although the theory of principal bundles plays an important role in the study of "g"-structures, the two notions are different. a "g"-structure is a principal subbundle of the tangent frame bundle, but the fact that the "g"-structure bundle "consists of tangent frames" is regarded as part of the data. for example, consider two riemannian metrics on r. the associated o("n")-structures are isomorphic if and only if the metrics are isometric. but, since r is contractible, the underlying o("n")-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces are trivial bundles.