This is probably a naive question, but here goes. To motivate my question, I'll consider a unit circle in $\mathbb C$ or $\mathbb R^2$. This is a compact Lie group equipped with the usual exponential map. However, any deformation no matter how smooth of the unit circle makes it lose the group closure property (say under complex multiplication) and it ceases to be a Lie group.
My question is why is this so? How does one prove, for example, that it is not possible to have a Lie group which is an ellipse (or any other shape) in $\mathbb C$ or $\mathbb R^2$? Why is it not possible to have a "complex multiplication like" binary operator that takes two points on an ellipse and returns a third without affecting other group properties?
Thanks.