Manifolds 17 | Examples of Smooth Maps [dark version]

TL;DR

Smoothness between manifolds is verified by converting the map into coordinate expressions using charts and checking ordinary differentiability in Euclidean space.

Briefing Cornell Notes

Briefing

Smooth maps between manifolds can be checked by reducing the problem to ordinary differentiability between Euclidean spaces using charts. With that framework in place, two canonical constructions become concrete examples: the inclusion of the 2-sphere into 3-space and the quotient map from the 2-sphere to real projective space.

The first example starts with the 2-dimensional sphere S2 viewed as a subset of R3. The inclusion (canonical injection) map I sends each point x on S2 to the same point in R3. Topologically it is automatically continuous, but smoothness requires verifying differentiability in local coordinates. Using the standard smooth structure on S2 built from hemispherical charts—specifically the southern hemisphere chart H3−—the inclusion becomes, in coordinates, essentially the identity: after applying the chart inverse on S2 and the corresponding chart on R3, the coordinate expression just passes from two coordinates (on the chart domain) to three coordinates in a differentiable way. The missing third coordinate is reconstructed using a square root, and because the resulting coordinate map has well-defined derivatives, it is differentiable and in fact C∞. The same reasoning works across all hemispheres, so the inclusion map I: S2 → R3 is a smooth map.

The second example uses the quotient construction of real projective space P2. By definition, P2 is obtained from S2 by identifying antipodal points: x is equivalent to y if y equals x or −x. This identification yields a canonical projection q: S2 → P2, which is continuous by construction. Smoothness again comes down to charts. On S2 one uses the same southern hemisphere chart H3−, while on P2 one uses a chart (denoted K in the transcript) that corresponds to the region where the third coordinate X3 is nonzero. In that chart, the coordinates on P2 can be represented by ratios such as X1/X3 and X2/X3, which are legitimate precisely because division by X3 is allowed only when X3 ≠ 0.

To test smoothness, the coordinate expression for q is formed as a map between Euclidean spaces: K ∘ q ∘ (H3−)^{-1}. In coordinates, this composition produces a vector in R2 whose components are obtained by dividing the relevant sphere coordinates by the square-root expression coming from (H3−)^{-1}. Since these coordinate functions are smooth on the chart domain (where the denominator never vanishes), all partial derivatives exist and the map is C∞. Covering the remaining points of P2 requires using the other charts as well, but the same local argument applies.

Taken together, the inclusion S2 ⊂ R3 and the quotient projection S2 → P2 both qualify as smooth maps, illustrating how smoothness on manifolds is verified through local Euclidean coordinate computations rather than global geometric intuition alone.

Cornell Notes

Smoothness of maps between manifolds is checked by translating the map into coordinates via charts, turning the question into ordinary differentiability between Euclidean spaces. For the inclusion I: S2 → R3, using the southern hemisphere chart H3− on S2 shows the coordinate form is differentiable and actually C∞, with the third coordinate recovered using a square root. For the quotient projection q: S2 → P2, where antipodal points on S2 are identified, a chart on P2 is chosen so that X3 ≠ 0 and coordinates can be written using ratios like X1/X3 and X2/X3. In that coordinate region, the composed map K ∘ q ∘ (H3−)^{-1} becomes a smooth (C∞) function because the denominators never vanish. Using the corresponding charts to cover all points yields smoothness globally.

Why does the inclusion map I: S2 → R3 become a smooth map once charts are used?

Because in local coordinates the inclusion reduces to a differentiable coordinate transformation. Using the southern hemisphere chart H3− on S2, (H3−)^{-1} reconstructs the missing third coordinate from the first two using a square root. In the corresponding local coordinates on R3, the inclusion does not introduce any non-smooth operations; it effectively passes from the 2D chart coordinates to 3D coordinates in a way whose components have well-defined derivatives. This makes the coordinate expression differentiable and, in fact, C∞. The same argument repeats on other hemispherical charts to cover all of S2.

What is the quotient map q: S2 → P2 doing geometrically, and why is it continuous?

P2 is formed by identifying antipodal points on S2: x is equivalent to y if y equals x or −x. The canonical projection q sends each point on S2 to its equivalence class in P2. Continuity follows directly from the quotient-space construction: the projection map is continuous by definition of the quotient topology.

How does choosing a chart on P2 make the smoothness test manageable?

The P2 chart used in the transcript corresponds to the region where the sphere coordinate X3 is nonzero. In that region, P2 coordinates can be represented by ratios such as X1/X3 and X2/X3 (interpretable as slopes). Because X3 ≠ 0 on the chart domain, division is legitimate and produces smooth coordinate functions. This avoids singularities that would occur if the denominator could vanish.

What coordinate expression is used to test whether q is smooth, and what makes it C∞?

Smoothness is tested by forming the coordinate map K ∘ q ∘ (H3−)^{-1}, which turns the manifold map q into a map between Euclidean spaces (R2 to R2). After applying (H3−)^{-1}, the coordinates involve a square-root expression for X3; then q passes to the equivalence class, and K converts that class into ratios like X1/X3 and X2/X3. On the chart domain where X3 ≠ 0, these functions are smooth, so the resulting coordinate map is C∞.

Why does the argument need multiple charts even if one chart shows C∞ behavior?

A single chart only covers the subset of the manifold where its coordinate formulas are valid. For S2 and P2, the chosen chart relies on conditions like X3 ≠ 0. Points where X3 = 0 are not covered, so additional charts (e.g., other hemispherical charts on S2 and corresponding charts on P2) are required. On each chart region, the same style of computation shows the coordinate expression is C∞, and together these cover the entire manifold.

Review Questions

How does the coordinate map K ∘ q ∘ (H3−)^{-1} demonstrate smoothness of the quotient projection q: S2 → P2?
What role does the condition X3 ≠ 0 play in defining the P2 chart and ensuring the coordinate functions are smooth?
Why does recovering a missing coordinate via a square root not automatically create a smoothness problem in the inclusion example?

Key Points

1
Smoothness between manifolds is verified by converting the map into coordinate expressions using charts and checking ordinary differentiability in Euclidean space.
2
The inclusion map I: S2 → R3 is smooth because, in hemispherical coordinates, it becomes a differentiable (indeed C∞) coordinate transformation.
3
On S2, the southern hemisphere chart H3− reconstructs the third coordinate using a square root, yielding well-defined derivatives on its chart domain.
4
Real projective space P2 is constructed by identifying antipodal points on S2, producing a canonical projection q: S2 → P2 that is continuous by quotient topology.
5
A chart on P2 is chosen so that X3 ≠ 0, allowing coordinates to be written as ratios like X1/X3 and X2/X3 without division by zero.
6
The quotient map q is smooth because the coordinate expression K ∘ q ∘ (H3−)^{-1} is C∞ on each chart region where denominators stay nonzero.
7
Covering the entire manifold requires repeating the coordinate argument across multiple charts so every point lies in some valid coordinate domain.

Highlights

The inclusion S2 ⊂ R3 becomes a C∞ map once the southern hemisphere chart turns it into a simple differentiable coordinate reconstruction.

Real projective space P2 arises from S2 by identifying x with −x, and the quotient projection q inherits smoothness through chart-based coordinate checks.

Smoothness on P2 hinges on using charts where X3 ≠ 0, making coordinate formulas like X1/X3 and X2/X3 legitimate and smooth.

The decisive test for q is the Euclidean coordinate composition K ∘ q ∘ (H3−)^{-1}, whose components remain smooth on the chart domain.

Topics

Smooth Maps
Charts
Inclusion Map
Quotient Projection
Real Projective Space