I have problems to come up with a proof of this, but i am pretty sure that it is correct: Let be a compact n-manifold with boundary the disjoint union . By Whitney's embedding theorem we can embed the boundary into . Is it true that we can extend this to an embedding of into ? And secondly in a way such that and .

I think the first claim should proof well, but i am not sure about the second. Grateful for any answers.

banach