Manifolds 13 | Examples of Smooth Manifolds [dark version]

TL;DR

The n-sphere S^n is given a C∞ smooth structure by verifying that all transition maps between its standard hemisphere-based charts are C∞ diffeomorphisms.

Briefing Cornell Notes

Briefing

A concrete check of the transition maps shows why the n-dimensional sphere carries a C∞ smooth structure: every overlap map between its standard stereographic-type charts is a C∞ diffeomorphism. The key step is not just choosing charts, but verifying that when two charts overlap, the coordinate change map—built as a composition of one chart’s inverse with the other chart—has derivatives of all orders and a smooth inverse. That verification matters because smooth manifolds are defined by compatibility of charts, not by the sets alone.

The construction starts with the n-sphere S^n as a subset of R^{n+1}. Using the atlas from the earlier discussion, each chart comes from a hemisphere U_i^± together with a map h_i^± that projects away one coordinate. Intuitively, for S^2 in R^3, a hemisphere maps to a disk by dropping the third coordinate; in general, h_i^± takes a vector with n+1 components and returns an n-component vector by omitting the i-th component. To prove this atlas is C∞, the focus shifts to transition maps on chart overlaps: given two overlapping chart domains U_i^+ and U_j^+, the overlap is mapped by both charts to corresponding regions in R^n, and the transition map Ω is defined by Ω = h_j^+ ∘ (h_i^+)^{-1} restricted to the overlap.

A worked example with n = 2, i = 3, and j = 1 makes the mechanism explicit. Starting from a point (x1′, x2′) in the left chart’s coordinate disk, applying (h_3^+)^{-1} reconstructs the missing coordinate using the sphere constraint, producing a vector in R^3 whose omitted component equals √(1 − ||x′||^2). Then applying h_1^+ drops the first coordinate, leaving a new pair (x2′, √(1 − ||x′||^2)). This resulting coordinate change is the transition map Ω. The crucial conclusion is that Ω is differentiable on the open overlap domain, its inverse is differentiable as well, and derivatives of any order exist—so Ω is a C∞ diffeomorphism. Because the same reasoning works for any dimension and any choice of i and j, all transition maps in the atlas are C∞, making it a C∞ atlas.

From there, the smooth structure is completed by extending the C∞ atlas to a maximal C∞ atlas, yielding a C∞ smooth manifold structure on S^n. The discussion then adds simpler examples to build intuition. First, R^n is a smooth manifold with the standard smooth structure: one chart suffices, using the identity map from R^n to itself, and maximal extension supplies the full C∞ structure. Finally, the graph of a continuously differentiable function f ∈ C^1(R) is treated as a one-dimensional manifold G_f ⊂ R×R. Using a single chart that sends (x, f(x)) to x gives a smooth structure on G_f, and because this graph sits inside R^2 with the standard smooth structure, it becomes a submanifold—setting up the next topic: submanifolds of R^n.

Cornell Notes

The sphere S^n becomes a C∞ smooth manifold once its standard charts are shown to be compatible: every transition map between overlapping charts is a C∞ diffeomorphism. Charts are built from hemispheres U_i^± and maps h_i^± that project away one coordinate, turning each hemisphere into a disk in R^n. On overlaps, the transition map Ω is computed as h_j^+ ∘ (h_i^+)^{-1}; the inverse reconstructs the missing coordinate using the sphere equation, introducing terms like √(1 − ||x′||^2). In the worked S^2 example, this produces an explicit smooth coordinate change whose inverse is also smooth, with derivatives of all orders. Extending the resulting C∞ atlas maximally yields the sphere’s C∞ smooth structure, and the same framework motivates R^n and graphs of C^1 functions as smooth manifolds and submanifolds.

Why does proving smoothness for S^n require checking transition maps, not just choosing charts?

A smooth manifold structure depends on how coordinates relate on overlaps. Even if each chart individually looks smooth (hemisphere to disk via coordinate omission), the atlas is only C∞ if every coordinate change map on U_i^± ∩ U_j^± is C∞ with a C∞ inverse. Those coordinate changes are the transition maps Ω = h_j^± ∘ (h_i^±)^{-1} restricted to the overlap domain.

How do the charts h_i^± for S^n work in practice?

Each chart comes from a hemisphere U_i^± on the sphere and a map h_i^± that omits the i-th coordinate. Concretely, h_i^± takes a vector in R^{n+1} with n+1 components and returns an n-component vector by dropping the i-th component. For S^2 in R^3, this means a hemisphere maps to a disk by removing one coordinate.

In the S^2 example (n = 2, i = 3, j = 1), what does the transition map Ω do to a point (x1′, x2′)?

Starting with (x1′, x2′) in the left chart’s coordinates, (h_3^+)^{-1} reconstructs the missing third coordinate using the sphere constraint: the omitted component becomes √(1 − ||x′||^2). Then h_1^+ drops the first coordinate, producing (x2′, √(1 − ||x′||^2)). That coordinate transformation is Ω.

What guarantees Ω is a C∞ diffeomorphism on the overlap?

The overlap domain is open, and the transition map is built from smooth operations: reconstructing the missing coordinate via √(1 − ||x′||^2) and then projecting by coordinate omission. The inverse transition map is obtained by reversing the same steps, and the discussion notes that derivatives of any order exist. Together, that yields a C∞ diffeomorphism.

How do the simpler examples (R^n and graphs G_f) fit the same atlas logic?

For R^n, a single chart suffices: use the identity map from R^n to R^n, then extend to a maximal C∞ atlas (the standard smooth structure). For a C^1 function f, the graph G_f = {(x, f(x))} ⊂ R×R is treated as a 1-dimensional manifold: one chart maps (x, f(x)) to x, giving a smooth structure on G_f. Because G_f sits inside R^2 with its standard smooth structure, it functions as a submanifold.

Review Questions

What is the formal definition of a transition map Ω between two charts, and why is it central to defining a C∞ atlas?
In the S^2 coordinate change example, where does the square-root term √(1 − ||x′||^2) come from?
How does the graph of a C^1 function f ∈ C^1(R) become a 1-dimensional smooth manifold, and what ambient space does it sit in?

Key Points

1
The n-sphere S^n is given a C∞ smooth structure by verifying that all transition maps between its standard hemisphere-based charts are C∞ diffeomorphisms.
2
Each chart h_i^± projects away one coordinate, mapping a hemisphere U_i^± to a disk in R^n.
3
On chart overlaps, the transition map is computed as Ω = h_j^+ ∘ (h_i^+)^{-1}, where the inverse reconstructs the omitted coordinate using the sphere equation.
4
In the worked S^2 case, the transition map sends (x1′, x2′) to (x2′, √(1 − ||x′||^2)), and the inverse is smooth as well.
5
All transition maps are C∞ for any dimension n and any indices i, j, so the atlas is a C∞ atlas.
6
A maximal C∞ atlas extension turns the C∞ atlas into the standard C∞ smooth manifold structure on S^n.
7
R^n carries the standard smooth structure via the identity chart, and the graph G_f of a C^1 function becomes a 1-dimensional smooth manifold that sits as a submanifold of R^2.

Highlights

Smoothness on S^n hinges on transition maps: coordinate changes on overlaps must be C∞ with C∞ inverses.

The missing coordinate on the sphere is reconstructed using the constraint, producing terms like √(1 − ||x′||^2) in the transition map.

For S^2, the explicit overlap map sends (x1′, x2′) to (x2′, √(1 − ||x′||^2)).

Graphs of C^1 functions form 1-dimensional smooth manifolds via a chart that projects (x, f(x)) back to x.

R^n’s standard smooth structure comes from a single identity chart extended to a maximal C∞ atlas.

Topics

Smooth Manifolds
Sphere Atlases
Transition Maps
C∞ Diffeomorphisms
Submanifolds

Mentioned

C∞
C1