Let $\mathcal H$ be a complex Hilbert space with scalar product $\langle \cdot,\cdot\rangle$. Let us consider on $\mathcal H$ the $1$-form defined for any $\psi\in \mathcal H$ by $$ \widetilde \omega_\psi\colon [\gamma]_\psi \in T_\psi \mathcal H\mapsto \Im\langle \psi,\gamma'(0)\rangle\in \mathbb R\,. $$ Let's $\omega=\iota^\ast \widetilde \omega$, where $\iota\colon \mathcal S\mathcal H\hookrightarrow \mathcal H$ is the canonical inclusion of the Hilbert sphere $\mathcal S\mathcal H=\{\psi\in \mathcal H\mid \langle \psi,\psi\rangle=1\}$. Thus, $\omega$ is a $1$-form on $\mathcal S\mathcal H$.
Let $P\colon [0,1]\to \mathbb P\mathcal H$ be a closed curve on the complex Hilbert space $\mathbb P\mathcal H$. Let us assume that $P([0,1])\subseteq U_{P(0)}$, where for any $\psi\in \mathcal S\mathcal H$, $U_\psi=\Pi(\mathcal S\mathcal H\setminus \psi^\perp)$ is the domain of the coordinate chart $$ u_\psi \colon \chi\in U_\psi\mapsto \frac{\chi}{\langle \psi,\chi\rangle}-\psi\in \psi^\perp\,. $$ Let $s$ be a local section of the projection $\Pi\colon \mathcal S\mathcal H\to \mathbb P\mathcal H$ defined on $U_\psi$. Let us define $A=s^\ast \omega$, so that $A$ is a $1$-form on $U_\psi$.
Now I have trouble in calculating $$ \int_P A\,. $$
My attempt First, $$ \int_P A=\int_P s^\ast\iota^\ast \widetilde \omega=\int_0^1 P^\ast s^\ast \iota^\ast \widetilde \omega=\int_0^1 (\iota\circ s\circ P)^\ast \widetilde \omega\,. $$ Since $\widetilde \omega$ is not written in terms of differentials ($\mathcal H$ can be infinite-dimensional), I don't know how to calculate the pullback, and so how to continue...