I know that the Klein bottle is a $C^{\infty}$ manifold (obtained as the quotient space of the action $(x,y) \longrightarrow (x+1/2,-y)$ on the torus).
I have to find the diffeomorphim between the so obained Klein bottle and the connected sum of two projective spaces. I know it can be seen "cutting and pasting" the square which represents the Klein bottle $aba^{-1}b$, however I would like to find an esplicit diffeomormism, is that possible?
