I want to try and show that a map between two manifold product spaces is an isomorphism. Im just a bit confused as to what 'isomorphism' means in this sense. At first I thought it was equivalent to the map being a homeomorphism but then I read some things saying it required the map to be linear and so on.
Any chance someone would be able to explicitly define what an isomorphism between two manifolds means?
In the class of differentiable manifolds an isomorphism is a diffeomorphism, that is to say an homeo $f$ such that both $f$ and its inverse are differentiable.
In general, when we speak about isomorphism it is understood the class of morphisms we are working with (for example continuous maps, differentiable maps, linear maps, holomorphic maps, polynomial maps et cetera...).
For example, for complex manifolds, isomorphisms are biholomorphisms; for algebraic manifolds isomorphisms are biregular maps; for PL-manifolds isomorphism are PL-homeos et cetera.
Thus, if the manifolds you are dealing with have linear structure, you may want to require isomorphisms to be linear. But it depends on the context.