I'm reading Differential Forms in Algebraic Topology by Bott and Tu. I reached the point where the book defines the normal bundle of a submanifold and uses the tubular neighborhood theorem. I can't for the life of me understand the definition. The book uses the following definition:
If $ S $ is a submanifold in $ M $, then normal bundle $ N $ of $ S $ in $ M $ is the vector bundle on $ S $ defined by the exact sequence $$ 0\rightarrow T_{S} \rightarrow \left.T_{M}\right|_{S} \rightarrow N \rightarrow 0, $$ where $ \left.T_{M}\right|_{S} $ is the restriction of the tangent bundle of $ M $ to $ S$.
The book doesn't explain what the maps are. Surprisingly I could not find a clear definition on the web either. I think $ T_{S} $ is mapped to corresponding vectors in $ \left.T_{M}\right|_{S} $, but I don't know how to make this map well-defined. Some PDFs online suggest using adapted charts on $ M $ in which $ S $ is defined by the vanishing of a fixed number of coordinates. I have no idea what the other map in the exact sequence is.
Can you please show me how to $ N $ is defined, how it gets its topology and becomes a bundle on $ S $?
I don't know Riemannian Geometry, so please don't use it.
As a side note, I found the book to be terse. Everyone swears by how clear it is, but I find it missing a lot of details and it has very few examples. I studied Hatcher's Algebraic Topology and Tu's Introduction to Manifold, so I think I'm prepared.
Thank you. I have been banging my head against this for most of the day.
To amplify Rene's comment, which really amounts to the answer: At each point $p$ of the submanifold $S$, the tangent space to $S$, $T_S(p)$, is a vector subspace of the tangent space to $M$, $T_M(p)$. The quotient $T_M(p)/T_S(p)$ is the fiber $N(p)$ of the normal bundle at $p$.