pull-back

[Omar Z]

Hello,

I would like to know if there is the recommended way to deal with a pull-back covariant derivative.

Let me explain the setting (which is the standard one of a non-linear sigma model):

Suppose I have a map
phi: M -> G
and that G has its own connection \nabla_a (give greek indices to M and latin indices to G).
The pull-back of phi induces a connection D_\mu acting on a vector space over M locally isomorphic to TG. I would like to use the same indices for both TG and the pull-back bundle.

Furthermore, I'd like to automatize with xact the geodesic expansion described (for example) here:
https://golem.ph.utexas.edu/~distler/blog/archives/002532.html

Best,
OZ

[Omar Z]

So I fixed the problem thanks to a simple solution suggested by two gentleman colleagues.

The trick is to define the two manifolds (M,G), each with its own covariant derivative.

Then the map phi:M->G can be defined as a tensor living on the product MxG with an index on TG.

For simplicity I'm assuming that the connection on M is already the pull-back one, alternatively that can be introduced separately.

The construction is "almost" geometrically correct as a fiber bundle!