orthonormal frame sentence in Hindi
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- The orthonormal frame bundle of a rank " k"
- Variations on the above example include the orthonormal frame bundle of a Riemannian manifold.
- Over, one replaces by the orthogonal group, and by restricting to orthonormal frames, one gets the identity
- Similarly, the Darboux frame on a surface is an orthonormal frame whose first two vectors are tangent to the surface.
- If " M " is orientable, then one also has the oriented orthonormal frame bundle F SO " M ".
- For instance, in relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection.
- 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.
- In Euclidean geometry, Cartesian coordinates are affine coordinates relative to an "'orthonormal frame "', that is an affine frame such that is an orthonormal basis.
- For example, if " M " is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of " M ".
- The orthonormal frame bundle is just a reduction of the structure group of F GL ( " M " ) to the orthogonal group O ( " n " ).
- For example, a pair of Riemannian manifolds are ( locally ) equivalent if and only if their bundles of orthonormal frames are ( locally ) isomorphic " G "-structures.
- 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 ".
- 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.
- 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.
- 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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