In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x) = V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X × V over X. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.
Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.
Hi, this is my first post. Congratulations to a very interesting forum.
The issue, I want to bring up, has probably been discussed already, but hopefully at least not in exactly the same way. I have read the book "Road to Reality" from Roger Penrose, which by the way I found quite interesting...
Hey, I'm a little confused on the definition of a principle bundle. The basic question:
"Do elements of the structure group, G, have to act on elements of the fiber, G, from the right?"
I've read a bunch of papers that seem to imply that the fiber bundle structure group elements could act...
I need help solving the following problem:
Let M,N be differentiable manifolds, and f\in C^\infty(M,N). We say that the fields X\in \mathfrak{X}(M) and Y \in \mathfrak{X}(N) are f-related if and only if f_{*p}(X(p))=Y_{f(p)} for all p\in M.
Prove that:
(a) X and Y are f-related if and only if...
I try to understand the notion of connections on fibre bundles from the lecture notes http://arxiv.org/abs/math-ph/9902027" by George Svetlichny.
On page 27 stands what the attached picture shows.
(I don't know whether the attached picture will be visible, so I copy the text here:
)...
Hello,
I've 2 qustions :
1.Calculus the tangent bundel TM and the cotangent bundles T*M like a bundles associates to the principal bundle B(M) of the reper of TM
2.If M has a riemannian structure set up the principale bundels B'(M) of orthonormal system of TM in case of M=S^2 , we can...
i was wondering what happens where an infinite number of lightbeams meet in one point/knot in open space or anywhere..
how come they do'nt mess up one another? is this on account of their being massless?
..anything we see, any lightbeam that reaches our eye is crossed infinite times by beams...