This post is based on John Lee’s “Introduction to Smooth Manifolds” and lectures by Prof. Richard Bamler at UC Berkeley.

\[\def\acts{\curvearrowright}\]

The action of the circle \(\mathbb{S}^1\) on the higher dimensional sphere \(\mathbb{S}^{2n+1}\) is one of the most profound objects in differential geometry. We study the consequences of this action here.

First, we note that the manifold \(\mathbb{S}^{2n+1} \subset \mathbb{R}^{2n+2} \subset \mathbb{C}^{n+1}\) consists of elements of the form:

\[\begin{align} \{(z_1, \dots, z_{n+1} ) \in \mathbb{C}^{n+1} | |z_1|^2 + \dots + |z_{n+1}|^2 = 1\} \end{align}\]

The circle \(\mathbb{S}^1\) acts on the sphere in the following way. We have that \(\mathbb{S}^1 \acts \mathbb{S}^{2n+1}\) as:

\[\begin{align} \lambda \cdot (z_1, \dots, z_{n+1}) = (\lambda z_1, \dots, \lambda z_n) \end{align}\]

The orbits of this action are \(\{ ( \lambda z_1, \dots, \lambda z_{n+1}) : \lambda \in S^1 \}\) for any \((z_1, \dots, z_{n+1} ) \in \mathbb{S}^{2n+1}\) and the orbit space is:

\[\begin{align} \mathbb{S}^1 \backslash \mathbb{S}^{2n+1} = \{ [z_1 , \dots, z_n ] \} = \mathbb{C} P^n \end{align}\]

Here is a visualization of the situation:

This is remarkable! The action of the circle on the \(2n+1\)-dimensional sphere splits the sphere into complex projective spaces of \(n\)-dimension. Further, we can check that the map:

\[\begin{align} \pi: \mathbb{S}^{2n+1} \rightarrow \mathbb{C} P^n \\ \pi(z_1, \dots, z_{n+1}) = [z_1, \dots, z_{n+1} ] \end{align}\]

is a submersion.

We can also take another perspective: that of vector fields. For any \(\lambda \in \mathbb{S}^1\), we can define the map \(\phi_{\lambda} : \mathbb{S}^{2n+1} \rightarrow \mathbb{S}^{2n+1}\) that takes \(\phi_{\lambda}(z_1,\dots, z_{n+1} ) = (\lambda z_1, \dots,\lambda z_{n+1} )\). However, this can simply be seen as the restriction of the following diagonal matrix:

\[\begin{align} \begin{bmatrix} \lambda & 0 & \dots & 0 \\ 0 & \lambda & \dots & 0 \\ \dots & \dots & \dots \\ 0 & 0 & \dots & \lambda \end{bmatrix} \in U(n+1) \end{align}\]

and we can see that we can view the derivative of \(\phi_{\lambda}\) as a velocity vector:

\[\begin{align} \frac{d}{dt} \phi_{e^{it}} (z_1, \dots, z_{n+1} ) = (i e^{it} z_1, \dots, i e^{it} z_{n+1} ) = i \phi_{e^{it}} (z_1,\dots, z_{n+1}) \end{align}\]

Further, we see that \(i \phi_{e^{it}} (\bar{z})\) and \(\phi_{e^{it}} (\vec{z})\) are orthogonal, and thus \(i \phi_{e^{it}} (\vec{z}) \in T_{\phi_{e^{it}}(\vec{z})}\mathbb{S}^{2n+1}\), the tangent space to the sphere at the point \(\phi_{e^{it}}(\bar{z})\).

Further, we can view \(X_{\vec{z}} = i \vec{z}\) can be viewed as a vector field on \(\mathbb{S}^{2n+1}\) and therefore:

\[\begin{align} \frac{d}{dt} \phi_{e^{it}} (\vec{z}) = X_{\phi_{e^{it}}(\vec{z})} \end{align}\]

which lets us compute \(\theta_{t}(\vec{z}) = \phi_{e^{it}} (\vec{z})\), the flow of the vector field \(X\).

We can now consider the \(3\)-dimensional example (i.e. \(n = 1\)), and we see that \(\mathbb{S}^1 \acts \mathbb{S}^{3}\), and thus we have:

\[\begin{align} \pi : \mathbb{S}^3 \rightarrow \mathbb{C} P^1 \cong S^2 \end{align}\]