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I am unhappy with Lie derivative. In the previous version it defined the Lie derivative as a secondary notion, using the differential and the Cartan homotopy formula (for which I finally created an entry). I have added a bit mentioning vector fields etc. and a formula using derivatives for forms but this is still not the right thing. Namely, in my understanding the Lie derivative is a fundamental notion and should not be defined using other differential operators, but by the “fisherman’s derivative” formula. Second it makes sense not only for differential forms but for any geometric quantities associated to the (co)frame bundle, and in particular to any kind of tensors, not necessarily contravariant or antisymmetrized. For this one has a prerequisite which will require some work in $n$Lab. Namely to a vector field, one associated the flow, not necessarily defined for all times, but for small times. Then for any $t$ one has a diffeomorphism, which is used in the fisherman’s formula. But fisherman’s formula requires the pullback and the pullback is usually defined for forms while for general tensor fields one may need combination of pullbacks and pushforwards. However, for diffeomorphisms, one can define pullback in both cases, and pullback for time $t$ flow corresponds to the pushforward for time $-t$. To define such general pullback it is convenient to work with associated bundles for frame or coframe bundle and define it in the formalism of associated bundles. In the coframe case, this is in Sternberg’s Lectures on differential geometry (what returns me back into great memories of the summer 1987/1988 when I studied that book). So there is much work to do, to add details on those. If somebody has comments or shortcuts to this let me know.
However, there is a scientific question here as well: what about when frame bundle is replaced by higher jet bundles, and one takes some higher differential operator for functions and wants to do a similar program – are there nontrivial extensions of Lie derivative business to higher derivatives which does not reduce to the composition of usual Lie derivatives ?
I have added some material on coframe bundle into frame bundle. Please check.
For those like me who were mystified by “fisherman’s derivative”, see page 4 here. The terminology is apparently due to Vladimir Arnold, who conjures a fisherman observing objects in a river flowing past him as he stands on the shore. The idea is to take the time derivative of objects like tensor fields as they are transported along the integral flow of a given vector field.
I made some mistakes in the text (for frame bundle), I will correct and add tonight or tomorrow, once I get a free minute.
I have sent you an English copy by email. It is a shame that they charge scans of so low scan quality. In the same time, the Russian original is free:
<p>How strange; I just hit a pay wall myself when I clicked on the link. Oh well, let me give a different link. It turns out “fisherman’s derivative” is easily googleable, and appears for example in Arnold’s Mathematical Methods of Classical Mechanics (p. 198).</p>
<p>Try <a href="http://iopscience.iop.org/1063-7869/47/12/B08">this</a>. Or if that still won’t work, there’s <a href="http://books.google.com/books?id=UhYyuUEUFrUC&pg=PA331&lpg=PA331&dq=%22fisherman%27s+derivative%22&source=bl&ots=YO5sASNqsx&sig=SQ5BHM5ZtvpE_xLG8yNcIIMDPwA&hl=en&ei=Rs1kTpuOCZLAgQfR_OXECg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDMQ6AEwBA#v=onepage&q=%22fisherman%27s%20derivative%22&f=false">this</a>, or <a href="http://books.google.com/books?id=Pd8-s6rOt_cC&pg=PA198&lpg=PA198&dq=%22fisherman%27s+derivative%22&source=bl&ots=uKkhsHJKPB&sig=fUJF7BOFYM0zoIfsvyaAO9Bhz80&hl=en&ei=Rs1kTpuOCZLAgQfR_OXECg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CDkQ6AEwBg#v=onepage&q=%22fisherman%27s%20derivative%22&f=false">this</a>, or, in a humorous vein, <a href="http://books.google.com/books?id=vQR0mN1dgUEC&pg=PA71&lpg=PA71&dq=%22fisherman%27s+derivative%22&source=bl&ots=R4N950CCx4&sig=TaJII4X7hf5hk_ioW9hF1ET7irE&hl=en&ei=bM5kTsGRIIrVgQeq87SnCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwADgK#v=onepage&q=%22fisherman%27s%20derivative%22&f=false">this</a>.</p>
Added:
Cartan introduced Lie derivatives of differential forms and derived Cartan’s magic formula in
Extension to arbitrary tensor fields was given in
The term “Lie derivative” (Liesche Ableitung) is due to van Dantzig, who also suggested a definition using the flow of a vector field:
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