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
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.
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.
These examples all have two features in common, namely:
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|
|Date of creation||2013-03-22 13:26:06|
|Last modified on||2013-03-22 13:26:06|
|Last modified by||antonio (1116)|