Over a small neighborhood U of a manifold, a vector bundle is spanned by the local sections defined on U. For example, in a coordinate chart U with coordinates (x_1,...,x_n), every smooth vector field can be written as a sum sum_(i)f_ipartial/partialx_i where f_i are smooth functions. The n vector fields partial/partialx_i span the space of vector fields, considered as a module over the ring of smooth real-valued functions. On this coordinate chart U, the tangent bundle can be written U×R^n. This is a trivialization of the tangent bundle.


In general, a vector bundle of bundle rank r is spanned locally by r independent bundle sections. Every point has a neighborhood U and r sections defined on U, such that over every point in U the fibers are spanned by those r sections.

Similarly, for a fiber bundle, near every point p in M, there is a neighborhood U such that the bundle over U is U×F, where F is the fiber.

A bundle is a set of trivializations that cover the base manifold. The trivializations are put together to form a bundle with its transition functions.

See also

Bundle, Fiber Bundle, Manifold, Transition Function, Vector Bundle

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Trivialization." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications