Bundle Map

A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose p:X->M and q:Y->N are two bundles, then


is a bundle map if there is a map f:M->N such that q(F(x))=f(p(x)) for all x in X. In particular, the fiber bundle of X over a point m in M, gets mapped to the fiber of Y over f(m) in N.


In the language of category theory, the above diagram commutes. To be more precise, the induced map between fibers has to be a map in the category of the fiber. For instance, in a bundle map between vector bundles the fiber over m in M is mapped to the fiber over f(m) in M by a linear transformation.

For example, when f:M->N is a smooth map between smooth manifolds then df:TM->TN is the differential, which is a bundle map between the tangent bundles. Over any point in m in M, the tangent vectors at m get mapped to tangent vectors at f(m) in N by the Jacobian.

See also

Bundle, Commutative Diagram, Fiber Bundle, Jacobian, Principal Bundle, Vector Bundle

This entry contributed by Todd Rowland

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

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