Smooth Sphere, Projective Space, and Tori

The Smooth Sphere

The previous page constructed the abstract apparatus of smooth manifolds — smooth atlases, smooth structures, and the proposition that any smooth atlas determines a unique smooth structure containing it. The standard smooth structure on \(\mathbb{R}^n\) and the inherited smooth structures on finite-dimensional vector spaces and on \(GL(n, \mathbb{R})\) were the only concrete examples treated there. Beginning with this page, the apparatus is put to work on the genuinely curved manifolds that drive both classical geometry and modern applications: spheres, projective spaces, and tori.

Each of the three examples below is already a topological manifold by the constructions of the topological-manifolds page, where graph charts, quotient charts, and product charts were exhibited and the three manifold conditions verified. The work that remains is the smooth-compatibility check: we revisit the explicit charts already in hand and verify that their transition maps, viewed as maps between open subsets of \(\mathbb{R}^n\), are smooth in the ordinary calculus sense. Once that check is complete, the smooth structure generated by the atlas is the smooth structure we adopt, and the topological manifold has been promoted to a smooth one.

The first example is the \(n\)-sphere.

Recall: The Hemisphere Atlas

Recall from the topological construction of \(\mathbb{S}^n\) that the \(n\)-sphere \[ \mathbb{S}^n = \left\{ x = (x^1, \ldots, x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots + (x^{n+1})^2 = 1 \right\} \] is covered by \(2(n+1)\) hemisphere charts. For each \(i \in \{1, \ldots, n+1\}\) and each sign \(\epsilon \in \{+, -\}\), the open hemisphere \[ U_i^\epsilon = \{ x \in \mathbb{S}^n : \epsilon \cdot x^i > 0 \} \] is mapped onto the open unit ball \[ \mathbb{B}^n = \{u \in \mathbb{R}^n : |u| < 1\} \] by the chart \[ \varphi_i^\epsilon : U_i^\epsilon \to \mathbb{B}^n, \qquad \varphi_i^\epsilon(x^1, \ldots, x^{n+1}) = (x^1, \ldots, \widehat{x^i}, \ldots, x^{n+1}), \] where the hat denotes omission of the \(i\)-th entry. In words, the chart simply drops the \(i\)-th coordinate of an ambient point on the sphere and reports the remaining \(n\) coordinates as a point of the unit ball. The inverse map performs the converse operation: given \(n\) numbers in the ball, it reconstructs the dropped coordinate from the sphere equation \(x^i = \epsilon\sqrt{1 - \sum_{k \ne i}(x^k)^2}\) and inserts it back at position \(i\), \[ (\varphi_i^\epsilon)^{-1}(u^1, \ldots, u^n) = \bigl(u^1, \ldots, u^{i-1},\, \epsilon\sqrt{1 - |u|^2},\, u^i, \ldots, u^n\bigr). \] The sign \(\epsilon\) determines which of the two solutions of the sphere equation is selected — the positive or negative root. The collection \[ \mathcal{A}_{\mathbb{S}^n} = \{(U_i^\epsilon, \varphi_i^\epsilon) : i = 1, \ldots, n+1,\ \epsilon \in \{+, -\}\} \] is the topological atlas whose smooth compatibility we will now verify.

Transition Maps Between Hemisphere Charts

Smooth compatibility of two charts is a condition on their transition map. Fix two charts \((U_i^\epsilon, \varphi_i^\epsilon)\) and \((U_j^{\epsilon'}, \varphi_j^{\epsilon'})\) on \(\mathbb{S}^n\), and consider the transition map \(\varphi_j^{\epsilon'} \circ (\varphi_i^\epsilon)^{-1}\). This map first reads a parameter point of \(\mathbb{B}^n\) back to a point on the sphere via \((\varphi_i^\epsilon)^{-1}\), and then re-reads that sphere point through the second chart \(\varphi_j^{\epsilon'}\). The result is a map between open subsets of \(\mathbb{R}^n\) — exactly the kind of map to which the ordinary multivariable calculus definition of smoothness applies. Three cases arise, organized by the relation between the indices \(i\) and \(j\) and the signs \(\epsilon, \epsilon'\).

Case 1: \(i = j\) with opposite signs.
When the two charts use the same coordinate index but opposite signs, their domains are the antipodal hemispheres \(U_i^+\) and \(U_i^-\). One consists of the points of \(\mathbb{S}^n\) with \(x^i > 0\) and the other of those with \(x^i < 0\); the two conditions cannot hold simultaneously, so \(U_i^+ \cap U_i^- = \emptyset\). With an empty overlap there is no transition map to write down, and the smooth-compatibility condition is satisfied vacuously by its first clause.

Case 2: \(i = j\) with the same sign.
Here the two "charts" coincide as labeled elements of the atlas: \((U_i^\epsilon, \varphi_i^\epsilon)\) and \((U_j^{\epsilon'}, \varphi_j^{\epsilon'})\) are literally the same chart. The transition map \(\varphi_i^\epsilon \circ (\varphi_i^\epsilon)^{-1}\) is then the identity on \(\mathbb{B}^n\), which is a diffeomorphism. We include this case for completeness — every pair of charts in the atlas, including the pair consisting of a chart with itself, must be checked — but the verification is immediate.

Case 3: \(i \ne j\).
This is the substantive case, and it is the only one that produces a transition map with nontrivial content. We compute \(\varphi_j^{\epsilon'} \circ (\varphi_i^\epsilon)^{-1}\) explicitly, treating the subcase \(i < j\) first and then the subcase \(i > j\); the two computations are structurally identical but differ in bookkeeping, so writing both out makes the index tracking transparent.

Subcase 3a: \(i < j\).
Start with a point \(u = (u^1, \ldots, u^n) \in \mathbb{R}^n\) lying in the domain \(\varphi_i^\epsilon(U_i^\epsilon \cap U_j^{\epsilon'})\) of the transition. Apply \((\varphi_i^\epsilon)^{-1}\) first. By construction, this map reconstructs a point of \(\mathbb{S}^n\) by inserting the radical \(\epsilon\sqrt{1 - |u|^2}\) at the \(i\)-th coordinate position and shifting the remaining entries of \(u\) to fill the other \(n\) positions: \[ (\varphi_i^\epsilon)^{-1}(u^1, \ldots, u^n) = \bigl(\underbrace{u^1, \ldots, u^{i-1}}_{\text{positions } 1, \ldots, i-1},\, \underbrace{\epsilon\sqrt{1 - |u|^2}}_{\text{position } i},\, \underbrace{u^i, \ldots, u^n}_{\text{positions } i+1, \ldots, n+1}\bigr). \] Two things to notice. First, the radical sits at position \(i\) in the ambient \((n+1)\)-tuple, but it carries the value \(\epsilon\sqrt{1 - |u|^2}\) — a value computed from \(u\), not a value inherited directly from any entry of \(u\). Second, the original parameter entry \(u^k\) for \(k \ge i\) gets pushed to position \(k+1\) of the ambient tuple, because the radical insertion has used up position \(i\).

Now apply \(\varphi_j^{\epsilon'}\), which drops the \(j\)-th coordinate of the ambient tuple. Since we assumed \(i < j\), position \(j\) lies strictly to the right of position \(i\) where the radical sits — so the radical is preserved, and what gets dropped is one of the entries pushed there from \(u\). Specifically, position \(j\) of the ambient tuple holds the parameter entry \(u^{j-1}\) (because, as noted above, parameter entry \(u^k\) for \(k \ge i\) sits at ambient position \(k+1\), and the choice \(k = j-1 \ge i\) gives ambient position \(j\)). Dropping that entry yields \[ \varphi_j^{\epsilon'} \circ (\varphi_i^\epsilon)^{-1}(u^1, \ldots, u^n) = \bigl(u^1, \ldots, u^{i-1},\, \epsilon\sqrt{1 - |u|^2},\, u^i, \ldots, \widehat{u^{j-1}}, \ldots, u^n\bigr), \] where the radical sits at the \(i\)-th position of the output and the hat at the \((j-1)\)-st position indicates the omitted parameter entry. The two signs \(\epsilon\) and \(\epsilon'\) need not agree: the sign \(\epsilon\) determines which hemisphere \((\varphi_i^\epsilon)^{-1}\) reconstructs into, and \(\epsilon'\) plays no role in the formula at all, since dropping a coordinate does not depend on the sign of the chart that drops it.

Subcase 3b: \(i > j\).
The same procedure applies, but now position \(j\) lies strictly to the left of position \(i\). When \((\varphi_i^\epsilon)^{-1}\) inserts the radical at position \(i\) of the ambient tuple, the entries to the left of position \(i\) (which include position \(j\)) are unaffected: the parameter entry \(u^k\) for \(k < i\) sits at ambient position \(k\). Applying \(\varphi_j^{\epsilon'}\), the entry dropped is \(u^j\) — at parameter index \(j\), now also at ambient position \(j\). The radical at ambient position \(i\) survives, and the entries pushed past it from positions \(i, \ldots, n\) of \(u\) survive as well. The result is \[ \varphi_j^{\epsilon'} \circ (\varphi_i^\epsilon)^{-1}(u^1, \ldots, u^n) = \bigl(u^1, \ldots, \widehat{u^j}, \ldots, u^{i-1},\, \epsilon\sqrt{1 - |u|^2},\, u^i, \ldots, u^n\bigr), \] with the radical now at the \((i-1)\)-st position of the output (one position has been opened up by the omission to its left) and the omission at the \(j\)-th position.

In both subcases, the structure of the output is the same: a tuple of \(n\) entries, of which \(n - 1\) are coordinate projections \(u \mapsto u^k\) and the remaining one is the radical \(\epsilon\sqrt{1 - |u|^2}\). The exact positions of these entries depend on the bookkeeping above, but the form of the entries does not. It is on this common form that the smoothness argument now operates.

Smoothness of the Transition Maps

Every entry of the transition tuple computed above is one of two types: a coordinate projection \(u \mapsto u^k\), or the radical \(u \mapsto \epsilon\sqrt{1 - |u|^2}\). The smoothness of the transition map reduces, then, to the smoothness of these two kinds of map on the relevant domain.

Coordinate projections.
The map \(u = (u^1, \ldots, u^n) \mapsto u^k\) selecting the \(k\)-th coordinate is linear, and any linear map on \(\mathbb{R}^n\) is smooth in the calculus sense: its first partial derivatives are constants and all higher derivatives are zero. There is no constraint on the domain — projections are smooth on all of \(\mathbb{R}^n\).

The radical.
The map \(u \mapsto \epsilon\sqrt{1 - |u|^2}\) is a composition of three pieces:

• The polynomial \(u \mapsto 1 - |u|^2 = 1 - \sum_k (u^k)^2\) is smooth on all of \(\mathbb{R}^n\), since polynomials are smooth.
• The square root \(t \mapsto \sqrt{t}\) is smooth on the open half-line \(\{t > 0\}\). Its first derivative \(\tfrac{1}{2}t^{-1/2}\), second derivative \(-\tfrac{1}{4}t^{-3/2}\), and all higher derivatives are continuous on \(\{t > 0\}\). They diverge as \(t \to 0^+\), which is why \(\sqrt{t}\) fails to be smooth at the boundary point \(t = 0\); but the open half-line where it is smooth is exactly what we need.
• Multiplication by the sign \(\epsilon \in \{+, -\}\) is a linear map, hence smooth.

Composition preserves smoothness, so the radical is smooth on the set where the inner polynomial is strictly positive: \[ \{u \in \mathbb{R}^n : 1 - |u|^2 > 0\} = \mathbb{B}^n. \] The transition domain \(\varphi_i^\epsilon(U_i^\epsilon \cap U_j^{\epsilon'})\) is a subset of \(\mathbb{B}^n\) — indeed, the codomain of every hemisphere chart is \(\mathbb{B}^n\) — so the radical is smooth on this domain as well.

Conclusion.
Each entry of the transition tuple is smooth on its domain, so the transition tuple itself is smooth.

The reverse transition \(\varphi_i^\epsilon \circ (\varphi_j^{\epsilon'})^{-1}\) is obtained by the same computation with the roles of \(i\) and \(j\) exchanged, and it produces a tuple of exactly the same form: \(n-1\) coordinate projections and one radical \(\epsilon'\sqrt{1 - |u|^2}\). The argument for its smoothness is identical, with \(\epsilon\) replaced by \(\epsilon'\). The two transitions are mutual inverses that are both smooth, so each is a diffeomorphism, and the charts \((U_i^\epsilon, \varphi_i^\epsilon)\) and \((U_j^{\epsilon'}, \varphi_j^{\epsilon'})\) are smoothly compatible.

Since the choice of charts was arbitrary, every pair of charts in \(\mathcal{A}_{\mathbb{S}^n}\) is smoothly compatible, and \(\mathcal{A}_{\mathbb{S}^n}\) is a smooth atlas on \(\mathbb{S}^n\).

The Smooth Structure on the Sphere

Having a smooth atlas in hand, the smooth structure follows automatically. The smooth-structure-from-atlas proposition from the previous page asserts that any smooth atlas on a topological manifold is contained in a unique maximal smooth atlas; that maximal atlas is, by definition, a smooth structure. So once an atlas has been verified to be smoothly compatible — as \(\mathcal{A}_{\mathbb{S}^n}\) just has — there is nothing further to construct.

The smooth sphere will recur throughout the remainder of the manifold series and beyond. The circle \(\mathbb{S}^1\) and the three-sphere \(\mathbb{S}^3\) carry natural group structures that make them compact Lie groups, and the smooth structure constructed here is the one that interacts compatibly with those group operations — a connection developed when the manifold series reaches Lie groups. In applied settings, neural networks that respect rotational symmetry — used, for instance, to process signals on the surface of the Earth (\(\mathbb{S}^2\)) or three-dimensional orientation data (\(\mathbb{S}^3\)) — are built on the smoothness apparatus established here: their convolution layers and differential operators are defined on the smooth manifold, not on the underlying topological space.

Smooth Projective Spaces

The second example is real projective space \(\mathbb{RP}^n\). Topologically, it was built by an act of identification: points of \(\mathbb{R}^{n+1} \setminus \{0\}\) related by nonzero scalar multiplication were declared equivalent, and the resulting quotient was equipped with the quotient topology. The chart system that exhibited \(\mathbb{RP}^n\) as a topological manifold was a collection of affine charts indexed by the \(n+1\) coordinate directions of the ambient space; each chart corresponded to the affine hyperplane on which one coordinate is set to \(1\). The construction of a smooth structure on \(\mathbb{RP}^n\) takes that same affine atlas and verifies that its transition maps are smooth as maps between open subsets of \(\mathbb{R}^n\).

Recall: The Affine Atlas

Recall from the topological construction of \(\mathbb{RP}^n\) that \[ \mathbb{RP}^n = \bigl( \mathbb{R}^{n+1} \setminus \{0\} \bigr) / \sim, \qquad x \sim y \iff y = \lambda x \text{ for some } \lambda \in \mathbb{R} \setminus \{0\}, \] with equivalence classes denoted \([x] = [x^1, \ldots, x^{n+1}]\). For each index \(i \in \{1, \ldots, n+1\}\), the chart domain consists of those equivalence classes whose representatives have nonzero \(i\)-th coordinate, \[ U_i = \{[x] \in \mathbb{RP}^n : x^i \ne 0\}, \] a well-defined condition because the equivalence relation rescales all coordinates by the same nonzero factor and so preserves the property of being nonzero. The chart map records the ratios of the other coordinates to the \(i\)-th: \[ \varphi_i : U_i \to \mathbb{R}^n, \qquad \varphi_i\bigl([x^1, \ldots, x^{n+1}]\bigr) = \left( \frac{x^1}{x^i}, \ldots, \widehat{\frac{x^i}{x^i}}, \ldots, \frac{x^{n+1}}{x^i} \right), \] where the hat indicates omission of the \(i\)-th ratio (which would equal \(1\)). The inverse map promotes a parameter tuple \((u^1, \ldots, u^n) \in \mathbb{R}^n\) to the equivalence class of an \((n+1)\)-tuple by inserting a \(1\) at position \(i\), \[ \varphi_i^{-1}(u^1, \ldots, u^n) = [u^1, \ldots, u^{i-1}, 1, u^i, \ldots, u^n], \] so that the \(i\)-th coordinate of any representative of this class is \(1\) and the other coordinates are the entries of \(u\) in their original order. The collection \[ \mathcal{A}_{\mathbb{RP}^n} = \{(U_i, \varphi_i) : i = 1, \ldots, n+1\} \] is the topological atlas whose smooth compatibility we now verify.

Computing the Transition Map

Fix two distinct indices \(i, j \in \{1, \ldots, n+1\}\) — the case \(i = j\) collapses to a single chart and gives an identity transition, exactly as for the sphere. The overlap \(U_i \cap U_j\) consists of equivalence classes whose representatives have both \(x^i \ne 0\) and \(x^j \ne 0\). The transition map \(\varphi_j \circ \varphi_i^{-1}\) reads a parameter tuple \(u\) into \(\mathbb{RP}^n\) via \(\varphi_i^{-1}\) and then rereads the resulting class via \(\varphi_j\). Carrying this out for the subcase \(i > j\) gives the following.

Start with \(u = (u^1, \ldots, u^n) \in \varphi_i(U_i \cap U_j)\). The image \(\varphi_i^{-1}(u) = [u^1, \ldots, u^{i-1}, 1, u^i, \ldots, u^n]\) is the equivalence class of the ambient \((n+1)\)-tuple \[ \bigl(\underbrace{u^1, \ldots, u^{i-1}}_{\text{positions } 1, \ldots, i-1},\, \underbrace{1}_{\text{position } i},\, \underbrace{u^i, \ldots, u^n}_{\text{positions } i+1, \ldots, n+1}\bigr). \] Two observations on indexing. First, the constant \(1\) sits at ambient position \(i\). Second, because position \(j\) lies strictly to the left of position \(i\) (we assumed \(i > j\)), the entry at ambient position \(j\) is unaffected by the insertion: it is simply the parameter entry \(u^j\) at parameter index \(j\). For \(u\) to lie in \(\varphi_i(U_i \cap U_j)\), the class must lie in \(U_j\) — equivalently, its \(j\)-th ambient coordinate must be nonzero — so \(u^j \ne 0\) on this domain.

Now apply \(\varphi_j\), which divides every ambient coordinate by the \(j\)-th and omits the resulting \(1\) at position \(j\). The \(j\)-th coordinate here is \(u^j\), so the division is by \(u^j\); the omission removes one entry from the \((n+1)\)-tuple, leaving \(n\) entries in the output. Performing the division position by position and dropping the omitted slot gives the formula \[ \varphi_j \circ \varphi_i^{-1}(u^1, \ldots, u^n) = \left( \frac{u^1}{u^j}, \ldots, \frac{u^{j-1}}{u^j}, \frac{u^{j+1}}{u^j}, \ldots, \frac{u^{i-1}}{u^j}, \frac{1}{u^j}, \frac{u^i}{u^j}, \ldots, \frac{u^n}{u^j} \right). \] Three position checkpoints are worth pausing on. The entries \(u^k / u^j\) for \(k < j\) and \(k > i-1\) come from dividing the unchanged parameter entries by \(u^j\). The entry \(1/u^j\) is the image of the constant \(1\) inserted at ambient position \(i\); this is the only place a constant appears, and it appears precisely because position \(i\) (where the chart \(\varphi_i\) put a \(1\)) survives the omission of position \(j\). The omission of the slot at position \(j\) removes the entry that would have been \(u^j / u^j = 1\) — the omission is automatic from the definition of \(\varphi_j\), which never reports its own \(j\)-th ratio.

The subcase \(i < j\) yields a structurally identical formula obtained by interchanging the roles of the inserted constant and the omitted ratio; we omit the parallel bookkeeping. In both subcases the form of the output is the same: a tuple of \(n\) entries, of which one is the reciprocal \(1/u^j\) and the remaining \(n-1\) are rational functions of the form \(u^k / u^j\). It is on this common form that the smoothness argument operates.

Smoothness of the Transition Map

Each entry of the transition tuple is a rational function of \((u^1, \ldots, u^n)\) — either a reciprocal \(1/u^j\) or a quotient \(u^k / u^j\). Rational functions are smooth on the open set where their denominators are nonzero, and the denominator \(u^j\) is nonzero exactly on the domain \(\varphi_i(U_i \cap U_j)\): this domain consists, by definition, of parameter tuples \(u\) such that the corresponding ambient point has \(j\)-th coordinate \(u^j \ne 0\). So every entry of the transition tuple is smooth on the transition domain, and the transition map itself is smooth.

The reverse transition \(\varphi_i \circ \varphi_j^{-1}\) is obtained by interchanging the roles of \(i\) and \(j\) in the computation above. Its formula has the same shape — \(n-1\) quotients and one reciprocal, with denominators now equal to one of the parameter entries — and the same rational-function argument shows that it is smooth on its domain. The two transitions are mutual inverses that are both smooth, so each is a diffeomorphism, and the charts \((U_i, \varphi_i)\) and \((U_j, \varphi_j)\) are smoothly compatible.

Since the indices were arbitrary, every pair of charts in \(\mathcal{A}_{\mathbb{RP}^n}\) is smoothly compatible, and \(\mathcal{A}_{\mathbb{RP}^n}\) is a smooth atlas on \(\mathbb{RP}^n\).

The Smooth Structure on Projective Space

As with the sphere, the smooth-structure-from-atlas proposition converts the smooth atlas into a smooth structure with no further work.

Real projective space is the simplest of a broader family of manifolds — the Grassmann manifolds — whose points are the \(k\)-dimensional linear subspaces of an ambient Euclidean space. In this language, \(\mathbb{RP}^n = G_1(\mathbb{R}^{n+1})\): a point of \(\mathbb{RP}^n\) is a line through the origin in \(\mathbb{R}^{n+1}\), and a line is a one-dimensional subspace. The next page constructs the smooth structure on \(G_k(\mathbb{R}^{m})\) for arbitrary \(k\) and \(m\) using a more flexible technique that does not require a topology to be in hand at the outset; specializing that construction to \(k = 1\) and \(m = n+1\) recovers the smooth structure on \(\mathbb{RP}^n\) constructed here.

Smooth Product Manifolds and Tori

The third example is the simplest of the three in its construction and the broadest in its applications: given any finite collection of smooth manifolds, their Cartesian product is again a smooth manifold, with smooth structure built factor-by-factor from the smooth structures of the factors. The construction is the smooth counterpart of the topological product, in which finite Cartesian products of topological manifolds were shown to inherit a topological-manifold structure from their factors. The transition-map smoothness check is mechanical: it reduces to the smoothness of the transitions of the factors, which is already in hand by hypothesis.

Recall: Product Charts

Let \(M_1, \ldots, M_k\) be topological manifolds of dimensions \(n_1, \ldots, n_k\). The Cartesian product \(M = M_1 \times \cdots \times M_k\), equipped with the product topology, is a topological manifold of dimension \(n_1 + \cdots + n_k\). The atlas exhibited in the topological setting consists of product charts: if \((U_j, \varphi_j)\) is a chart on \(M_j\) for each \(j\), the product \[ (U_1 \times \cdots \times U_k,\ \varphi_1 \times \cdots \times \varphi_k) \] is a chart on \(M\), where the product map sends \[ (q_1, \ldots, q_k) \in U_1 \times \cdots \times U_k \mapsto \bigl(\varphi_1(q_1), \ldots, \varphi_k(q_k)\bigr) \in \mathbb{R}^{n_1} \times \cdots \times \mathbb{R}^{n_k} = \mathbb{R}^{n_1 + \cdots + n_k}. \] The atlas \[ \mathcal{A}_M = \bigl\{(U_1 \times \cdots \times U_k,\ \varphi_1 \times \cdots \times \varphi_k) : (U_j, \varphi_j) \in \mathcal{A}_j \text{ for each } j\bigr\} \] obtained by taking one chart from the atlas \(\mathcal{A}_j\) of each factor in every possible combination is the topological atlas of \(M\). When each \(M_j\) is in fact a smooth manifold, each \(\mathcal{A}_j\) may be taken to be its smooth structure (or any smooth atlas in it), and the question is whether \(\mathcal{A}_M\) is a smooth atlas on \(M\).

Transition Maps Factor by Factor

Fix two product charts \[ (U_1 \times \cdots \times U_k,\ \varphi_1 \times \cdots \times \varphi_k), \qquad (V_1 \times \cdots \times V_k,\ \psi_1 \times \cdots \times \psi_k), \] where \((U_j, \varphi_j)\) and \((V_j, \psi_j)\) are charts on \(M_j\). Their overlap is the product \[ (U_1 \cap V_1) \times \cdots \times (U_k \cap V_k), \] a fact that follows from the definition of the product topology and the elementary set-theoretic identity \((A_1 \times \cdots) \cap (B_1 \times \cdots) = (A_1 \cap B_1) \times \cdots\). On this overlap, the transition map \((\psi_1 \times \cdots \times \psi_k) \circ (\varphi_1 \times \cdots \times \varphi_k)^{-1}\) acts factor by factor: given a parameter tuple \((u_1, \ldots, u_k)\) with \(u_j \in \varphi_j(U_j \cap V_j) \subseteq \mathbb{R}^{n_j}\), apply \((\varphi_1 \times \cdots \times \varphi_k)^{-1}\) to obtain \((\varphi_1^{-1}(u_1), \ldots, \varphi_k^{-1}(u_k))\), and then apply \(\psi_1 \times \cdots \times \psi_k\) to obtain \((\psi_1(\varphi_1^{-1}(u_1)), \ldots, \psi_k(\varphi_k^{-1}(u_k)))\). Each component depends only on the corresponding factor, so the transition decomposes as \[ (\psi_1 \times \cdots \times \psi_k) \circ (\varphi_1 \times \cdots \times \varphi_k)^{-1} = (\psi_1 \circ \varphi_1^{-1}) \times \cdots \times (\psi_k \circ \varphi_k^{-1}). \] The transition of the product chart is the product of the transitions of the factors.

Smoothness of the Product Transition

By hypothesis, each \(M_j\) is a smooth manifold and \((U_j, \varphi_j), (V_j, \psi_j)\) are charts in its smooth structure, so the factor transition \(\psi_j \circ \varphi_j^{-1}\) is smooth on its domain \(\varphi_j(U_j \cap V_j) \subseteq \mathbb{R}^{n_j}\). The product transition is, then, a map \[ \prod_j \varphi_j(U_j \cap V_j) \to \prod_j \psi_j(U_j \cap V_j), \] between open subsets of \(\mathbb{R}^{n_1 + \cdots + n_k}\), whose component-wise structure is dictated by the product decomposition above. Each component of the output depends only on the variables of one factor — those belonging to \(\varphi_j(U_j \cap V_j)\) — and on those variables it coincides with a component of the factor transition \(\psi_j \circ \varphi_j^{-1}\). Smoothness of a map between Euclidean open sets is equivalent to smoothness of each component function in all of its input variables; here, partial derivatives with respect to variables of factor \(j' \ne j\) all vanish (the component is constant in those variables), and partial derivatives with respect to the variables of factor \(j\) are the partial derivatives of the smooth function \(\psi_j \circ \varphi_j^{-1}\). All partial derivatives of all orders exist and are continuous, so the product transition is smooth.

The reverse transition has the same form with \(\varphi_j\) and \(\psi_j\) interchanged in each factor, and is smooth by the same argument. The two transitions are mutual inverses that are both smooth — hence diffeomorphisms — so any two product charts in \(\mathcal{A}_M\) are smoothly compatible. The atlas \(\mathcal{A}_M\) is a smooth atlas on \(M\).

The Product Smooth Manifold Structure

The smooth-structure-from-atlas proposition converts this atlas into a smooth structure.

The Smooth Torus

The most important application of the product construction at this stage is the torus. The circle \(\mathbb{S}^1\) is a smooth \(1\)-manifold by the standard smooth structure constructed earlier on this page (as the case \(n = 1\) of the smooth sphere). The \(n\)-torus is the \(n\)-fold product of the circle with itself, \[ \mathbb{T}^n = \underbrace{\mathbb{S}^1 \times \mathbb{S}^1 \times \cdots \times \mathbb{S}^1}_{n \text{ factors}}, \] and the theorem above equips it with the product smooth manifold structure: each factor contributes the standard smooth structure on \(\mathbb{S}^1\), and the product atlas built from the hemisphere charts of each circle factor is a smooth atlas on \(\mathbb{T}^n\). The dimension is \(n\) (one from each circle factor), matching the dimension already established at the topological level.

The torus appears repeatedly throughout differential geometry and its applications. As a compact smooth manifold — a finite product of compact smooth manifolds is compact — it serves as a natural setting for global analysis. It carries a group structure inherited componentwise from the circle group \(\mathbb{S}^1 \subseteq \mathbb{C}^*\), and that group structure is smooth in a sense that the manifold series will make precise: the torus is the prototypical example of a compact abelian Lie group, and the smooth structure constructed here is the one that makes its group operations compatible with the manifold structure. The same product construction yields the smooth structures on configuration spaces built from circles (pendulum systems, coupled oscillators), on the natural parameter spaces of multivariate periodic phenomena, and on quotient spaces \(\mathbb{R}^n / \mathbb{Z}^n\) arising in number theory and dynamical systems.