- #1
center o bass
- 560
- 2
Say one one have a projection map ##\pi : M^5 \to M^4## which in adapted coordinates are of the form
$$ \pi(x^\mu, x^4) = x^\mu$$
where ##\mu = 0,1,2,3##. Now if one ##M^4## introduce an orthonormal frame ##\left\{ e_\mu, e_4\right\}## where ##e_\mu## are tangential to ##M^4## and ##e_4## orthogonal to it. The corresponding dual basis is ##\left\{\omega^\mu, \omega^4\right\}##. If one now considers ##\omega^\nu## to be part of the cotangent space ##T^*_pM^4## and one take the exterior derivative
$$d ^{^{(4)}}\omega^\nu$$
Where ##^{^{(4)}}\omega^\nu## denotes that i consider the covector a part of ##T_p^*M^4##. Now if one considers the same covector as a covector in ##T^*_pM^5## and we take the exterior derivative
$$d ^{^{(5)}}\omega^\nu$$
would it then be correct to say that
$$\pi^* \left( d ^{^{(4)}}\omega^\nu\right) = d \pi^* \left(^{^{(4)}}\omega^\nu\right) = d^{^{(5)}}\omega^\nu$$
due to the commutation of the pullback ##\pi^*## with the exterior derivative?
$$ \pi(x^\mu, x^4) = x^\mu$$
where ##\mu = 0,1,2,3##. Now if one ##M^4## introduce an orthonormal frame ##\left\{ e_\mu, e_4\right\}## where ##e_\mu## are tangential to ##M^4## and ##e_4## orthogonal to it. The corresponding dual basis is ##\left\{\omega^\mu, \omega^4\right\}##. If one now considers ##\omega^\nu## to be part of the cotangent space ##T^*_pM^4## and one take the exterior derivative
$$d ^{^{(4)}}\omega^\nu$$
Where ##^{^{(4)}}\omega^\nu## denotes that i consider the covector a part of ##T_p^*M^4##. Now if one considers the same covector as a covector in ##T^*_pM^5## and we take the exterior derivative
$$d ^{^{(5)}}\omega^\nu$$
would it then be correct to say that
$$\pi^* \left( d ^{^{(4)}}\omega^\nu\right) = d \pi^* \left(^{^{(4)}}\omega^\nu\right) = d^{^{(5)}}\omega^\nu$$
due to the commutation of the pullback ##\pi^*## with the exterior derivative?