Information AboutVector Bundle |
| CATEGORIES ABOUT VECTOR BUNDLE | |
| differential topology | |
| algebraic topology | |
| complex analysis | |
| vector bundles | |
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This article deals mostly with ''real'' vector bundles, with finite-dimensional fibers. '''''Complex''''' vector bundles are important in many cases, too; they are a special case, meaning that they can be seen as extra structure on an underlying real bundle. DEFINITION AND FIRST CONSEQUENCES A real vector bundle is given by the following data:
satisfying the following compatibility condition: for every point in ''X'' there is an open neighborhood ''U'', a Natural Number ''n'', and a Homeomorphism φ : ''U'' × R''n'' → π−1(''U'') such that for every point ''x'' in ''U'':
The open neighborhood ''U'' together with the homeomorphism φ is called a local trivialization of the bundle. The local trivialization shows that "locally" the map π looks like the projection of ''U'' × '''R'''''n'' on ''U''. A vector bundle is called trivial if there is a "global trivialization", i.e. if it looks like the projection ''X'' × '''R'''''n'' → ''X''. Every vector bundle π : ''E'' → ''X'' is Surjective , since vector spaces cannot be Empty . Every fiber π−1({''x''}) is a finite-dimensional real vector space and hence has a Dimension ''d''''x''. The function ''x'' ''d''''x'' is Locally Constant , i.e. it is constant on all Connected Components of ''X''. If it is constant globally on ''X'', we call this dimension the rank of the vector bundle. Vector bundles of rank 1 are called Line Bundle s. VECTOR BUNDLE MORPHISMS A Morphism from the vector bundle π1 : ''E''1 → ''X''1 to the vector bundle π2 : ''E''2 → ''X''2 is given by a pair of continuous maps ''f'' : ''E''1 → ''E''2 and ''g'' : ''X''1 → ''X''2 such that
The class of all vector bundles together with bundle morphisms forms a Category . Restricting to smooth manifolds and smooth bundle morphisms we obtain the category of smooth vector bundles. We can also consider the category of all vector bundles over a fixed base space ''X''. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the Identity Map on ''X''. That is, bundle morphisms for which the following diagram Commutes : (Note that this category is ''not'' Abelian ; the Kernel of a morphism of vector bundles is in general not a vector bundle in any natural way.) SECTIONS AND LOCALLY FREE SHEAVES Given a vector bundle π : ''E'' → ''X'' and an open subset ''U'' of ''X'', we can consider sections of π on ''U'', i.e. continuous functions ''s'' : ''U'' → ''E'' with π''s'' = id''U''. Essentially, a section assigns to every point of ''U'' a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but Vector Field s on that manifold. Let ''F''(''U'') be the set of all sections on ''U''. ''F''(''U'') always contains at least one element, namely the zero section: the function ''s'' that maps every element ''x'' of ''U'' to the zero element of the vector space π−1({''x''}). With the pointwise addition and scalar multiplication of sections, ''F''(''U'') becomes itself a real vector space. The collection of these vector spaces is a Sheaf of vector spaces on ''X''. If ''s'' is an element of ''F''(''U'') and α : ''U'' → R is a continuous map, then α''s'' is in ''F''(''U''). We see that ''F''(''U'') is a Module over the ring of continuous real-valued functions on ''U''. Furthermore, if O''X'' denotes the structure sheaf of continuous real-valued functions on ''X'', then ''F'' becomes a sheaf of O''X''-modules. Not every sheaf of O''X''-modules arises in this fashion from a vector bundle: only the Locally Free ones do. (The reason: locally we are looking for sections of a projection ''U'' × R''n'' → ''U''; these are precisely the continuous functions ''U'' → R''n'', and such a function is an ''n''-tuple of continuous functions ''U'' → R.) Even more: the category of real vector bundles on ''X'' is Equivalent to the category of locally free and finitely generated sheaves of O''X''-modules. So we can think of the vector bundles as sitting inside the category of sheaves of O''X''-modules; this latter category is abelian, so this is where we can compute kernels of morphisms of vector bundles. OPERATIONS ON VECTOR BUNDLES Two vector bundles on ''X'', over the same field, have a Whitney sum, with fibre at any point the Direct Sum of fibres. In a similar way, ''fibrewise'' Tensor Product and Dual Space bundles may be introduced. VARIANTS AND GENERALIZATIONS Vector bundles are special Fiber Bundle s, loosely speaking those where the fibers are vector spaces. Smooth vector bundles are defined by requiring that ''E'' and ''X'' be s. Replacing real vector spaces with Complex ones, we obtain complex vector bundles. This is a special case of Reduction Of The Structure Group Of A Bundle . Vector spaces over other Topological Field s may also be used, but that is comparatively rare. If we allow arbitrary Banach Space s in the local trivialization (rather than only R''n''), we obtain '''Banach bundles'''. REFERENCES
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