For example, a pair of Riemannian manifolds are ( locally ) equivalent if and only if their bundles of orthonormal frames are ( locally ) isomorphic " G "-structures.
12.
Since the group of rotations in the plane SO ( 2 ) acts simply transitively on oriented orthonormal frames in the plane, it follows that it also acts on the frame or circle bundles of " M ".
13.
At a given point " p " a general frame may be made orthonormal by orthonormalization; in fact this can be done smoothly, so that the existence of a moving frame implies the existence of a moving orthonormal frame.
14.
In relativity and in Riemannian geometry, the most useful kind of moving frames are the "'orthogonal "'and "'orthonormal frames "', that is, frames consisting of orthogonal ( unit ) vectors at each point.
15.
If, now, E _ 1, E _ 2, \ ldots, E _ m is a local orthonormal frame ( of tangent vector fields ) on the same open subset of " M ", then we can define the mean curvatures of the immersion by
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