I couldn't find a duplicate, although I think is a very common question.
Given two charts, ($U_{1},φ_{1}$), ($U_{2},φ_{2}$), on a n-dimensional topological manifold M, such that: $U_{1} \cap U_{2}\neq \emptyset$, we get transition maps:
$φ_{1}\circ φ_{2}^{-1} : φ_{2}(U_{1}\cap U_{2}) \rightarrow φ_{1}(U_{1}\cap U_{2})$, and
$φ_{2}\circ φ_{1}^{-1} : φ_{1}(U_{1}\cap U_{2}) \rightarrow φ_{2}(U_{1}\cap U_{2})$
Two charts, as above, are called compatible if the transition maps, as above, are homeomorphisms. If $U_{1} \cap U_{2} = \emptyset$, then they are compatible.
My question is, why do we need this behavior? In addition, if we want to define C$^{\infty}$-compatible charts, why do we need to take transition maps to be smooth, Euclidian mappings?
Best Answer
Imagine taking two nice pieces of tissue and gluing them together with the goal to obtain a larger piece of tissue which is still nice. Then you will put part of the second piece of tissue over part of the first one, make it so that there are no wrinkles and then glue. This basically what is happening with manifolds: you are gluing patches of euclidean space together in such a way that the resulting object is "nice", i.e. is again locally euclidean. If you want nicer objects, you will have to take better behaved gluings (you could accept that the resulting piece of tissue forms an angle somewhere, or you could ask for it to be smooth everywhere...)