Tubular Neighborhoods VS Product Neighborhood (of a submanifold)

Question

Yes, I am asking for differences in definitions. I find people answering such questions more useful, than getting lost in terminology of each individual book/notes. So, I will appreciate your answers and thoughts.

Answer 1

Given a submanifold $X\subset M$, there exists a vector bundle $NX\rightarrow X$, the normal bundle, and a diffeomorphism $NX\rightarrow U$ where $U$ is an open neighborhood of $X$. The diffeomorphism takes the zero section of $NX$ diffeomorphically to $X\subset M$ (here I mean that $(x,0)\in NX$ is taken to $x\in X$). The vector bundle $NX$ is defined by $TM\bigr|_X/TX$. We say that $U$ is a tubular neighborhood of $X$. Now the the vector bundle $NX$ can be trivial: Then the tubular neighborhood is a product neighborhood: There is a diffeomorphism $X\times \mathbb{R}^{\dim M-\dim X}\rightarrow U$ which takes $X\times\{0\}$ to $X\subset U$.

An example of a submanifold which does not admit a product neighborhood is the center circle in the mobius strip. The normal bundle is not trivial.