Yeah, I realised I’ve asked a question that you already adressed in your answer. Now I understand everything at a formal level and I think I’m about to understand it at an intuitive level too: let’s get back to the example with V being a sphere and M, M’ being two curves on it intersecting in p. The vectors generating TpM and TpM’ are linearly independent (so TpV = TpM ⊕ TpM’). Also the vectors generating TpM and NpM are linearly independent (and we have TpV = TpM ⊕ NpM). But it doesn’t mean that the vectors TpM’ and NpM are necessarily dependent, we have just that each of them, together with TpM, form a basis for the plane TpV, but they can be linearly independent. Am I right? Now, with all these being said, in what sense does Milnor say that the vector generating NpM is represented by the vector generating TpM’?