Consider the submanifold $M$ of $\mathbb R^8$, with coordinates $(x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4)$, defined by the following equations $$x_1^2+y_1^2+x_2^2+y_2^2=1,$$ $$x_3^2+y_3^2+x_4^2+y_4^2=1,$$ $$x_1x_2+y_1y_2+x_3x_4+y_3y_4 = 0.$$
Consider a loop (closed curve) $\gamma$ and assume that $\gamma$ is contractible in $M$.
What I want to prove is the following:
The integral $$\int_\gamma x_1dx_2+y_1dy_2+x_3dx_4+y_3dy_4$$ is bounded by a constant not dependent on $\gamma$. I believe that such an integral is bounded by $\pi$.
I am not sure if such a constant exists. This integral appeared in a research problem where I was computing the holonomy of certain vector bundles, where it does seem that such a bound exists.
I also believe there is some trick using symplectic geometry to find this bound. If $\Gamma$ is a disc with boundary $\gamma$, then $$\int_\gamma x_1dx_2+y_1dy_2+x_3dx_4+y_3dy_4 =\int_\Gamma dx_1dx_2+dy_1dy_2+dx_3dx_4+dy_3dy_4,$$ by the Stokes theorem, the integral of a symplectic form.
Any ideas are welcome. Thank you in advance.
