According to Wikipedia, the definition of normal bundle is defined as,
Defintion $1$. [Normal bundle]
Let $(M,g)$ be a Riemannian manifold, and $S\subset M$ a Riemannian submanifold.
- For a given $p \in S$ , a vector $n \in T_pM$ to be normal to $S$ if whenever $g(n,v)=0$ for all $v \in T_pS$. Then the set $N_pS$ of all such $n$ is called the normal space to $S$ at $p$.
- $NS:= \coprod _{s \in S}N_pS$ is called the total space of normal bundle to $S$ at $p$. ............. $(*)$
From the above definition, I understand
$$N_p S := \left\{ n \in T_pM : g(n,v)=0~ for~ all~~ v ~\in T_pS \right\}$$
For example, let $M=\mathbb{R}^2$ and $S=S^1$ and pick a point $p\in S$. Then, since $S$ is embedded in $M$, naturally $p\in M$. and $T_pM$ is also well defined. then since the vector $n$ is easily constructed ( clearly the every vector $v$ who lives in $T_pM$ is perpendicular to $n$, literally, $g(n,v)=0$. ) and like the second bullet, $NS$ is also well defined for any point $p \in M$. (The following image is just visualizing my description. the pink vector is one of element of $N_pS $ )
Meanwhile, how about the formal definition of normal bundle? Even though such definition is slightly different from each source, essentially, the basic idea seems to use a quotient space, However, I think that the formal definition contradicts the first definition, $(*)$. To begin with, based on wikipedia description,
one can define a normal bundle of $N$ in $M$, by at each point of $N$, taking the quotient space of the tangent space on $M$ by the tangent space on $N$. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot...
I will define for convenience,
Defintion2. [General definition of normal bundle] Let $(M,g)$ be a Riemannian manifold, and $S\subset M$ a Riemannian submanifold. Then the quotient space $TM/TS$ is called a normal bundle to $S$ at $p$.
For example, also consider $M=\mathbb{R}^2$ and $S=S^1$ and metric tensor still is Riemannian metric, $g$. if I pick two vector bundle $v,w \in TM$, the equivalence relation ~ is given,
$$v\sim w ~if~and~only~if~ v-w \in TS ...... (**)$$
Then we can viusalize both vector and $v$ and $w$ like the below picture,
and since $\sim$ is equivalent relation, we consider a representation $\nu \in TM/TS $. Then the representation $\nu$ would be a normal bundle. Of course, when considering $(**)$, the representation is descirbed
$$\nu=\left\{ w + \alpha u : w \in TM , \alpha \in \mathbb{R}, u \in TS \right\}$$
However, when comparing to the first definition, $\nu $ clearly does not represent normal vector. Obviously, for any point $p \in S$ , then $w+u$ is not perpendicular to the tangential vector $T_pS$
Therefore, I cannot understand why the formal definition of normal bundle , Defintion2, is a reasonable statement when considering Definition1.