Vector bundle - Wikipedia
Vector bundles are special fiber bundles, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example sphere bundles are fibered by spheres.
Vector Bundle -- from Wolfram MathWorld
5 天之前 · A vector bundle is special class of fiber bundle in which the fiber is a vector space V. Technically, a little more is required; namely, if f:E->B is a bundle with fiber R^n, to be a vector bundle, all of the fibers f^ (-1) (x) for x in B need to have a coherent vector space structure.
A vector bundle on a space B (generally called the base space) is a space E (generally called the total space) together with a map p:E B and the structure of a vector space on each fiber p −1 (x) for x ∈ B, satisfying the extra condition:
vector spaces give rise to tensor products of vector bundles, which in turn give prod- uct operations in both real and complex K–theory similar to cup product in ordinary cohomology.
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very \non-linear". However, there exist many smooth manifolds which admit very nice \partial...
Vector bundle - Encyclopedia of Mathematics
2023年12月23日 · The concept of a vector bundle arose as an extension of the tangent bundle and the normal bundle in differential geometry; by now it has become a basic tool for studies in various branches of mathematics: differential and algebraic topology, the theory of linear connections, algebraic geometry, the theory of (pseudo-) differential operators, etc.
These vector spaces “vary smoothly” over the manifold. The notion of a vector bundle is a topological extrapolation of this idea. Let B be a topological space. To begin with, let’s define the “category of spaces over B,” Top=B. An object is just a map E ! B.
VECTOR BUNDLES. 1. The definition of a vector bundle The de nitions below can be applied to any geometric category that contains vector spaces (topological spaces, smooth varieties, algebraic varieties, etc). We need to be able to say what is a variety and a morphism of varieties, we need topology on our varieties, and we need direct product.
3 Introduction to vector bundles - Oxford Academic
To a first approximation, a vector bundle (over a given manifold) is a family of vector spaces that is parametrized by the points in the manifold. Some sort of vector bundle appears in most every aspect of differential geometry. Chapter 3 introduces these objects and gives the …
We may denote a vector bundle by ˇ: E! M(and suppress the vector space) or as E. If the dimension of the vector space is mthen the bundle is often called an m-plane bundle. A 1-plane bundle is also called a line bundle. A bundle over a manifold is trivial if it is simply the Cartesian product of the manifold and a vector space. The ...