Let $S^{2n+1}$ be the unit sphere considered as a subset of $\mathbb{C}^{n+1}$ and define an action of $S^1 \subset \mathbb{C}$ by usual (complex) scalar multiplication. The quotient space is the well known complex projective space $\mathbb{C}P^n$. Denote by $\pi : S^{2n+1} \to \mathbb{C}P^n$ the natural projection and by $[z_1 : \cdots : z_{n+1}]$ the image of $(z_1, \dots, z_{n+1})$ under $\pi$.

For $1 \leq i \leq n+1$, let $U_i = \{ [z_1 : \cdots : z_{n+1}] \in \mathbb{C}P^n : z_i \neq 0\}$ and endow $\mathbb{C}P^n$ with a differentiable structure making the $U_i$ open submanifolds.

I already proved $\pi^{-1}(U_i)$ is homeomorphic to $U_i \times S^1$ via the map $\psi_i : \pi^{-1}(U_i) \to U_i \times S^1$ given by $\psi_i(z) = \left( \pi(z), \frac{z_i}{|z_i|} \right)$.

I am now trying to prove $\psi_i^{-1} : U_i \times S^1 \to \pi^{-1}(U_i)$ is differentiable. A formula for $\psi_i^{-1}$ is known:

$\psi_i^{-1}(x, a) = \frac{a |z_i|}{z_i} \cdot z$, for a chosen $z \in \pi^{-1}(x)$.

Am I overthinking this or it is obvious?

  • $\begingroup$ Which Hopf map are you referring to? $S^3 \to \mathbb{CP}^1$? $\endgroup$ – Michael Albanese Jan 11 '16 at 2:06
  • $\begingroup$ $S^{2n+1} \to \mathbb{C}P^n$ for arbitrary $n \geq 1$. $\endgroup$ – Eduardo Longa Jan 11 '16 at 2:10
  • $\begingroup$ But $S^3 \to \mathbb{C}P^1$ is fine. $\endgroup$ – Eduardo Longa Jan 11 '16 at 2:12

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