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Suppose $M$ is a smooth manifold. If $X:M \to TM$ is a smooth vector field, then we can define a map $\iota_X: \Omega^k(M) \to \Omega^{k-1}(M) $ as a map which takes $\omega\in \Omega^k(M)$ to $\iota_X(\omega)\in \Omega^{k-1}(M)$ whose action is given by $$\iota_X(\omega)(X_1, \ldots, X_{k-1}) = \omega(X,X_1, \ldots, X_{k-1}).$$ Now if $ G:M \to \Lambda^2(TM)$ is a bivector field, how to define a map $\iota_G: \Omega^k(M) \to \Omega^{k-2}(M)$.

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One choice is to define $$ \iota_{U\wedge V}(\omega)(X_1,\cdots,X_{k-2})=\omega(U,V,X_1,\cdots,X_{k-2}) $$ for vector fields $U,V$. Of course, one must show that $\iota$ is well defined and uniquely determined by this expression.

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  • $\begingroup$ What is the domain of $U$ and $V$, is it whole of $M$ or some open subsets of $M$? $\endgroup$
    – Uncool
    May 4 at 10:20
  • $\begingroup$ Everything is local, so one could use either local or global objects; the resulting product will behave the same. $\endgroup$
    – Kajelad
    May 5 at 3:53
  • $\begingroup$ Thankyou, are there any other definitions $\endgroup$
    – Uncool
    Jun 12 at 12:00

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