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.
Read more about this topic: Frame Bundle
Other articles related to "tangent frame bundle, frame bundle, bundle, frame, tangent":
... 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 ...
Famous quotes containing the words bundle and/or frame:
“We styled ourselves the Knights of the Umbrella and the Bundle; for, wherever we went ... the umbrella and the bundle went with us; for we wished to be ready to digress at any moment. We made it our home nowhere in particular, but everywhere where our umbrella and bundle were.”
—Henry David Thoreau (1817–1862)
“A cold and searching wind drives away all contagion, and nothing can withstand it but what has a virtue in it, and accordingly, whatever we meet with in cold and bleak places, as the tops of mountains, we respect for a sort of sturdy innocence, a Puritan toughness. All things beside seem to be called in for shelter, and what stays out must be part of the original frame of the universe, and of such valor as God himself.”
—Henry David Thoreau (1817–1862)