reduction of structure group (original) (raw)

Given a fiber bundle p:E→B with typical fiber F and structure group G (henceforth called an (F,G)-bundle over B), we say that the bundle admits a reduction of its structure group to H, where H<G is a subgroup, if it is isomorphic to an (F,H)-bundle over B.

For the following examples, let E be an n-dimensional vector bundle, so thatF≅ℝn with G=G⁢L⁢(n,ℝ), the general linear group acting as usual.

Example 2

Set H=G⁢L+⁢(n,ℝ), the subgroup of G⁢L⁢(n,ℝ) consisting of matrices with positive determinant. A reduction to H is equivalent to an orientation of the vector bundle. In the case where B is a smooth manifold and E=T⁢B is its tangent bundle, this coincides with other definitions of an orientation of B.

Example 3

When B 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 O⁢(n).

Example 4

Let n=2⁢m be even, and let H=G⁢L⁢(m,ℂ), the group of invertible complex matrices, embedded in G⁢L⁢(n,ℝ) by means of the usual identification of ℂ with ℝ2.A reduction to H is called a complex structure on the vector bundle, and it is equivalent to a continuous fiberwise choice of an endomorphism J satisfying J2=-I.

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 ℂm such that the transition functions are holomorphic.

Example 5

Let H=G⁢L⁢(1,ℝ)×G⁢L⁢(n-1,ℝ), embedded in G⁢L⁢(n,ℝ) by(A,B)↦A⊕B. A reduction to H is equivalent to the existence of a splitting E≅E1⊕E2, where E1 is a line bundle. More generally, a reduction to G⁢L⁢(k,ℝ)×G⁢L⁢(n-k,ℝ) is equivalent to a splitting E≅E1⊕E2, where E1 is a k-plane bundle.

Remark 2

These examples all have two features in common, namely:

• •
  the subgroup H can be interpreted as being precisely the subgroup of G which preserves a particular structure, and,
• •
  a reduction to H is equivalent to a continuous fiber-by-fiber choice of a structure of the same kind.

For example, O⁢(n) is the subgroup of G⁢L⁢(n,ℝ) which preserves the standard inner product of ℝn, and reduction of structure to O⁢(n) 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 x∈B: we simply choose a neighborhood U on which the bundle is trivial, and with respect to a trivialization p-1⁢(U)≅ℝn×U, we can let the inner product on each p-1⁢(y) be the standard inner product. However, if we make these choices locally around every point in B, 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 O⁢(n)means.

The same explanation holds for subgroups preserving other kinds of structure.