Lie-derivative of tensor field along tensor field

Question

What is the natural notion of the Lie-derivative of a tensor field along another tensor field, and where can I find an exposition of that?

Answer 1

This can be done in certain special cases besides the usual Lie derivatives along a vector field. More precisely, let $X$ and $Y$ be tensor fields over a manifold $\mathscr{M}$. The Lie derivative $\mathscr{L}_X Y$ of $Y$ along $X$ can be defined in the following cases:

1.) The usual one when $X$ is a vector field. If $Y$ is also a vector field, $\mathscr{L}_X Y=[X,Y]$ is the Lie bracket of $X$ and $Y$. All other extensions can be defined by analogy with the Lie bracket with the aid of the Leibniz rule for derivations;

2.) When both $X$ and $Y$ are completely anti-symmetric contravariant tensor fields (i.e. $p$-vector fields), $\mathscr{L}_X Y=[X,Y]$, where $[\cdot,\cdot]$ is the Schouten-Nijenhuis bracket;

3.) When $X$ and $Y$ are vector-valued differential forms, $\mathscr{L}_X Y=[X,Y]$, where $[\cdot,\cdot]$ is the Frölicher-Nijenhuis bracket.

In complete generality, your problem should be seen as (part of) the problem of finding all bilinear, graded-symmetric natural operators from tensor fields of a certain rank to tensor fields of an appropriate rank. The above special cases of this problem are treated from this point of view in the book by I. Kolár, P. W. Michor and Jan Slovák, Natural Operations in Differential Geometry (Springer-Verlag, 1993), Section 30, pp. 250ff.