Canonical line bundle

The canonical or tautological line bundle on a projective space appears frequently in mathematics, often in the study of characteristic classes. Note that there is possible confusion with the theory of the canonical class in algebraic geometry; for which reason the name "tautological" is preferred in some contexts. See also tautological bundle.

Definition

Form the cartesian product $mathbf\{R\}\; P^n\; imesmathbf\{R\}^\{n+1\}$, with the first factor denoting real projective "n"-space. We consider the subset

: We have an obvious projection map $pi:E(gamma^n)\; omathbf\{R\}P^n$, with $(\{pm;x\},v)mapsto\{pm;x\}$. Each fibre of $pi$ is then the line through $x$ and $-x$ inside Euclidean ("n"+1)-space. Giving each fibre the induced vector space structure we obtain the bundle:$gamma^n:=(E(gamma^n)\; omathbf\{R\}P^n),$ the canonical line bundle over $mathbf\{R\}P^n$.

Facts

*$gamma^n$ is locally trivial but not trivial, for $ngeq\; 1$.

In fact, it is straightforward to show that, for $n=1$, the canonical line bundle is none other than the well-known bundle whose total space is the Möbius strip. For a full proof of the above fact, see [J. Milnor & J. Stasheff, "Characteristic Classes", Princeton, 1974.] .

ee also

*Stiefel-Whitney class.

References

* [M+S] J. Milnor & J. Stasheff, "Characteristic Classes", Princeton, 1974.