Julian Henry

Sphere Eversion Vol. III

Manifolds and Immersions

7 min read

Sphere Eversion: I · II · III · IV · V · VI

Calculus enters. Charts, tangent vectors, the Jacobian that must not drop rank, and the first honest definition of the word “eversion”.

In Vol. I we built topological spaces, continuity, and paths; in Vol. II, homotopies and $\pi_{1}$. Topology alone cannot see a crease: a homotopy is allowed to squash a sphere flat, fold it, and unfold it. The eversion problem is only interesting once we ask for smooth maps whose derivative never degenerates. This volume adds exactly that much calculus.

Contents
  1. The two-sphere, locally flat
  2. Immersions, and the matrix that must not drop rank
  3. The homotopy that is not regular

The two-sphere, locally flat

It is instructive to say what a manifold is and what makes one differentiable, then to put differentiable coordinates on $S^2$ and take a derivative at $(1,0,0)$.
A manifold is a space that is Euclidean in the small. Formally, a smooth $n$-manifold is a Hausdorff space covered by charts $\psi_{\alpha}: U_{\alpha}\to\mathbb{R}^n$ (homeomorphisms from open sets onto open subsets) whose transition maps $\psi_{\beta}\circ\psi_{\alpha}^{-1}$ are $C^\infty$, meaning smooth: partial derivatives of every order exist and are continuous. You already know one: the surface of the Earth. No single paper map covers it, but an atlas does, and the overlap rules are smooth.

The 2-sphere is the set of unit vectors,

\[S^2=\bigl\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=1\bigr\}.\]

Spherical coordinates give one chart, and a rotated copy gives a second (you need at least two: the azimuth $\varphi$ is not a global coordinate). On the complement of a meridian,

\[\mathbf{r}(\theta,\varphi)=\bigl(\sin\theta\cos\varphi,\;\sin\theta\sin\varphi,\;\cos\theta\bigr),\qquad \theta\in(0,\pi),\;\varphi\in(-\pi,\pi).\]

A point is not a function of $(\theta,\varphi)$. Differentiate the point and you get $\partial_{\theta}(1,0,0)=\mathbf{0}$, which says nothing. The object that has derivatives is the parametrization $\mathbf{r}$, the inverse of a chart:

\[\begin{aligned} \mathbf{r}_\theta&=(\cos\theta\cos\varphi,\;\cos\theta\sin\varphi,\;-\sin\theta),\\ \mathbf{r}_\varphi&=(-\sin\theta\sin\varphi,\;\sin\theta\cos\varphi,\;0). \end{aligned}\]

At $(1,0,0)$ one has $(\theta,\varphi)=(\pi/2,\,0)$, so

\[\mathbf{r}_\theta=(0,0,-1),\qquad \mathbf{r}_\varphi=(0,1,0),\qquad \mathbf{r}_\theta\times\mathbf{r}_\varphi=(1,0,0)=\mathbf{r}.\]

The tangent plane $T_{p}S^2$ is the set of velocities of curves on the sphere passing through $p$; it is the plane through the origin perpendicular to $p$. The two vectors above are a basis of $T_{(1,0,0)}S^2$, the $yz$-plane. Drag the sliders; the blue and orange arrows point along $\mathbf{r}_{\theta}$ and $\mathbf{r}_{\varphi}$.

r, rθ, rφ

The sphere is the image of $\mathbf{r}$. Blue is $\mathbf{r}_{\theta}$, orange is $\mathbf{r}_{\varphi}$, gold is their cross product $\mathbf{r}_{\theta}\times\mathbf{r}_{\varphi}=\sin\theta\,\mathbf{r}$. At the poles $\sin\theta=0$ and the chart is singular; the sphere is not. That distinction is the whole subject.

It is instructive to see a nonzero derivative, say for a curve that wraps around the sphere.
Take the equator as a curve $\gamma:[0,2\pi]\to S^2$, $$\gamma(t)=(\cos t,\;\sin t,\;0).$$ Then $\gamma'(t)=(-\sin t,\;\cos t,\;0)\neq\mathbf{0}$. At $t=0$ this is $(0,1,0)$, which is $\mathbf{r}_{\varphi}$ at $(1,0,0)$. Wrapping around $S^2$ means the image of $\gamma$ is a closed loop that is not contractible in $S^2\setminus\lbrace\text{two poles}\rbrace$. We will need that loop.

Immersions, and the matrix that must not drop rank

It is worth asking what an immersion is, what a Jacobian is, and what happens when the Jacobian determinant is zero or switches sign.
For $f:\mathbb{R}^n\to\mathbb{R}^m$ the Jacobian is the $m\times n$ matrix of first partials: rows are outputs, columns are inputs. The differential $df_{p}$ sends the velocity of a curve $c$ through $p$ to the velocity of $f\circ c$ through $f(p)$. It is a linear map $T_{p}M\to T_{f(p)}N$, and in coordinates its matrix is the Jacobian. An immersion $f:M\to N$ is a smooth map whose differential is injective at every $p$. For $f:S^2\to\mathbb{R}^3$, in local coordinates $(u,v)$, that is the statement that the $3\times 2$ matrix $$ J_f=\begin{pmatrix} \partial_u f_1 & \partial_v f_1 \\ \partial_u f_2 & \partial_v f_2 \\ \partial_u f_3 & \partial_v f_3 \end{pmatrix} $$ has rank $2$ everywhere, equivalently $\mathbf{f}_{u}\times\mathbf{f}_{v}\neq\mathbf{0}$. There is no single determinant to watch: the components of that cross product are, up to sign, the three $2\times 2$ minors of $J_{f}$ (the determinants of the $2\times 2$ matrices made by keeping two of its three rows), and rank $2$ means they never vanish together.

For a map $\mathbb{R}^n\to\mathbb{R}^n$, $\det J<0$ means the map reverses orientation (it turns a right-handed frame into a left-handed one, as a mirror does), and a continuous $\det J$ can change sign only by passing through $0$. The complex exponential $f(x,y)=(e^x\cos y,\,e^x\sin y)$ has $\det J=e^{2x}>0$ everywhere, so it is a local diffeomorphism with no critical point at all. That step is the inverse function theorem: a smooth map with $\det J\neq0$ at a point has a smooth inverse near that point, which is what local diffeomorphism means (a diffeomorphism is a smooth bijection with a smooth inverse). Where $\det J$ vanishes, the map is not a local diffeomorphism there.

Self-intersection is allowed. A figure-eight $\gamma(t)=(\sin 2t,\,\sin t)$ is an immersion ($\gamma’$ never vanishes), and it crosses itself. An embedding is an immersion that is a homeomorphism onto its image; when the domain is compact, as $S^2$ is, any injective immersion is one, because a continuous bijection from a compact space onto a Hausdorff space always has a continuous inverse. The standard sphere $\iota(p)=p$ is an embedding. Mid-eversion surfaces are immersions and not embeddings.

Step through three curves. Watch the arrow, not the picture.

γγ′(t)

The gold arrow is the velocity $\gamma'(t)$, drawn at true relative length. An immersion is a curve whose arrow never shrinks to a point. Self-crossings are allowed; stopping is not.

The homotopy that is not regular

Smale's theorem says that any two $C^2$ immersions of $S^2$ in $E^3$ are regularly homotopic ($C^2$: twice continuously differentiable; $E^3$: his notation for $\mathbb{R}^3$). It is instructive to test the obvious proof, $H(x,t)=(1-t)f_{0}(x)+t f_{1}(x)$, and to ask what its domain $S^2\times[0,1]$ looks like.
$S^2\times[0,1]$ is a thickened sphere: a spherical shell. Each point of $S^2$ grows an interval. A homotopy of maps $S^2\to\mathbb{R}^3$ is a single map $$F:S^2\times[0,1]\to\mathbb{R}^3.$$ It is a regular homotopy when each slice $F(\,\cdot\,,s)$ is an immersion. Straight-line interpolation between immersions is a homotopy of smooth maps, but in general not a regular homotopy. It routinely drops rank.
Definition (regular homotopy). A homotopy $F:S^2\times[0,1]\to\mathbb{R}^3$ such that every slice $f_{s}=F(\cdot,s)$ is an immersion and the derivative $df_{s}$ varies continuously with $s$. Equivalently: a path in the space $\operatorname{Imm}(S^2,\mathbb{R}^3)$ with the $C^1$ topology. That is Vol. II's homotopy, with one extra rule: no slice may crease.

The sock-push, shoving the north pole straight down through the south pole, is the interpolation from $\iota$ to the reflection in the equatorial plane,

\[F_s(\theta,\varphi)=\bigl(\sin\theta\cos\varphi,\;\sin\theta\sin\varphi,\;(1-2s)\cos\theta\bigr).\]

A short computation:

\[\mathbf{F}_\theta\times\mathbf{F}_\varphi=\bigl((1-2s)\sin^2\theta\cos\varphi,\;(1-2s)\sin^2\theta\sin\varphi,\;\sin\theta\cos\theta\bigr),\] \[\bigl\|\mathbf{F}_\theta\times\mathbf{F}_\varphi\bigr\|=\lvert\sin\theta\rvert\sqrt{(1-2s)^2\sin^2\theta+\cos^2\theta}.\]

At $s=\tfrac12$ this is $\lvert\sin\theta\cos\theta\rvert$, which is zero all along the equator. The image is a disk covered twice, with a fold on the boundary. That is a crease. Colour in the canvas is $\lVert\mathbf{F}_{\theta}\times\mathbf{F}_{\varphi}\rVert$: blue is a healthy tangent plane, orange is rank drop. (The poles are orange at every $s$ for the boring reason that the chart is singular there; watch the equator.)

min ‖Fθ × Fφ‖
rank 2
rank drop

$F_{s}(\theta,\varphi)=(\sin\theta\cos\varphi,\,\sin\theta\sin\varphi,\,(1-2s)\cos\theta)$. This is not an eversion. At $s=1$ you have reflected the sphere through the $xy$-plane, and at $s=1/2$ you have left $\operatorname{Imm}(S^2,\mathbb{R}^3)$.

An eversion is a regular homotopy from the inclusion $\iota(p)=p$ to the antipodal embedding $\alpha(p)=-p$. The antipodal map $x\mapsto -x$ reverses the orientation of $\mathbb{R}^3$ ($\det(-I)=(-1)^3=-1$), and the outward normal of the image sphere at $-p$ is $-p$, while the pushed tangent frame produces the opposite normal. Inside has become outside. $F_{s}$ ends at the reflection $\rho(x,y,z)=(x,y,-z)$, which is $\alpha$ followed by a half-turn about the $z$-axis. Rotating an immersion keeps it an immersion, so undoing that half-turn gradually is a regular homotopy from $\rho$ to $\alpha$, and reaching $\rho$ regularly would be as good as reaching $\alpha$. $F_{s}$ gets there, but it cheats: at $s=\tfrac12$ it leaves $\operatorname{Imm}(S^2,\mathbb{R}^3)$.

So the question is sharp now. $\iota$ and $\alpha$ are both immersions (both embeddings, in fact). Is there a path between them in $\operatorname{Imm}(S^2,\mathbb{R}^3)$? The crease homotopy is not one. Before attacking the sphere we drop a dimension and ask the same question of circles in the plane, where the answer is no and the reason is an integer.

Cite this volume 
@misc{henry2026circlingthespherevoliii,
  author       = {Henry, Julian},
  title        = {Sphere Eversion, Vol. III: Manifolds and Immersions},
  year         = {2026},
  month        = {August},
  howpublished = {\url{https://juleshenry.github.io/blog/2026/08/17/circling-the-sphere-Vol-III}},
  note         = {Blog post}
}