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It seems that the tangent bundle of a hypersurface of Euclidean space is the bundle induced from the tangent bundle of the unit sphere under Gauss mapping. Is this true?
The reason I think this is that tangent space at a point on the surface can be parallel translated to the tangent space on the sphere at the unit normal.
If this is true - which it seems to be - then there is an induced Levi-Cevita connection on the hypersurface under the Gauss mapping. What are the conditions under which this induced connection is the same as the connection that the hypersurface inherits from the embedding?
The reason I think this is that tangent space at a point on the surface can be parallel translated to the tangent space on the sphere at the unit normal.
If this is true - which it seems to be - then there is an induced Levi-Cevita connection on the hypersurface under the Gauss mapping. What are the conditions under which this induced connection is the same as the connection that the hypersurface inherits from the embedding?