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Mathematics
Differential Geometry
Question of Cartan's structure equation (in Principal bundle)
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[QUOTE="lavinia, post: 6831681, member: 243745"] For a general connection on a principal bundle the bracket of horizontal vector fields is not necessarily vertical. However, if the connection is not flat(if the curvature 2 form is not zero) there is always some pair of horizontal vector fields whose bracket product has a non-zero vertical component. I know of a case where the bracket product of horizontal vector fields is always vertical. On 2 dimensional orientable surfaces with a Riemannian metric, the bracket product of horizontal vector fields is vertical for a unique connection on the tangent unit circle bundle (which is a principal SO(2) bundle). This is called the Riemannian connection in Singer and Thorpe's Lecture Notes on Elementary Topology and Geometry. I would guess that this generalizes to a unique connection on the bundle of tangent oriented orthonormal frames on an n dimensional oriented manifold with a Riemannian metric. I would also guess that these connections correspond to the Levi-Civita connection on a Riemannian manifold and the connections where bracket products of horizontal vector fields have horizontal components correspond to connections with non-zero torsion. [/QUOTE]
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Differential Geometry
Question of Cartan's structure equation (in Principal bundle)
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