Lie derivative with respect to anything else

[jfy4]

Hi,

I have been looking around, and I can't seem to find a slightly different version of the lie derivative where the lie derivative is taken with respect to a tensor field, rather than a vector field. That is, a quantity which measures the change in a vector field, along the "flow" of a tensor field.

I am not asking about the lie derivative of a tensor, which is the change in a tensor field through the flow of a vector field.

does such a derivative exist? is this a reasonable question?

Also, would a different, but correct way to describe the Lie derivative be "it measures the change in a tensor field with respect to the change in a vector field" ?

Thanks,

[Eynstone]

I can't think of a definition of the lie derivative with respect to a covector off my head.
However, we may talk about the lie derivative with respect to a totally contravariant tensor.We could define it as the tensor product of component-wise lie derivatives. Such a quantity could be another tensor.