# Non trivial section of the Bott bundle

Consider the following subbundle of $\mathbb{C}P^1\times \mathbb{C}^2$, $$V=\{(x,v)\in \mathbb{C}P^1\times \mathbb{C}^2:\; v\in x\}$$ where we consider $x\in \mathbb{C}P^1$ as a complex "line" passing through the origin. This is called the Bott bundle and can also be seen as a bundle over $S^2$ via the identification of $\mathbb{C}P^1$ as the Riemann sphere. What I'm actually trying to do is to construct a finite Parseval frame of $\Gamma(V)$ (looked as a Hilbert module over $C(\mathbb{C}P^1)$) but in order to even consider this I need to know how to construct a single non-trivial (meaning non zero) section at the very least.

I tried considering $\mathbb{C}P^1=U_1\cup U_2$, where $U_i$ are the canonical open subsets in $\mathbb{C}P^1$, and then trying to extend a couple of functions from $\mathbb{C}\simeq U_1\rightarrow V|_{U_1}$ to functions $\mathbb{C}P^1\rightarrow V$ but these didn't actually formed a Parseval frame (was aiming for a frame with 2 elements). Do sections for this bundle have any particular form I could use to search my desired Parseval frame?

The couple of sections I considered but didn't result in a frame are the following:

Let $U_1=\{[x:1]\in \mathbb{C}P^1:\; x\in \mathbb{C}\}$ and $U_2=\{[1:y]\in \mathbb{C}P^1:\; y\in \mathbb{C}\}$. Now consider \begin{align*} \tilde{\xi_1}: U_1&\rightarrow V|_{U_1}\\ [x:1]&\mapsto \left([x:1],\frac{(x,1)}{\|(x,1)\|_\text{max}^2}\right) \end{align*} We can extend this to a section in $\mathbb{C}P^1$ by considering $\xi_1:\mathbb{C}P^1\rightarrow V$ as $\xi_1(x)=\tilde{\xi_1}(x)$ if $x\in U_1$ and $0$ outisde $U_1$. Similarly we can define $\xi_2$ for $U_2$. The fact that we have a norm squared I feel is required to extend $\tilde{\xi_i}$ but is also in the way for $\{\xi_i\}$ to be a Parseval frame.

Note: A Parseval frame for a Hilbert module $M$ over a C*-algebra $A$ is a family $\{\xi_\lambda\}_{\lambda\in\Lambda}$ such that for any $\xi\in M$ we have $$\xi=\sum_{\lambda\in\Lambda}\xi_\lambda\langle\xi_\lambda ,\xi\rangle$$ If $\Lambda$ is finite we say that the Parseval frame is finite.

I finally found a frame for it, consider \begin{align*} \xi_i:\mathbb{C}P^1 \rightarrow V\\ [x_1:x_2]&\mapsto \left([x_1:x_2],\frac{\overline{x_i}}{|x_1|^2+|x_2|^2}(x_1,x_2) \right) \end{align*} It's not hard to verify that $\{\xi_1,\xi_2\}$ is a Parseval frame for $\Gamma(V)$ if we consider the inner product inherited from $\Gamma(\mathbb{C}P^1\times \mathbb{C}^2)$.