meaning of DefCovD with a VBundle parameter

[Daniel Mahler]

The help for DefCovD says:

DefCovD[covd[-a], {post, pre}] defines covd to be a connection on the \

tangent bundle of index a. DefCovD[covd[-a], vbundle, {post, pre}] \

defines covd to be a connection on the tangent bundle of index a and \

the inner vbundle (which must have the same base manifold)

I am not clear on what the meaning of DefCovD[covd[-a], vbundle, {post, pre}] .

The above defined covd takes an index that only ranges over the tangent bundle of the base manifold.

What does difference does the additional vbundle make?

I played around a bit with such a derivative

and could not see any difference in behaviour from a derivative defined without the vbundle.

thanks

Daniel

[Alfonso García-Parrado Gómez-Lobo]

Dear Daniel,

The meaning of the additional vbundle is that you are defining a
a covariant derivative on the vector bundle vbunle (inner vbundle). This
means that you will be able to use certain formulae defined for this
derivative when it acts on tensor with indices in the vbundle. For
example if T[-A] is a tensor in a vector bundle the output of

ChangeCovD[cd[-a]@T[-A]]

is different if cd[-a] was defined just as a covariant derivative in the
tangent bundle where the index a lives or as a covariant derivative in
the vector bundle which T[-A] was defined on.


Alfonso.