Communications in Mathematical Physics

, Volume 78, Issue 4, pp 455-478

First online:

Symmetry and bifurcations of momentum mappings

  • Judith M. ArmsAffiliated withDepartment of Mathematics, University of Washington
  • , Jerrold E. MarsdenAffiliated withDepartment of Mathematics, University of California
  • , Vincent MoncriefAffiliated withDepartment of Physics, Yale University

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The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.