- Canonical line bundle
The canonical or tautological
line bundle on aprojective space appears frequently inmathematics , often in the study ofcharacteristic class es. Note that there is possible confusion with the theory of thecanonical class inalgebraic geometry ; for which reason the name "tautological" is preferred in some contexts. See alsotautological 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 thesubset :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.
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