"Clutching construction" is not a widely recognized term in standard architectural or construction terminology as of my last knowledge update in October 2021. It could potentially refer to specific techniques or methods in a niche area of construction, or it may have emerged as a new concept or terminology after my last update.
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A vector bundle over can be constructed by gluing two trivial bundles over its hemispheres with a transition map . Extend this map constantly in an equatorial collar to obtain local vector bundle trivializations. A transition map into the special orthogonal group gives compatible fiber orientations and a fiber metric. Homotopies of transition maps give isomorphic bundles.