Chaning the primary metric of a manifold

[Ali Seraj]

Dear all,

I have a bisic question:

I have a maniforld, say S2 on which I have two metrics gamma[-A,-B] and Omega[-A,-B]. Since gamma is defined first, it is the primary metric on the manifold.

However, in the second part of my code, I want to switch the primary metric to Omega. 

What is the best way to do that?

I tried the following but didn't work:

Unprotect[MetricsOfVBundle]
MetricsOfVBundle[TangentS2] = {Omega, gamma}
Protect[MetricsOfVBundle]

Thanks in advance.

Best regards,

Ali

[Jose]

Hi,

This is a tricky operation. Once indices have been moved up and down with one metric, changing the name of the first metric will effectively render all previous computations incorrect. This can be considered a limitation of xTensor: indices do not remember with which metric they were raised or lowered. We always assume it was with the first metric. Compare for example with the notation of basis indices, say {2, B} which always carry the basis B they are associated to. Something like this could have been done. It would have been more flexible, but it would have also complicated considerably the language.

So I'd say there is no good way to do this. I'd just restart the whole computation using the new metric.

Cheers,

Jose.