"Drawable" Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use:

When our base space is $\mathbb{S}^1$ and we assign to each $p \in \mathbb{S}^1$ a copy of $\mathbb{R}$ and make either a Cylinder (trivial) or a Möbius Bundle.
We can also consider a trivial bundle $\mathbb{S}^1 \times \mathbb{R}^2$ like in this post.

What are other good examples of (non-trivial?) vector bundles that can easily be explained/drawn? Also are there any surfaces on which the tangent bundle is used for something interesting? Although I understand what vector/tangent bundles are, I don't quite see the motivation for studying them yet. If you have any examples of general fiber bundles that can be simply explained, that would be appreciated too.

To provide more context, I am an undergraduate giving a talk to undergraduates in a differential geometry course that focuses on smooth surfaces. It is a small-stakes talk. I would rather the audience walk away with an intuitive sense of what a vector bundle is (a picture in their head) rather than just knowing the definition. Vector bundles themselves are not exactly part of what we've been talking about in the course, though we've touched on ideas relating to tangent bundles.

Solution 1:

On the vector bundle side, after discussing the cylinder and Möbius bundle and then moving to tangent bundles, you can explore the tangent bundles on $S^2$ and $T^2$ by looking at vector fields on them.

The tangent bundle of $T^2$ is trivial because there are two, linearly independent non-vanishing sections (forming a basis at every point) given by a vector field pointing in one direction around the torus and a vector field pointing in the other.

The tangent bundle of $S^2$, by contrast, has no non-vanishing sections, let alone two linearly independent ones (this is the hairy ball theorem), so the tangent bundle of $S^2$ is non-trivial.

On the fiber bundle side, bundles over the circle are nice to visualize. You can see a lot of the bundle part happening without the picture becoming hard to visualize.

A circle bundle can be trivialized over all but one point, so all the action can be viewed in how you "glue" the bundle at that one point.

The cylinder and Möbius bundle are $\mathbb{R}$ bundles over the circle, where the the Möbius bundle is glued by the map $x \mapsto -x$ (and the cylinder is glued by the identity).

The torus and Klein bottle are circle bundles over the circle, where the torus is glued by the identity and the Klein bottle is glued by a map $S^1 \rightarrow S^1$ that "goes backwards", e.g. $\theta \mapsto -\theta$ (or alternatively, given by a reflection in $\mathbb{R}^2$ restricted to $S^1$).

If you're extremely ambitious, you could draw 2-dimensional vector bundles over $S^2$ and discuss how they can be trivialized over the upper hemisphere and over the lower hemisphere. Then the bundle is determined by how you glue these together over the equator. If your class has done examples with atlases for $S^2$, you could then figure out what this gluing is in the case of $TS^2$.

Solution 2:

Let $M$ be a Riemannian manifold. The normal bundle of a submanifold $S \subset M$ is the the vector bundle $(S, NS)$ where $N_pS$ consists of the elements of $T_pM$ that are orthogonal to $T_p S$. For example, if $M = \mathbb{R}^m$ and $S = S^{m-1} \subset M$, then the normal bundle consists of the lines in the radial directions at each $p\in S^{m-1}$. For surfaces the normal bundle is pretty easy to draw.

To motivate vector bundles to an undergraduate audience, I suggest starting with the idea of a tangent vector field. On Euclidean space or on a surface ideas like tangent vector fields and normal vector fields are fairly intuitive to define, and they are eminently applicable to a variety of situations (describing the flow of some smooth process, for example). But the usual definition of a vector field on a surface depends on how the surface is embedded in Euclidean space. Smooth manifolds are generally not presented as embedded in some sort of ambient space; to generalize the notion of a tangent vector field it is required to have a way to define a collection of tangent vectors that varies smoothly over the manifold without referring to any ambient space or coordinates. The tangent bundle is precisely what allows you to do this. (If you need to look into this in more detail, the keyword is section of a fiber bundle.)

Once you have the notion of a vector bundle you can do a variety of things other than define vector fields. For instance, with a Riemannian manifold you can define the length of a curve, and then you get into useful ideas like geodesics. With the tangent bundle you can discuss the orientability of a manifold (if you know some homology theory, you can do this with a fiber bundle instead). If the fibers of your fiber bundle have some extraordinary structure, you can think about acting on the fibers; for example, on a $S^1$-bundle you can think about what it means to apply a $S^1$-rotation to every point of the manifold. The gist is that interesting manifolds often have more structure than the bare minimum the definition of a manifold implies, and fiber bundles are structures that contain this sort of information in a coordinate-free way.