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Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $(P,M,G,\omega)$ be a vector bundle with connection form $\omega$.

Let $X,Y$ be smooth vector fields on $P$. I am confused about the meaning of notation $X(\omega(Y))$.

I know what it means when we have a real valued differential form in place of $\omega$.

Suppose $\omega$ is a real valued differential form then $\omega(Y)$ is a real valued smooth function on $P$ sending $p$ to $\omega(p)Y(p)$. As $X$ is a vector field, this takes a real valued smooth function to another real valued smooth function. Thus, $X(\omega(Y))$ is a real valued smooth function.

What would this mean when $\omega $ is a $\mathfrak{g}$ valued $1$ form. I guess $X(\omega(Y))$ is a $\mathfrak{g}$ valued function on $P$.

Any reference is welcome.

This notation is used in structure equation of a connection form.

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$\omega$ is a $1$-form, we deduce that for every $x\in P$, $\omega_x(Y(x))\in{\cal g}$ and $h_Y:P\rightarrow {\cal g}$ defined by $x\rightarrow\omega_x(Y(x))$, $X\omega(Y)$ is the differential of $h_Y$ evaluated at $X$, $dh_Y.X(x)$.

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  • $\begingroup$ Can you give reference, please. $\endgroup$ – user312648 Dec 29 '17 at 18:41
  • $\begingroup$ perhaps Kobayashi Nomizu $\endgroup$ – Tsemo Aristide Dec 29 '17 at 18:41
  • $\begingroup$ I could not see where they have explained this notation, I am sure by g you mean $\mathfrak{g}$ $\endgroup$ – user312648 Dec 29 '17 at 18:43
  • $\begingroup$ It is the the Lie algebra, where the connection form takes its values. $\endgroup$ – Tsemo Aristide Dec 29 '17 at 18:44
  • $\begingroup$ Yes I understand. Can you give some reference where this is mentioned in detailed. $\endgroup$ – user312648 Dec 29 '17 at 18:52

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