\) with \(p \in U\), let \(\tau : \mathbb{R}^n \to \mathbb{R}^n\) be the translation \(\tau(y) = y - \varphi(p)\). Then \(\tau \circ \varphi : U \to \tau(\widehat{U})\) is again a homeomorphism onto an open subset of \(\mathbb{R}^n\), and \((\tau \circ \varphi)(p) = 0\), so \((U, \tau \circ \varphi)\) is a chart centered at \(p\). It is therefore harmless, and frequently convenient, to assume in proofs that a chart through \(p\) is centered at \(p\).

The second refinement classifies a chart by the shape of its codomain in \(\mathbb{R}^n\).

The terminology is descriptive rather than restrictive. Every coordinate domain can be shrunk to a coordinate ball or cube around any of its points. Given \((U, \varphi)\) and \(p \in U\), choose an open ball (resp. cube) \(B \subseteq \widehat{U}\) containing \(\varphi(p)\). The restricted chart \((\varphi^{-1}(B), \varphi|_{\varphi^{-1}(B)})\) is a coordinate ball (resp. cube) around \(p\). The choice of shape is a matter of convenience for the argument at hand.

Atlases and the Covering Question

A single chart describes only an open piece of a manifold. To capture the global structure, we collect charts into an atlas.

Every topological manifold admits an atlas. The locally Euclidean condition provides a chart around each point, and the collection of all such charts is automatically an atlas. The interesting questions concern how small an atlas can be and what the relations among its charts look like.

Dimension and the Number of Charts

A common source of initial confusion is the relationship between the dimension of a manifold and the number of charts needed to cover it. These two quantities are essentially independent, and conflating them obscures the geometric content.

The dimension \(n\) is a local invariant. It measures the size of the Euclidean space that each point's neighborhood looks like. The number of charts in a minimal atlas is a global topological invariant. It reflects how the manifold is shaped overall.

Consider \(\mathbb{R}^n\) itself as a manifold. Regardless of how large \(n\) is, \(\mathbb{R}^n\) is covered by a single chart, namely the identity map \(\mathrm{id} : \mathbb{R}^n \to \mathbb{R}^n\). A thousand-dimensional Euclidean space is a one-chart manifold, just as the real line is. In a machine learning context, this is the situation of an ordinary feature vector space: a high-dimensional ambient space, but one with a globally trivial chart structure.

Contrast this with the \(n\)-sphere \(\mathbb{S}^n\). For any \(n \ge 1\), \(\mathbb{S}^n\) cannot be covered by a single chart. The reason is topological. By Heine-Borel, the sphere \(\mathbb{S}^n\) is compact, being closed and bounded in \(\mathbb{R}^{n+1}\), while no nonempty open subset of \(\mathbb{R}^n\) is compact. Such a set would have to be closed as well as open, hence all of \(\mathbb{R}^n\) by connectedness, and \(\mathbb{R}^n\) is unbounded. A homeomorphism would map the compact \(\mathbb{S}^n\) onto a compact open subset of \(\mathbb{R}^n\), which cannot exist. The phenomenon is independent of \(n\). The sphere \(\mathbb{S}^{1000}\) requires multiple charts for the same reason \(\mathbb{S}^1\) does. The cause is not dimension but global topology.

The cardinality of a minimal atlas is itself a coarse measure of complexity. What truly governs the geometry of a manifold is not how many charts are used but how the charts relate where they overlap. We turn to this now.

Overlaps and Transition Maps

For a connected manifold covered by two or more nonempty charts, the charts must overlap. Indeed, if \(M\) is connected and \(\mathcal{A} = \{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}\) is an atlas with \(|A| \ge 2\) whose chart domains are nonempty and pairwise disjoint, fix any \(\alpha_0 \in A\). The sets \(U_{\alpha_0}\) and \(\bigsqcup_{\beta \ne \alpha_0} U_\beta\) are nonempty, disjoint, open, and their union is \(M\), contradicting connectedness. Thus an atlas of a connected manifold with two or more nonempty charts must involve genuine overlaps.

Overlap regions are where the descriptive power of an atlas is concentrated. Suppose \((U, \varphi)\) and \((V, \psi)\) are two charts with \(U \cap V \ne \emptyset\). A point \(p \in U \cap V\) is then assigned two different coordinate representations: \(\varphi(p) \in \mathbb{R}^n\) under the first chart, and \(\psi(p) \in \mathbb{R}^n\) under the second. The relation between these two representations is captured by a single map between open subsets of \(\mathbb{R}^n\).

The transition map \(\psi \circ \varphi^{-1}\) is automatically a homeomorphism. The inverse \(\varphi^{-1}\) is a homeomorphism from \(\varphi(U \cap V)\) to \(U \cap V\), and \(\psi\) is a homeomorphism from \(U \cap V\) to \(\psi(U \cap V)\), so the composition is a homeomorphism between open subsets of \(\mathbb{R}^n\).

At the topological level, this is as much as can be said. The transition maps of a topological atlas are homeomorphisms, that is, continuous bijections with continuous inverses. They need not be differentiable, let alone smooth. That gap is the reason a purely topological manifold structure is insufficient for calculus. There is no well-defined notion of a smooth function on \(M\) unless the transition maps themselves are required to be smooth. We turn to the construction of smooth structures in the upcoming page on smooth manifolds.

The remaining sections of this page exhibit the four most important families of topological manifolds, organized by the technique used to construct them: graphs of continuous functions, spheres, projective spaces, and products. Each family will reappear later carrying smooth structure, and the smooth versions will be built directly on top of the topological constructions given here.

Graphs of Continuous Functions

The first construction we examine is the most elementary of the four: the graph of a continuous function. As topological manifolds, graphs are essentially trivial. They are covered by a single chart, and that chart is already built into the definition. The reason for treating them carefully is that the technique they introduce, sometimes called the graph chart construction, becomes the workhorse of the next section.

Setup

Let \(U \subseteq \mathbb{R}^n\) be an open subset and \(f : U \to \mathbb{R}^k\) a continuous function. The graph of \(f\) is the subset \[ \Gamma(f) = \{(x, f(x)) \in \mathbb{R}^{n+k} : x \in U\} \subseteq \mathbb{R}^{n+k}, \] equipped with the subspace topology inherited from \(\mathbb{R}^{n+k}\). Geometrically, \(\Gamma(f)\) is the \(n\)-dimensional "tilted copy" of \(U\) sitting inside \(\mathbb{R}^{n+k}\) that records the value of \(f\) at every point of \(U\).

The Graph as a Manifold

The graph admits a single, natural chart. We project away the \(f\)-coordinates to recover the input \(x \in U\). This single map covers all of \(\Gamma(f)\), so a single-chart atlas suffices to make \(\Gamma(f)\) a topological \(n\)-manifold.

The Graph Chart Construction

The argument just given is more general than the statement suggests. What we have actually shown is that whenever a subset \(S \subseteq \mathbb{R}^{n+k}\) can be written as the graph of a continuous function on some open subset of \(\mathbb{R}^n\), the projection forgetting the last \(k\) coordinates serves as a chart for \(S\). This observation has a name in the manifold literature: the graph chart construction.

In its full form, the construction allows the role of the "input" coordinates and the "output" coordinates to be played by any partition of the ambient coordinates of \(\mathbb{R}^{n+k}\). For instance, a subset of \(\mathbb{R}^3\) that is locally the graph of a continuous function of \(y, z\), that is, locally of the form \(x = g(y, z)\), can be charted by the projection \((x, y, z) \mapsto (y, z)\). The relevant condition is not which coordinates are "input" and which are "output", but rather that the set is locally a graph of some continuous function from one coordinate set to the rest.

This flexibility is exactly what is required to chart the sphere. The sphere \(\mathbb{S}^n \subseteq \mathbb{R}^{n+1}\) is not globally the graph of any single function, but every open hemisphere is locally a graph. Once one coordinate is singled out, the remaining \(n\) coordinates serve as the input, and the chosen coordinate is determined by the constraint \(\sum (x^i)^2 = 1\). The explicit construction occupies the next section.

Spheres

The sphere \(\mathbb{S}^n\) is the second of the four constructions we examine, and in many ways the most important. It is the standard example of a manifold that genuinely requires multiple charts. The chart construction we give here applies the graph chart machinery of the previous section directly, and it is the one used throughout differential geometry.

Spheres are also the most concretely visualizable non-Euclidean manifolds, and they will reappear repeatedly: as Lie groups (\(\mathbb{S}^1, \mathbb{S}^3\)), as factors of tori \(\mathbb{T}^n = (\mathbb{S}^1)^n\), and as base spaces for fundamental constructions in geometric deep learning, where rotation-equivariant networks process data living on \(\mathbb{S}^2\) or \(\mathbb{S}^3\).

Setup

The \(n\)-sphere is the subset of \(\mathbb{R}^{n+1}\) consisting of unit vectors: \[ \mathbb{S}^n = \left\{ x = (x^1, \ldots, x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots + (x^{n+1})^2 = 1 \right\}. \] We equip \(\mathbb{S}^n\) with the subspace topology inherited from \(\mathbb{R}^{n+1}\). The Hausdorff and second-countable conditions are automatic. Any subspace of \(\mathbb{R}^{n+1}\) inherits both properties from the ambient space. What requires work is the third manifold condition. We must exhibit an explicit atlas that witnesses local Euclidean structure of dimension \(n\) at every point.

The Hemisphere Charts

The idea is to view \(\mathbb{S}^n\) as a union of open hemispheres, each of which is a graph of a continuous function over an open ball in \(\mathbb{R}^n\). There are \(2(n+1)\) hemispheres in total, two for each of the \(n+1\) coordinate directions in the ambient space.

For each index \(i \in \{1, \ldots, n+1\}\) and each sign \(\epsilon \in \{+1, -1\}\), define the open hemisphere \[ U_i^\epsilon = \{ x \in \mathbb{S}^n : \epsilon \cdot x^i \gt 0 \}. \]

The set \(U_i^\epsilon\) consists of the points on the sphere where the \(i\)-th coordinate has the specified sign. Geometrically, it is one of the two open hemispheres whose pole is on the \(i\)-th coordinate axis. Each \(U_i^\epsilon\) is open in \(\mathbb{S}^n\), being the intersection of \(\mathbb{S}^n\) with the open half-space \(\{x \in \mathbb{R}^{n+1} : \epsilon \cdot x^i \gt 0\}\).

Define the chart map \[ \varphi_i^\epsilon : U_i^\epsilon \to \mathbb{B}^n, \quad \varphi_i^\epsilon(x^1, \ldots, x^{n+1}) = (x^1, \ldots, \widehat{x^i}, \ldots, x^{n+1}), \] where \(\mathbb{B}^n = \{u \in \mathbb{R}^n : |u| \lt 1\}\) is the open unit ball in \(\mathbb{R}^n\), and the hat over \(x^i\) indicates that the \(i\)-th coordinate is omitted from the tuple.

The chart map sends a point on the hemisphere to its remaining \(n\) coordinates. These lie inside \(\mathbb{B}^n\) because \(x^i \ne 0\) on \(U_i^\epsilon\), so that \(\sum_{j \ne i} (x^j)^2 = 1 - (x^i)^2 \lt 1\). The omitted coordinate is in turn recovered from the others as \(x^i = \epsilon \sqrt{1 - \sum_{j \ne i} (x^j)^2}\).

Remarks on the Construction

Two observations about the hemisphere atlas are worth noting now, even though their full significance becomes apparent only later.

First, the number of charts, \(2(n+1)\), is not the minimum possible. The sphere \(\mathbb{S}^n\) can in fact be covered by just two charts using stereographic projection from the north and south poles. The hemisphere atlas is preferred at this stage because it is the most direct application of the graph chart construction and because its symmetry makes it easier to analyze. That symmetry treats every coordinate direction identically. The stereographic atlas is mentioned only for context. In differential geometry, the choice of atlas is often a matter of computational convenience rather than logical necessity, and several distinct atlases can describe the same manifold.

Second, the construction so far is purely topological. The transition maps between distinct charts \((U_i^\epsilon, \varphi_i^\epsilon)\) and \((U_j^{\epsilon'}, \varphi_j^{\epsilon'})\) on their overlap are homeomorphisms by the general theory of transition maps. Their explicit form, and the smoothness of that form, is needed only when we upgrade \(\mathbb{S}^n\) to a smooth manifold.

The transition formulas \(\varphi_j^{\epsilon'} \circ (\varphi_i^\epsilon)^{-1}\) involve square roots of the form \(\sqrt{1 - |u|^2}\), which are smooth on the relevant open subsets of \(\mathbb{R}^n\). We defer this verification to the future page on smooth structures of the sphere and other constructed manifolds. At the topological level treated here, the existence of the homeomorphism atlas is the entire content of the statement.

Projective Spaces

The third construction is qualitatively different from the previous two. Where graphs and spheres are presented as subsets of an ambient Euclidean space, real projective space \(\mathbb{RP}^n\) is built by an act of identification. Points of a larger space that are related by a specified equivalence are declared to be the same. This is the quotient construction, and projective space is its first nontrivial example in our development. The construction will recur in more elaborate forms throughout differential geometry, most directly in the future Grassmann manifold construction, of which \(\mathbb{RP}^n\) is the simplest case.

Setup

Real projective \(n\)-space is the set of one-dimensional linear subspaces (that is, lines through the origin) of \(\mathbb{R}^{n+1}\). Equivalently, it is the quotient of \(\mathbb{R}^{n+1} \setminus \{0\}\) by the equivalence relation that identifies nonzero scalar multiples: \[ \mathbb{RP}^n = \bigl( \mathbb{R}^{n+1} \setminus \{0\} \bigr) / \sim, \quad x \sim y \iff y = \lambda x \text{ for some } \lambda \in \mathbb{R} \setminus \{0\}. \] We write \([x] = [x^1, \ldots, x^{n+1}]\) for the equivalence class of \(x = (x^1, \ldots, x^{n+1})\), and \(\pi : \mathbb{R}^{n+1} \setminus \{0\} \to \mathbb{RP}^n\) for the quotient map sending \(x\) to \([x]\). The set \(\mathbb{RP}^n\) is endowed with the quotient topology induced by \(\pi\). A subset \(V \subseteq \mathbb{RP}^n\) is open if and only if \(\pi^{-1}(V)\) is open in \(\mathbb{R}^{n+1} \setminus \{0\}\).

Geometrically, a point of \(\mathbb{RP}^n\) is a line through the origin in \(\mathbb{R}^{n+1}\). The representative \(x\) is any nonzero point lying on the line, and the equivalence \(x \sim \lambda x\) says simply that any two such points represent the same line. This geometric picture motivates the chart construction below. To coordinate a line, we intersect it with a suitable affine hyperplane and record the intersection point.

The Affine Charts

For each index \(i \in \{1, \ldots, n+1\}\), the chart domain consists of those lines whose \(i\)-th coordinate is nonzero. Equivalently, it consists of the lines that meet the affine hyperplane \(\{y \in \mathbb{R}^{n+1} : y^i = 1\}\) in a unique point. Define \[ \widetilde{U}_i = \{ x \in \mathbb{R}^{n+1} \setminus \{0\} : x^i \ne 0 \}, \quad U_i = \pi(\widetilde{U}_i) \subseteq \mathbb{RP}^n. \]

The set \(\widetilde{U}_i\) is open in \(\mathbb{R}^{n+1} \setminus \{0\}\), and it is saturated under \(\sim\) (if \(x \in \widetilde{U}_i\) and \(\lambda \ne 0\), then \(\lambda x \in \widetilde{U}_i\)). It follows from the definition of the quotient topology that \(U_i\) is open in \(\mathbb{RP}^n\).

Define the chart map by recording the ratios of the other coordinates to the \(i\)-th: \[ \varphi_i : U_i \to \mathbb{R}^n, \quad \varphi_i\bigl([x^1, \ldots, x^{n+1}]\bigr) = \left( \frac{x^1}{x^i}, \ldots, \widehat{\frac{x^i}{x^i}}, \ldots, \frac{x^{n+1}}{x^i} \right), \] where the hat indicates that the \(i\)-th entry (which would equal \(1\)) is omitted, leaving \(n\) coordinates in the output.

Well-definedness.
The right-hand side is independent of the choice of representative. If \(y = \lambda x\) with \(\lambda \ne 0\), then \(y^j / y^i = (\lambda x^j) / (\lambda x^i) = x^j / x^i\) for every \(j\), so the value of \(\varphi_i\) on the equivalence class \([x]\) does not depend on which point of the line we used to compute it.

Geometric interpretation.
The image \(\varphi_i([x])\) records the unique point at which the line \([x]\) meets the affine hyperplane \(\{y \in \mathbb{R}^{n+1} : y^i = 1\}\). To see this, observe that the line \([x]\) intersects this hyperplane at the unique point \(x / x^i\), whose coordinates are \((x^1 / x^i, \ldots, x^i / x^i, \ldots, x^{n+1}/x^i) = (x^1/x^i, \ldots, 1, \ldots, x^{n+1}/x^i)\). Dropping the entry equal to \(1\), which sits in the \(i\)-th slot, leaves precisely the \(n\) numbers that form \(\varphi_i([x])\).

Compactness and Two Useful Descriptions

Projective space is compact. The quotient map restricts to a continuous surjection \(\mathbb{S}^n \to \mathbb{RP}^n\) (every line through the origin meets the sphere), and the continuous image of a compact space is compact. This argument also reveals \(\mathbb{RP}^n\) under a second guise: as the quotient of \(\mathbb{S}^n\) by the antipodal identification \(x \sim -x\). Both descriptions are useful and will appear in subsequent constructions.

The case \(n = 1\) deserves special mention. The space \(\mathbb{RP}^1\) is the set of lines through the origin in \(\mathbb{R}^2\). Every such line other than the \(x\)-axis meets the closed upper half of the unit circle in exactly one point, while the \(x\)-axis meets it at both endpoints \((1, 0)\) and \((-1, 0)\), which therefore represent the same line. Thus \(\mathbb{RP}^1\) is the closed half-circle with its two endpoints identified, hence homeomorphic to a circle, although the natural quotient map \(\mathbb{S}^1 \to \mathbb{RP}^1\) is two-to-one (each pair of antipodal points on \(\mathbb{S}^1\) maps to a single point of \(\mathbb{RP}^1\)). This double-cover relationship between \(\mathbb{S}^1\) and \(\mathbb{RP}^1\) generalizes to all \(n\). The antipodal quotient \(\mathbb{S}^n \to \mathbb{RP}^n\) is a two-to-one covering map for every \(n \ge 1\).

Projective space will reappear later in two important contexts. First, \(\mathbb{RP}^n\) is the simplest example of a Grassmann manifold. If \(G_k(\mathbb{R}^{n+1})\) denotes the space of \(k\)-dimensional linear subspaces of \(\mathbb{R}^{n+1}\), then \(\mathbb{RP}^n = G_1(\mathbb{R}^{n+1})\). Second, the smooth structure on \(\mathbb{RP}^n\) will be constructed in a later page. It is built directly from the affine charts above, by verifying that the transition maps \(\varphi_j \circ \varphi_i^{-1}\) are rational functions and hence smooth. The topological construction completed here is the foundation on which the smooth and the Grassmannian generalizations are built.

Products and Tori

The fourth and final construction is the most mechanical of the four. We take the Cartesian product of two manifolds and equip it with the product topology. The result is again a manifold, with dimension equal to the sum of the dimensions of the factors and an atlas built by taking products of charts. The recipe is routine, but it is the source of one of the most important examples in geometry, the torus \(\mathbb{T}^n = \mathbb{S}^1 \times \cdots \times \mathbb{S}^1\). It is also the technique by which higher-dimensional manifolds are assembled from low-dimensional ones.

The Product of Manifolds

Given topological manifolds \(M_1, \ldots, M_k\) of dimensions \(n_1, \ldots, n_k\) respectively, the Cartesian product \[ M = M_1 \times \cdots \times M_k \] is endowed with the product topology, in which a base of open sets consists of products \(V_1 \times \cdots \times V_k\) where each \(V_j\) is open in \(M_j\).

The Torus

The most important application of the product construction at this stage is the torus.

The terminology can be subtle. In everyday usage, "the torus" almost always refers specifically to \(\mathbb{T}^2 = \mathbb{S}^1 \times \mathbb{S}^1\), the two-dimensional doughnut surface. The higher-dimensional cases \(\mathbb{T}^n\) for \(n \ge 3\) are routinely called "\(n\)-tori" or simply "tori," but no concrete geometric picture in three-dimensional space is available for them.

The torus inherits from the product structure both an atlas and a great deal of geometric content. The atlas built from the product of the standard hemisphere atlases of each circle factor consists of \(4^n\) charts (each \(\mathbb{S}^1\) contributes \(4\) hemisphere charts, and the product takes one chart from each factor). A more economical alternative is the product of stereographic atlases. Each \(\mathbb{S}^1\) is then covered by \(2\) charts, and the product has \(2^n\) charts in total. The hemisphere product atlas is nonetheless the conceptually natural one. Each chart maps a product of open arcs in \(\mathbb{T}^n\) onto an open box in \(\mathbb{R}^n\), namely a product of \(n\) copies of the interval \((-1, 1)\).

Although our treatment here is purely topological, the torus is one of the most important examples in differential geometry for several reasons. The torus is a compact manifold (a finite product of compact manifolds is compact), so it serves as a natural setting for global analysis. It also carries a natural group structure inherited componentwise from \(\mathbb{S}^1 \subseteq \mathbb{C}^*\), which makes it the prototypical example of a compact abelian Lie group. That Lie-theoretic structure will be the subject of a later page in the manifold series. Finally, the torus appears throughout physics and applied mathematics: as the configuration space of pendulum systems, as the support of multivariate periodic Fourier series, and as a quotient space \(\mathbb{R}^n / \mathbb{Z}^n\) that arises naturally in number theory and dynamical systems.