Bundle Rank

The rank of a vector bundle is the dimension of its fiber. Equivalently, it is the maximum number of linearly independent local bundle sections in a trivialization. Naturally, the dimension here is measured in the appropriate category. For instance, a real line bundle has fibers isomorphic with R, and a complex line bundle has fibers isomorphic to C, but in both cases their rank is 1.

The rank of the tangent bundle of a real manifold M is equal to the dimension of M. The rank of a trivial bundle M×R^k is equal to k. There is no upper bound to the rank of a vector bundle over a fixed manifold M.

See also

Bundle Section, Dimension, Fiber, Manifold, Tangent Bundle, Vector Bundle

This entry contributed by Todd Rowland

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

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