Frame Bundle - Tangent Frame Bundle

The tangent frame bundle (or simply the frame bundle) of a smooth manifold M is the frame bundle associated to the tangent bundle of M. The frame bundle of M is often denoted FM or GL(M) rather than F(TM). If M is n-dimensional then the tangent bundle has rank n, so the frame bundle of M is a principal GLn(R) bundle over M.

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Tangent Frame Bundle - Solder Form
... The frame bundle of a manifold M is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of M ... Let x be a point of the manifold M and p a frame at x, so that is a linear isomorphism of Rn with the tangent space of M at x ... The solder form of FM is the Rn-valued 1-form θ defined by where ξ is a tangent vector to FM at the point (x,p), p-1TxM → Rn is the inverse of the frame map, and dπ is the differential of the ...

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