Bundle restrictions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:49:38Z http://mathoverflow.net/feeds/question/57182 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57182/bundle-restrictions Bundle restrictions Darius Alexander 2011-03-02T23:10:21Z 2011-03-02T23:10:21Z <p>I found the following definition in C. Isham's book "Modern Differential geometry for physicists":</p> <p>Let $(E,\pi,\mathcal{M})$ be a fiber bundle and $\mathcal{N}$ a subspace of the base space $\mathcal{M}$. Then the <em>restriction</em> of $(E,\pi,\mathcal{M})$ to $\mathcal{N}$ is defined to be the bundle $(\pi^{-1}(\mathcal{N}\ )\ ,\ \pi|_{\pi^{-1}(\mathcal{N})}\ ,\ \mathcal{N})$ with the same fiber.</p> <p>Having this definition, let $A=\sigma^*\Omega$ be a connection 1-form on base space $\mathcal{M}$, i.e. the pullback ($\sigma:\mathcal{M}\rightarrow E$ is a local section) of a well defined connection $\Omega$ on the principal bundle $E$ (also suppose $E$ has a metric compatible with $\Omega$). Let $f$ be the inclusion map of $\mathcal{N}$ in $\mathcal{M}$. With this, $f^*A$ is a 1-form on $\mathcal{N}$. </p> <p>I have the following (hopefully not silly) question: How is the connection <code>$f^*A$</code> related to the restriction bundle of $E$ to $\mathcal{N}$ ? Is there a connection $\Omega'$ on $\pi^{-1}(\mathcal{N})$ and a suitable local section $\sigma'$ such that $\sigma'^*\Omega'=f^*A$ ?</p> <p>Thank you</p>