Structure Groups and Transition Functions
Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group which acts continuously on the fiber space F on the left. We lose nothing if we require G to act effectively on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle (E, B, π, F) is a local trivialization such that for any two overlapping charts (Ui, φi) and (Uj, φj) the function
is given by
where tij : Ui ∩ Uj → G is a continuous map called a transition function. Two G-atlases are equivalent if their union is also a G-atlas. A G-bundle is a fiber bundle with an equivalence class of G-atlases. The group G is called the structure group of the bundle; the analogous term in physics is gauge group.
In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps.
The transition functions tij satisfy the following conditions
The third condition applies on triple overlaps Ui ∩ Uj ∩ Uk and is called the cocycle condition (see Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).
A principal G-bundle is a G-bundle where the fiber F is a principal homogeneous space for the left action of G itself (equivalently, one can specify that the action of G on the fiber F is free and transitive). In this case, it is often a matter of convenience to identify F with G and so obtain a (right) action of G on the principal bundle.
Read more about this topic: Fiber Bundle
Famous quotes containing the words structure, groups, transition and/or functions:
“... the structure of our public morality crashed to earth. Above its grave a tombstone read, “Be tolerant—even of evil.” Logically the next step would be to say to our commonwealth’s criminals, “I disagree that it’s all right to rob and murder, but naturally I respect your opinion.” Tolerance is only complacence when it makes no distinction between right and wrong.”
—Sarah Patton Boyle, U.S. civil rights activist and author. The Desegregated Heart, part 2, ch. 2 (1962)
“Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.”
—Germaine Greer (b. 1939)
“Power ceases in the instant of repose; it resides in the moment of transition from a past to a new state, in the shooting of the gulf, in the darting to an aim.”
—Ralph Waldo Emerson (1803–1882)
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)