Variations on the above example include the orthonormal frame bundle of a Riemannian manifold.
3.
Over, one replaces by the orthogonal group, and by restricting to orthonormal frames, one gets the identity
4.
Similarly, the Darboux frame on a surface is an orthonormal frame whose first two vectors are tangent to the surface.
5.
If " M " is orientable, then one also has the oriented orthonormal frame bundle F SO " M ".
6.
For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection.
7.
Applying this construction to each tangent space yields the orthonormal frame bundle of a ( pseudo-) Riemannian manifold and a frame field is a section of this bundle.
8.
In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an "'orthonormal frame "', that is an affine frame such that is an orthonormal basis.
9.
For example, if " M " is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of " M ".
10.
The orthonormal frame bundle is just a reduction of the structure group of F GL ( " M " ) to the orthogonal group O ( " n " ).
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