Associated Vector Bundle

Given a principal bundle pi:A->M, with fiber a Lie group G and base manifold M, and a group representation of G, say phi:G×V->V, then the associated vector bundle is


In particular, it is the quotient space A×V/G where (a,v)∼(ga,g^(-1)v).

This construction has many uses. For instance, any group representation of the orthogonal group gives rise to a bundle of tensors on a Riemannian manifold as the vector bundle associated to the frame bundle.

For example, pi:SO(3)->S^2 is the frame bundle on S^2, where

 pi([w_1; w_2; w_3])=w_1,

writing the special orthogonal matrix with rows w_i. It is a SO(2) bundle with the action defined by

 [costheta -sintheta; sintheta costheta]·A=[1 0 0; 0 costheta -sintheta; 0 sintheta costheta]A,

which preserves the map pi.

The tangent bundle is the associated vector bundle with the standard group representation of SO(2) on V=R^2, given by pairs (v,A), with v=(a,b) in R^2 and A in SO(3). Two pairs (v_1,A_1) and (v_2,A_2) represent the same tangent vector iff there is a g in SO(2) such that v_2=gv_1 and A_1=g·A_2.

See also

Associated Fiber Bundle, Frame Bundle, Group Action, Group Representation, Lie Group, Principal Bundle, Quotient Space

This entry contributed by Todd Rowland

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Cite this as:

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

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