Ehresmann Connection - Special Cases - Associated Bundles

Associated Bundles

An Ehresmann connection on a fibre bundle (endowed with a structure group) sometimes gives rise to an Ehresmann connection on an associated bundle. For instance, a (linear) connection in a vector bundle E, thought of giving a parallelism of E as above, induces a connection on the associated bundle of frames PE of E. Conversely, a connection in PE gives rise to a (linear) connection in E provided that the connection in PE is equivariant with respect to the action of the general linear group on the frames (and thus a principal connection). It is not always possible for an Ehresmann connection to induce, in a natural way, a connection on an associated bundle. For example, a non-equivariant Ehresmann connection on a bundle of frames of a vector bundle may not induce a connection on the vector bundle.

Suppose that E is an associated bundle of P, so that E = P ×G F. A G-connection on E is an Ehresmann connection such that the parallel transport map τ : FxFx′ is given by a G-transformation of the fibres (over sufficiently nearby points x and x′ in M joined by a curve).

Given a principal connection on P, one obtains a G-connection on the associated fibre bundle E = P ×G F via pullback.

Conversely, given a G-connection on E it is possible to recover the principal connection on the associated principal bundle P. To recover this principal connection, one introduces the notion of a frame on the typical fibre F. Since G is a finite-dimensional Lie group acting effectively on F, there must exist a finite configuration of points (y1,...,ym) within F such that the G-orbit R = {(gy1,...,gym) | gG} is a principal homogeneous space of G. One can think of R as giving a generalization of the notion of a frame for the G-action on F. Note that, since R is a principal homogeneous space for G, the fibre bundle E(R) associated to E with typical fibre R is (equivalent to) the principal bundle associated to E. But it is also a subbundle of the m-fold product bundle of E with itself. The distribution of horizontal spaces on E induces a distribution of spaces on this product bundle. Since the parallel transport maps associated to the connection are G-maps, they preserve the subspace E(R), and so the G-connection descends to a principal G-connection on E(R).

In summary, there is a one-to-one correspondence (up to equivalence) between the descents of principal connections to associated fibre bundles, and G-connections on associated fibre bundles. For this reason, in the category of fibre bundles with a structure group G, the principal connection contains all relevant information for G-connections on the associated bundles. Hence, unless there is an overriding reason to consider connections on associated bundles (as there is, for instance, in the case of Cartan connections) one usually works directly with the principal connection.

Read more about this topic:  Ehresmann Connection, Special Cases

Famous quotes containing the word bundles:

    He bundles every forkful in its place,
    And tags and numbers it for future reference,
    So he can find and easily dislodge it
    In the unloading. Silas does that well.
    He takes it out in bunches like birds’ nests.
    Robert Frost (1874–1963)