I am an engineer so I apologize for my math language, if it is not the most suitable.
How can I define addition and multiplication between two connected differentiable manifolds $m_1$ and $m_2$, i.e. a connected space (as in https://en.wikipedia.org/wiki/Connected_space) that is also a differentiable manifold (https://en.wikipedia.org/wiki/Differentiable_manifold).
My goal is to find a kind of "interpolation" of a vector field $v_1$ defined on $m_1$ and vector field $v_2$ defined on $m_2$. The two vector fields are actually governed by the same equation, but the manifold changes. I also do not want to solve (because it is time consuming on a PC) again the new vector field on the new manifold.
Or alternatively, given $m_1$ and $v_1$ if I perturb $m_1$ into $m_2$ by a "small" parameter, how will the vector field change?
If there isn't a simple/direct answer can you point me to some literature?