How can I calculate exterior covariant derivate of $\,\operatorname{End}(E)$-valued $p$-form; i.e., $\,d_D(\eta)\;$?
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What is $E$? What is your diff. geoemtry setting? Where do your specific problems lie? – Avitus Jun 19 '13 at 08:18
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E is bundle for example tangent bundle, cotangent bundle. End(E) is endomorphism bundle – user83077 Jun 19 '13 at 08:23
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All you need is to consider the definition of covariant derivative for the vector bundle End(E) – Avitus Jun 19 '13 at 08:36
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As Avitus has commented, the derivative of forms is the one induced by that of sections (cf. e.g. Wells p. 74). Let $D$ be a connection in $E$. If $s$ is a section of $E$ and $\Phi$ a section of $\mathrm{End}\, E$, then you can define an induced connection $D^{\mathrm{End}}$ in $\mathrm{End}\,E$ by $$(D^{\mathrm{End}}_X\Phi)(s) = D_X (\Phi(s)) - \Phi(D_X s)$$ for vector fields $X$ on the base. This comes about by viewing $\mathrm{End}\,E$ as the tensor product bundle $E \otimes E^*$. The connection in $E$ induces one in $E^*$ causing the sign in the above formula (see Wells p. 93), and there is a canonical way (see this site) to define a connection in a tensor product bundle, given connections in the factors.
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