The discussion centers on the necessity of F being a diffeomorphism in the context of F-related vector fields. Participants explore the implications of F being smooth and bijective but not a diffeomorphism, examining the consequences for vector fields on manifolds.
Participants express differing views on the necessity of F being a diffeomorphism, with some supporting its necessity for unique vector field correspondence and others questioning this requirement. The discussion remains unresolved regarding the implications of F being smooth and bijective without being a diffeomorphism.
There are limitations in the discussion regarding the assumptions made about the definitions of F-related vector fields and the implications of smoothness and bijectiveness without diffeomorphism. The discussion also reflects varying interpretations of the definitions presented in the referenced book.
Tinyboss said:If you post the definition I might be able to help. I left my manifolds book at school.
quasar987 said:The definition does not even require F-->N to be bijective. If F is not bijective, then a vector field on M might to push-foward to a vector field on N, and a vector field on N might not have an F-related vector field on M. But if F is a diffeo, then we are in the nice situation where to every vector field on N, then exists a unique F-related vector field on M given, of course, by the pushfoward by F^-1.