Let's see some work! You've posted a lot of questions and have yet to really show your work!
Which working definition are you using? Immersion by the inclusion map? Plain old that for each there is a chart such that and ? That last definition only works for a -dimensional submanifold of a -dimensional smooth manifold. Are we working with finite dimensional smooth manifolds?
There are a lot of questions that need to be answered!
An -dimensional manifold is a submanifold of another -dimensional manifold () if
a) ( is a subset of )
b) The inclusion map with is an embedding
(Which means that for each the differential is and that the inclusion map is a homomorphism)
So I only care for finite dimensional differential manifolds.
Well I obviously have (a). Could I use the Reverse function therem for the differential?