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.

Contents

1 Definition 2 Facts 3 See also 4 References

Definition

Form the cartesian product , with the first factor denoting real projective NaodW29-math30784404864acfc00000002-space. We consider the subset

We have an obvious projection map , with . Each fibre of π is then the line through x and - x inside Euclidean NaodW29-math30784404864acfc00000009-space. Giving each fibre the induced vector space structure we obtain the bundle

the canonical line bundle over .

Facts

• γⁿ is locally trivial but not trivial, for .

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 band. For a full proof of the above fact, see [M+S].

See also

• Stiefel-Whitney class.

References

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