Canonical line bundle
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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.
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Definition
Form the cartesian product <math>\mathbb R P^n\times\mathbb R^{n+1}</math>, with the first factor denoting real projective n-space. We consider the subset
- <math>E(\gamma^n):=\big\{(\{\pm\;x\},v)\in\mathbb RP^n\times\mathbb R^{n+1}:v=\lambda x,\;\lambda\in\mathbb R\big\}.</math>
We have an obvious projection map <math>\pi:E(\gamma^n)\to\mathbb RP^n</math>, with <math>(\{\pm\;x\},v)\mapsto\{\pm\;x\}</math>. Each fibre of <math>\pi</math> is then the line through <math>x</math> and <math>-x</math> inside Euclidean (n+1)-space. Giving each fibre the induced vector space structure we obtain the bundle
- <math>\gamma^n:=(E(\gamma^n)\to\mathbb RP^n),</math>
the canonical line bundle over <math>\mathbb RP^n</math>.
Facts
- <math>\gamma^n</math> is locally trivial but not trivial, for <math>n\geq 1</math>.
In fact, it is straightforward to show that, for <math>n=1</math>, 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
References
- [M+S] J. Milnor & J. Stasheff, Characteristic Classes, Princeton, 1974.