# The definition of normal bundle as a quotient.

Let $$f: N \to M$$ be a smooth immersion and let $$p \in M$$, $$W = f(V) \subset M$$ be an submanifold with $$q = f(p).$$

Then the sequence is split exact

$$T_qW \hookrightarrow T_qM \stackrel{\mu}\to T_qM/T_qW$$

Therefore $$T_qM \cong T_qW \oplus T_qM/T_qW.$$ Now apparently the quotient according to wikipedia on normal bundles, it can be identified with the normal space (as a fiber) $$N_qW.$$ That is $$NM \stackrel{\pi}\to M$$, $$N_qW = T_qM/T_qW = \pi^{-1}(p).$$

I don't know how this last conclusion is reached...

The only answer I am thinking of is that there is some surjective linear (canonical?) map $$\ell: T_qW \to N_qW$$, then it would yield the commutative diagram:

$$\begin{array}{cccccccc} T_qM & \xrightarrow{\mu} & T_qM/T_qW & \\ \downarrow & \swarrow \\ N_qW \end{array}$$

• Do you know the answer for one point? That is, if $L$ is a subspace of $\mathbb R^n$, how to identify $L^\perp$ to $\mathbb R^n /L$? Mar 24, 2019 at 3:52
• @ArcticChar, yes let's just reduce it to that. I actually don't know. The most I could write is $\{r + L: r\in \mathbb{R}^n \} \to \{ g(n,\ell) = 0, \ell \in L \}$. Mar 24, 2019 at 4:03

Since $$T_qW \subset T_qM$$, we can talk about its orthogonal complement $$N_q(W)$$ which has direct sum with $$T_qW \oplus N_qW \approx T_qW \oplus T_qM/T_qV.$$ By comparing the elements, it is not surprising that there is an identification between the spaces in question. I just don't know what it is, but my initial question appears to be answered.