reduction of structure group
Given a fiber bundle with typical fiber and structure group (henceforth called an -bundle over ), we say that the bundle admits a reduction of its structure group to , where is a subgroup, if it is isomorphic to an -bundle over
Equivalently, admits a reduction of structure group to if there is a choice of local trivializations covering such that the transition functions all belong to
Remark 1
Here, the action of on is the restriction of the -action; in particular, this means that an -bundle is automatically an -bundle. The bundle isomorphism in the definition then becomes meaningful in the category of -bundles over .
Example 1
Let be the trivial subgroup. Then, the existence of a reduction of structure group to is equivalent to the bundle being trivial.
For the following examples, let be an -dimensional vector bundle, so that with the general linear group acting as usual.
Example 2
Set the subgroup of consisting of matrices with positive determinant. A reduction to is equivalent to an orientation of the vector bundle. In the case where is a smooth manifold and is its tangent bundle, this coincides with other definitions of an orientation of .
Example 3
Set , the orthogonal group. A reduction to is called a Riemannian or Euclidean structure on the vector bundle. It coincides with a continuous fiberwise choice of a positive definite inner product, and for the case of the tangent bundle, with the usual notion of a Riemannian metric on a manifold.
When is paracompact, an argument with partitions of unity shows that a Riemannian structure always exists on any given vector bundle. For this reason, it is often convenient to start out assuming the structure group to be
Example 4
Let be even, and let the group of invertible complex matrices, embedded in by means of the usual identification of with A reduction to is called a complex structure on the vector bundle, and it is equivalent to a continuous fiberwise choice of an endomorphism satisfying
A complex structure on a tangent bundle is called an almost-complex structure on the manifold. This is to distinguish it from the more restrictive notion of a complex structure on a manifold, which requires the existence of an atlas with charts in such that the transition functions are holomorphic.
Example 5
Let embedded in by A reduction to is equivalent to the existence of a splitting where is a line bundle. More generally, a reduction to is equivalent to a splitting where is a -plane bundle.
Remark 2
These examples all have two features in common, namely:
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a reduction to is equivalent to a continuous fiber-by-fiber choice of a structure of the same kind.
For example, is the subgroup of which preserves the standard inner product of and reduction of structure to is equivalent to a fiberwise choice of inner products.
This is not a coincidence. The intuition behind this is as follows. There is no obstacle to choosing a fiberwise inner product in a neighborhood of any given point : we simply choose a neighborhood on which the bundle is trivial, and with respect to a trivialization , we can let the inner product on each be the standard inner product. However, if we make these choices locally around every point in , there is no guarantee that they “glue together” properly to yield a global continuous choice, unless the transition functions preserve the standard inner product. But this is precisely what reduction of structure to means.
The same explanation holds for subgroups preserving other kinds of structure.
Title | reduction of structure group |
Canonical name | ReductionOfStructureGroup |
Date of creation | 2013-03-22 13:26:06 |
Last modified on | 2013-03-22 13:26:06 |
Owner | antonio (1116) |
Last modified by | antonio (1116) |
Numerical id | 12 |
Author | antonio (1116) |
Entry type | Definition |
Classification | msc 55R10 |
Related topic | VectorBundle |
Related topic | FiberBundle |
Defines | Euclidean structure |
Defines | Riemannian structure |
Defines | complex structure |
Defines | almost-complex structure |