Examples
Examples for vector bundles, particularly the tangent bundle of a manifold:
- is an orientation, and this is possible if and only if the bundle is orientable
- is a volume form; since is a deformation retract, a volume form exists if and only if a bundle is orientable
- is a pseudo-volume form, and this is always possible
- is a Riemannian metric; as is the maximal compact subgroup (so the inclusion is a deformation retract), this is always possible
- is a pseudo-Riemannian metric; there is the topological obstruction to this reduction
- is an almost complex structure
- (where is the group of n×n invertible quaternionic matrices acting on on the left and Sp(1)=Spin(3) the group of unit quaternions acting on from the right) is an almost quaternionic structure
- (which is not an inclusion: it's a 2-fold covering space) is a spin structure.
- decomposes a vector bundle as a Whitney sum (direct sum) of sub-bundles of rank k and n − k.
Read more about this topic: Reduction Of The Structure Group
Famous quotes containing the word examples:
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
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—G.C. (Georg Christoph)