Bundle restrictions

2011-03-02T23:10:21Z

I found the following definition in C. Isham's book "Modern Differential geometry for physicists":

Let $(E,\pi,\mathcal{M})$ be a fiber bundle and $\mathcal{N}$ a subspace of the base space $\mathcal{M}$. Then the restriction 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.

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 $\mathcal{N}$.

I have the following (hopefully not silly) question: How is the connection $f^*A$ 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 $\sigma'^*\Omega'=f^*A$ ?

Thank you

Bundle restrictions - MathOverflow

Bundle restrictions