proof that transition functions of cotangent bundle are valid

In this entry, we shall verify that the transition functionsMathworldPlanetmathPlanetmath proposed for the cotangent bundleMathworldPlanetmath the three criteria required by the classical definition of a manifold.

The first criterion is the easiest to verify. If α=β, then σαα reduces to the identity and we have

(σαα(x1,,x2n))i=(σαα(x1,,xn))i=xi  1in
(σαα(x1,,x2n))i+n=j=1n(σαα(x1,,xn))ixjxj+n=j=1nxixjxj+n=xi+n  1in

Thus we see that σαα is the identity map, as required.

Next, we turn our attention to the third criterion — showing that σβγσαβ=σαγ . For clarity of notation let us define yi=(σαβ)i(x1,x2n). Then we have

(σβγσαβ)i(x1,,x2n) = (σβγ)i(y1,,y2n)
= (σβγ)i(y1,,yn)
= (σβγσαβ)i(x1,,xn)
= (σαγ)i(x1,,xn)
= (σαγ)i(x1,,x2n)

when 1in.

(σβγσαβ)i+n(x1,,x2n) = (σβγ)i+n(y1,,y2n)
= j=1n(σβγ(y1,,yn))iyjyj+n
= j=1nk=1n(σβγ(y1,,yn))iyj(σαβ(x1,,xn))jxkxn+k
= k=1n(σβγσαβ(x1,,xn))ixkxn+k
= k=1n(σαγ(x1,,xn))ixkxn+k
= σαγ(x1,,x2n)

when 1in.

Finally, the second criterion does not need to be checked because it is a consequence of the first and third criteria.

Title proof that transition functions of cotangent bundle are valid
Canonical name ProofThatTransitionFunctionsOfCotangentBundleAreValid
Date of creation 2013-03-22 14:52:25
Last modified on 2013-03-22 14:52:25
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 13
Author rspuzio (6075)
Entry type Proof
Classification msc 58A32