cotangent bundle is a bundle


Verifying the first criterion is simply a matter of writing it out:

(x1,,x2n)(π(x1,,x2n),ϕα(x1,,xn))=((x1,,xn),(xn+1,,x2n))

This is obviously a homeomorphism.

As for the second criterion,

j=1ngαβji(x1,,x2n)ϕβj(x1,,x2n) = j=n2n(σαβ(x1,xn))ixjxj+n
= σαβi+n(x1,,x2n)
= ϕαi(x1,,x2n)

The third criterion follows from the chain rule:

gαβjigβγkj=(σαβ(x1,xn))ixj(σβγ(x1,xn))jxk
Title cotangent bundle is a bundle
Canonical name CotangentBundleIsABundle
Date of creation 2013-03-22 14:54:31
Last modified on 2013-03-22 14:54:31
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Proof
Classification msc 58A32