Intuition of normal bundle to a manifold So i recently learned about fibre bundles and tangent bundles in particular. While the definition of tangent bundles seems quite intuitive, i really struggle to understand any definition of the normalbundles to a submanifold. I looked everywhere and not only did the definitions all somehow seem to be different, i just havent managed to understand either one of them to be honest.
Can someone please help me in understanding the quotientbundle definition of the normal bundle and maybe provide some geometric intuition?
Thank you very much!
 A: I don't think there is really that much good intuition in the non-Riemannian setting.
Imagine the circle $S^1$ sitting in $\mathbb{R}^2$. At every point, $z=e^{i\theta}\in S^1$, you have the tangent plane of $z$ in $\mathbb{R}^2$ spanned by your two favourite cardinal directions. The tangent space of $z$ in $S^1$ is just one-dimensional, though, and these two spaces are naturally comparable. Indeed, if you draw the situation, it's clear that the line $z+tiz$ should realise the tangent space. Now, $iz\mathbb{R}$ has an orthogonal complement, namely $z\mathbb{R}$. In this way, the tanget plane of $z$ in $\mathbb{R}^2$ naturally decomposes into the direct sum of the tangents of $z$ in $S^1$ and their orthogonal complement. It's this orthogonal complement that we refer to as the normal bundle.
Formally, you let $z=e^{i\theta}$ be a local chart, and compute the differential as a smooth map $(a,b)\to \mathbb{R}^2$ with $b-a<2\pi$. Then you'll find, of course that the differential is given by $\theta\mapsto ie^{i\theta}$, which is exactly the above identification of the tanget space of $z$.
You could do the same analysis in any dimension: If $M\subseteq \mathbb{R}^n$ is an embedded manifold, you can simply identify the tangent space of $M$ at any point, and the normal bundle is then the bundle of orthogonal complements.
