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,
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.