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# Trivialization

Over a small neighborhood of a manifold, a vector bundle is spanned by the local sections defined on . For example, in a coordinate chart with coordinates , every smooth vector field can be written as a sum where are smooth functions. The vector fields span the space of vector fields, considered as a module over the ring of smooth real-valued functions. On this coordinate chart , the tangent bundle can be written . This is a trivialization of the tangent bundle.

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

Similarly, for a fiber bundle, near every point , there is a neighborhood such that the bundle over is , where 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.

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

This entry contributed by Todd Rowland

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Rowland, Todd. "Trivialization." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Trivialization.html