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By means of the geometric algebra the general decomposition of SU(2) gaugepotential on the sphere bundle of a compact and oriented 4-dimensional manifoldis given. Using this decomposition theory the SU(2) Chern density has beenstudied in detail. It shows that the SU(2) Chern density can be expressed interms of the $\delta -$function $\delta (\phi) $. And one can find that thezero points of the vector fields $\phi$ are essential to the topologicalproperties of a manifold. It is shown that there exists the crucial case ofbranch process at the zero points. Based on the implicit function theorem andthe taylor expansion, the bifurcation of the Chern density is detailed in theneighborhoods of the bifurcation points of $\phi$. It is pointed out that,since the Chren density is a topological invariant, the sum topologicalchargers of the branches will remain constant during the bifurcation process.Comment: revtex, 21pages, no figur

The general decomposition theory of SU(2) gauge potential, topological structure and bifurcation of SU(2) Chern density