Apply the theorem of tubular neighborhood

Question

$M$ is a connected manifold, $N\subset M$ is a connected submanifold with nontrivial normal bundle, and dimM-dimN=1. How to prove $M-N$ is connected?

There is a hint to use the tubular neighborhood theorem. Any suggestion will be grateful!

Answer 1

Hint: this follows from the observation that if you have a non-trivial normal bundle it is non-orientable and hence the complement of the zero section in this normal bundle gives you a connected manifold. (namely the orientation covering for this line bundle: it is oriented and connected and covers the normal bundle. You might need some slight modifications when you write it out in detail.)