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

:$$E(gamma^n):=ig{({pm;x},v)inmathbf{R}P^n imesmathbf{R}^{n+1}:v=lambda x,;lambdainmathbf{R}ig}. We have an obvious projection map $$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.