Connections and General Structures
Connections on a Manifold with Vector Bundles
In the previous paragraphs we say that a connection on a manifold is a “connection on its tangent bundle", or the difference of two connections is an “-form with values in the bundle ". These are the concepts in vector bundles, a special type of fibre(fiber) bundles.
Given a connection on , then connection on (also denoted by ) is given by for all and .
In local coordinates , . If an -form locally and , then . Similarly in tensor calculus, Generally, if , then
Skech of the proof.. Obviously both and are anti-symmetric.
We need to check that , which is clear; and  ◻
Proof. From previous discussion we’ve shown for all and . Thus  ◻
Sketch of the proof. The uniqueness follows from the Lemma and Remark 3. above. If is any connection on , then is given by is a torsion free connection with the same geodesics. ◻
Torsion, Curvature and Bianchi Identity
Let be a connection on a manifold . We have the torsion and connection curvature tensor
Now we have the identities
Proof of the thoerem.
We have Note that , thus we have (reordering by cyclic permutations)
We have Note that and every canceling is due to Jacobian identity with cyclic permutations.
 ◻
Proof. By the Bianchi identities, we have Then the result holds by taking the trace of both sides. ◻