For a principal fiber bundle with a base $M$ and a structure group $G$: $P(M,G)$ there is a connection form $\omega$. Is it true that if a fiber bundle restriction $P(M,G) \to Q(M,H)$ is possible it also defines a connection form on the fiber bundle $Q(M,H)$? Thanks
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$\begingroup$ Which properties do you require the "restricted" connection to have? $\endgroup$– QfwfqJan 28, 2014 at 13:35
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$\begingroup$ Could you explain what is $H$, and what is "a fiber bundle restriction"? $\endgroup$– abxJan 28, 2014 at 13:47
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$\begingroup$ @abx $H$ is a subgroup of $G$ (I assume both $G$ and $H$ to be simple or semisimple) $\endgroup$– user124443Jan 28, 2014 at 14:08
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If $H$ is a reductive group (for example, compact or semisimple), then the representation $\mathfrak{g}$ splits over $H$, and you do indeed get an induced connection, by splitting $\omega$ into its part valued in $\mathfrak{h}$ part and its part valued in $\mathfrak{h}^{\perp}$.
answered Jan 28, 2014 at 13:52
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$\begingroup$ Thanks! Here is what I meant. Assuming that $G$ is a Lie group, the connection form can be written as $\omega = \omega ^ A T _ A + \omega ^ a t _a$, where $T_ A$ and $t _ a$ are generators of the Lie algebra of $G$ ($t _ a$ are the ones corresponding to $H$). It seems that in this case using a "projection" Lie(G) $\to$ Lie(H) a connection on $Q(M,H)$ can be defined. Is there something wrong with this construction? $\endgroup$ Jan 28, 2014 at 14:24
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$\begingroup$ I was thinking of $\omega$ as a 1-form on the bundle $P$. To get a connection, you need to explain why (in mathematics language) the 1-form you use is $H$-equivariant. So you need an $H$-equivariant projection $\mathfrak{g} \to \mathfrak{h}$. In physics language, you need to ensure that you maintain gauge invariance; a computation shows that again you need the projection to be $H$-equivariant. But you want $H$ to be a subgroup of $G$, so you need to have an $H$-equivariant linear map $\mathfrak{g}\to\mathfrak{h}$ which induces the identity map on $\mathfrak{h} \subset \mathfrak{g}$. $\endgroup$ Jan 28, 2014 at 14:49