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Is there a nice way to recast riemannian geometry in terms of principal bundles?

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Metric compatibility is about connections on the orthonormal frame bundle. The torsion of a connection is the exterior covariant derivative of the canonical 1-form on the total space of the frame bundle. Kobayashi and Nomizu’s books mostly use principal bundles as their language of choice.
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Everything can be phrased in this language but experience shows that it buys you very little to do so.
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The local aspects of Riemannian geometry largely boil down to the Levi-Civita connection --- which is indeed quite special to the Riemannian setting --- but of course the global aspects have a lot to do with the calculus of variations and elliptic PDE theory.  


Anyway, your question is probably best answered via the theory of G-structures.  A G-structure on a manifold M is a principal G-subbundle of the general frame bundle FM, where G is a (usually closed) subgroup of GL(n,R).  Specifying an O(n)-structure on M is equivalent to specifying a Riemannian metric on M.  A connection on an O(n)-structure is then equivalent to a metric-compatible connection on TM.  


To each G-structure B in FM, there is a G-equivariant function T\_B called the "intrinsic torsion" (an older name is "structure function") whose domain is B and whose codomain is a particular R-vector space equipped with a G-action that I won't explain here.  The function T\_B is zero if and only if B admits a torsion-free G-connection.  In this language, the existence part of the fundamental lemma of Riemannian geometry states that every O(n)-structure has T\_B = 0.
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You're looking for the Atiyah algebroid. Connections can be seen as sections of a short exact sequence("SES"), for instance a principle bundle is a SES given by G → P → M and a connection is given by sections σ : M → P. Levi-Civita connections are sections of a similar SES for vector bundles, I can't remember the details of this. You can use extension/reduction of structure group and frame bundles to get a map between short exact sequences, hence you get a map between sections of principle bundles and Levi-Civita connections
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If I'm not mistaken Frederic Schuller covers this in his lectures a bit.

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