Hello

I was asked in a test to do an integration of an exact form on a manifold. The first thing you think is to apply the Stokes theorem. The problem here is to determine the boundary of the manifold. Let $\displaystyle S=\{(x,y,z,w):x^2+y^2+z^2+w^2=2\}$ and let $\displaystyle T^2=\{(x,y,z,w):x^2+y^2=z^2+w^2=1\}$. Now, is the boundary of the component of $\displaystyle S-T^2$, such that $\displaystyle x^2+y^2<1$, empty?

My teacher says it's not empty, but a fellow PhD of mine says it's empty. I don't know who's right. Can anyone find a way to decide this without any doubt remaining?

Thanks in advance.