# Question regarding reduction of structure group

Let $$\pi:P \to M$$ be a principal $$G$$-bundle. Given a subgroup $$H$$ of $$G$$ one can consider the fibre bundle $$P_H :=P\times_{G}G/H$$ whose fibres are the coset space $$G/H$$.

We define a reduction of structure group to be a section of $$P_H \to M$$ which is also the same as a $$G$$-equivariant map from $$P \to G/H$$.

Here are my questions :

1. I would expect $$P_H$$ to be a principal $$H$$-bundle from the name. But the fibres are $$G/H$$. Am I understanding wrong?

Another interpretation of $$G$$-bundles are by specifying transition functions from $$U\cap V \to G$$ satisfying cocycle conditions.

1. I would expect reduction to $$H$$ should somehow mean the transition functions land in $$H$$ instead of the full $$G$$. How can I formulate this?

All spaces above are smooth manifolds and groups are Lie groups.

$$P_H$$ is not a principal $$H$$-bundle, since as you correctly say, it has fiber isomorphic to $$G/H$$. In fact, one can show that $$P_H$$ can be identified with $$P/H$$: this is proved for example in "Foundations of Differential Geometry - Volume I" by Kobayashi and Nomizu, Chapter I, Proposition 5.5.
Instead, associated with a $$G$$-equivariant map $$\phi:P\to G/H$$, the corresponding $$H$$-subbundle is $$\phi^{-1}(eH)$$. Using $$G$$-equivariance it's straightforward to show this is actually invariant under the right $$H$$-action, and that it defines a $$H$$-bundle over $$M$$. The "reduction" of the bundle is simply the inclusion map $$i:\phi^{-1}(eH)\to P$$.
To answer your second question, for an arbitrary trivialisation of $$P$$, there's no guarantee the transition functions will take values in $$H$$. However if you define a local trivialisation of $$\phi^{-1}(eH)$$, in the form of local sections $$s_U:U\subset M\to \phi^{-1}(eH)\subset P$$, and use these to also define a local trivialisation of $$P$$, then the transition functions will be $$H$$-valued.
• Thank you. I will work out the details. I want to ask a follow-up question if you don't mind. A metric is defined to be the reduction of structure group to the maximal compact subgroup of my starting G. Then I'm told that one defines energy of the metric $h$ as integration over $|Dh|^2$. Can you please explain what the "norm of $Dh$" means here? Feb 24, 2022 at 19:05