# Searching for a bound for the integral of a 1-form along a loop.

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.

• I'm not used to thinking about $5$-dimensional manifolds, but $M$ certainly contains the product of two tori, and I believe the $1$-cycles generating its homology cone off in $M$. I haven't checked this carefully at all. But this makes me a bit dubious about what you hope. Commented Mar 15, 2023 at 18:41