I have a exercise, but the exercise isn't written very well. Do you understand, what the statement should be?:
"M is a smooth submanifold of N iff every point has an open nbh , s.t. is a smooth submanifold of U".
It doesn't make sense to me, because p is a point in M and U_p a subset in N. And what about the set U?
Our definition of a submanifold: M is a submanifold of N if there exist a smooth embedding and immersion f:M-> N