Let $ M $ be a manifold and let $ X $ be a vector field defined on $ M $. Given some $ m\in M $, my book (R. Abraham, J. E. Marsden, Foundations of Mechanics) defines a flow box at $ m $ as a triple $ (U,\epsilon,\Phi) $ where:
- $ m\in U\subset M $ is an open set and $ \epsilon > 0 $ or $ \epsilon = +\infty $;
- $ \Phi\colon \left]-\epsilon,\epsilon\right[\times U\to M $ is a smooth map;
- for every $ m^\prime\in U $, the map $ t\mapsto \Phi_t(m) = \Phi(t,m) $ is an integral curve of $ X $ at $ m $;
- for every $ t\in \left]-\epsilon,\epsilon\right[ $ the set $ \Phi_t(U) = \{\Phi_t(m^\prime) : m^\prime\in U\} $ is open in $ M $ and the map $ m^\prime\mapsto \Phi_t(m^\prime) $ is a diffeomorphism.
I was wondering, why do we require condition 4.? Assuming only 1. 2. and 3. and the fact that the set $ \Phi_t(U) $ is open one can prove that if $ U\cap \Phi_t(U)\neq \emptyset $ then the map $$ \Phi_t\colon U\cap \Phi_{-t}(U)\to U\cap \Phi_{t}(U) $$ is a diffeomorphism. Isn't that enough? It seems to me that condition 4. is not explicitely used elsewhere in the book.
I was also wondering if we could prove that $ \Phi_t(U) $ is open using only 1. 2. and 3.