I'm reading a paper on symplectic geometry. I reached to a point where the author uses normal bundles ( which I know the definition but I've never work with). He actually decomposes a given normal bundle into a sum of two other normal bundles. I can't give my precise question and the normal bundles I'm talking about, because this will require to introduce a lot of stuff. But i think he uses the following argument:
Let M be a smooth manifold, and let C and P be two submanifold of M such that: $ C \subset P \subset M.$
Let N be the normal bundle of C in M,
$N_1:=$ the normal bundle of C in P.
$N_2:=$ the normal bundle of P in M, restricted to C.
Then, we get $N = N_1 \oplus N_2.$
My questions are: first, what is the meaning of restricting a normal bundle to a submanifold (as is used to define the normal bundle $N_2)$, and the second one is, is the above argument true, namely is $N = N_1 \oplus N_2.$ ?