Tatuological Line Bundle

Definition

Tautological line bundle over ${\mathbb{RP}}^n$ is a bundle ${\gamma }_n^1$ such that

where $E({\gamma }_n^1)=\left\{(\left\{x,-x\right\},v)\in {\mathbb{RP}}^n\times {\mathbb{R}}^{n+1}|v\in \mathbb{R}x\right\}$ and $\pi$ is the projection on the first coordinate.

Example

Open Mobius band

Mobius band is a ${\gamma }_1^1$. Because we identified ${\mathbb{RP}}^1$ as ${\mathbb{S}}^1/x-x$, we can parametrized as $\theta \in \left[0,\pi \right]$(half-circle, because the other half is identified antipodally):

$$L_{\theta }=span\left\{(cos\theta ,sin\theta )\right\}.$$

• The fiber over $L_{\theta }$​ is the line through the origin in that direction.
• If you go around the circle once (from $\theta =0$ to $\theta =\pi$), the line comes back but flipped:

$$(\cos 0,\sin 0)=(1,0)\text{and}(\cos \pi ,\sin \pi )=(-1,0).$$

This flipping is exactly the half-twist that gives an open Möbius band.¹

where the red circle is a ${\mathbb{RP}}^1$.

Footnotes

1. ${\mathbb{RP}}^1$ means it has a parametrization. ↩