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Lichnerowicz definition of the Schouten-Nijenhuis bracket

Lichnerowicz defined the Schouten-Nijenhuis bracket implicitly as follows.
\begin{prop} For all $P\in\gX^p M$, $Q\in \gX^q M$, $\om\in\Om^{p+q-1} M$ \begin... ... \bracket{d(i_P\om)}{Q} +(-1)^p\bracket{d\om}{P\land Q} \end{equation}\end{prop}
With respect to our explicit construction this formula has the advantage of being well adapted and easy to use in ''global type'' computations.

Look at what happens, for example, when $X,Y\in\gX^1(M)$, $\om\in\Om^1 M$

\begin{displaymath} \bracket{\om}{[X,Y]} =\bracket{d(i_Y\om)}{X} - \bracket{d(i_X\om)}{Y} - \bracket{d \om}{X\land Y} \end{displaymath}

which you can rewrite as

\begin{displaymath} d\om(X,Y)=X(\om(Y))-Y(\om(X))-\om([X,Y]) \end{displaymath}

i.e. the formula for the differential of 1-form.

In our approach this formula needs a proof. It suffices to show that the bracket defined by [*] has the same algebraic properties as the Schouten-Nijenhuis bracket. Unicity then implies the claim.

Pawel Witkowski 2006-06-26