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I don't understand the geometry of what happens when you give a manifold a metric, in particular how the group structure reduces to the orthogonal group.

I've read the wikipedia article http://en.wikipedia.org/wiki/Reduction_of_the_structure_group a dozen times but I get stuck when it says that

I understand that the group [itex]G[/itex]structure is a requirement for constructing a Principal Bundle because we want to act on the fibres by representatives of some group [itex]G[/itex]. And it is this continuous, free and transitive right action that causes the fibre to be homeomorphic to the group.

However, giving a manifold a metric is equivalent to giving the tangent bundle a map (or a tensor) from [itex]TM\times TM \rightarrow \mathbb{R}[/itex]. The tangent bundle is associated with it's frame bundle (which is where the basis comes from) which is also a torsor, so I understand how the fibre bundle has an intimate relationship with a group. But now I get stuck, because my book says that now the Frames (i.e. fibres of the Frame Bundle) are

I've read the wikipedia article http://en.wikipedia.org/wiki/Reduction_of_the_structure_group a dozen times but I get stuck when it says that

...the reduction of the structure group is an [itex]H[/itex]-bundle [itex]B_H[/itex]such that the pushout [itex]B_H \times_H G[/itex]is isomorphic to [itex]B[/itex]...

I understand that the group [itex]G[/itex]structure is a requirement for constructing a Principal Bundle because we want to act on the fibres by representatives of some group [itex]G[/itex]. And it is this continuous, free and transitive right action that causes the fibre to be homeomorphic to the group.

However, giving a manifold a metric is equivalent to giving the tangent bundle a map (or a tensor) from [itex]TM\times TM \rightarrow \mathbb{R}[/itex]. The tangent bundle is associated with it's frame bundle (which is where the basis comes from) which is also a torsor, so I understand how the fibre bundle has an intimate relationship with a group. But now I get stuck, because my book says that now the Frames (i.e. fibres of the Frame Bundle) are

*required*to be orthonormal due to the presence of a metric. Why? What has caused the group to reduce from [itex]GL(n,\mathbb{R})[/itex] to [itex]\mathcal{O}[/itex]?
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