Local and Global Sections
Fiber bundles do not in general have such global sections, so it is also useful to define sections only locally. A local section of a fiber bundle is a continuous map s : U → E where U is an open set in B and π(s(x)) = x for all x in U. If (U, φ) is a local trivialization of E, where φ is a homeomorphism from π−1(U) to U × F (where F is the fiber), then local sections always exist over U in bijective correspondence with continuous maps from U to F. The (local) sections form a sheaf over B called the sheaf of sections of E.
The space of continuous sections of a fiber bundle E over U is sometimes denoted C(U,E), while the space of global sections of E is often denoted Γ(E) or Γ(B,E).
Read more about this topic: Section (fiber Bundle)
Famous quotes containing the words local, global and/or sections:
“Reporters for tabloid newspapers beat a path to the park entrance each summer when the national convention of nudists is held, but the cult’s requirement that visitors disrobe is an obstacle to complete coverage of nudist news. Local residents interested in the nudist movement but as yet unwilling to affiliate make observations from rowboats in Great Egg Harbor River.”
—For the State of New Jersey, U.S. public relief program (1935-1943)
“Ours is a brand—new world of allatonceness. “Time” has ceased, “space” has vanished. We now live in a global village ... a simultaneous happening.”
—Marshall McLuhan (1911–1980)
“That we can come here today and in the presence of thousands and tens of thousands of the survivors of the gallant army of Northern Virginia and their descendants, establish such an enduring monument by their hospitable welcome and acclaim, is conclusive proof of the uniting of the sections, and a universal confession that all that was done was well done, that the battle had to be fought, that the sections had to be tried, but that in the end, the result has inured to the common benefit of all.”
—William Howard Taft (1857–1930)