An oriented surface with a Riemannian metric has a natural action of the unit circle on its tangent bundle. Rotate the tangent vector through the angle theta in the positively direction.(adsbygoogle = window.adsbygoogle || []).push({});

Is there a natural action of SO(n) on the tangent bundle of an oriented Riemannian n-manifold?

Same question for any oriented vector bundle with Riemannian metric over a smooth manifold.

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# SO(n) actions on vector bundles

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