Communications in Mathematical Physics

, Volume 78, Issue 4, pp 455–478

Symmetry and bifurcations of momentum mappings

Authors

  • Judith M. Arms
    • Department of MathematicsUniversity of Washington
  • Jerrold E. Marsden
    • Department of MathematicsUniversity of California
  • Vincent Moncrief
    • Department of PhysicsYale University
Article

DOI: 10.1007/BF02046759

Cite this article as:
Arms, J.M., Marsden, J.E. & Moncrief, V. Commun.Math. Phys. (1981) 78: 455. doi:10.1007/BF02046759

Abstract

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.

Glossary of Symbols

(P,ω)

symplectic manifold

\(T_{x_0 } P\)

tangent space toP atx0εP

G,g

Lie group, Lie algebra

g·xg(x)

action ofG onP

ξP

infinitesimal generator of the action onP corresponding toξεg

J:P→g*

momentum mapping

\(dJ\left( {x_0 } \right):T_{x_0 } P \to \mathfrak{g}^ * \)

differential ofJ atx0

\(\mathbb{J}\)

complex structure onP

\(S_{x_0 } \)

slice for theG-action atx0

\(I_{x_0 } \)

isotropy group ofx0; {g εG|gx0=x0}

\(\mathcal{L}_{x_0 } ,\mathfrak{s}_{x_0 } \)

identity component of\(I_{x_0 } \), its Lie algebra [Eq. (10)]

«,»

(weak) metric onP

〈,〉

pairing between g* and g

\((,)_{x_0 } \)

inner product of g* depending onx0εP

\(dJ\left( {x_0 } \right)^ * :\mathfrak{g} \to T_{x_0 } P\)

adjoint ofdJ(x0) relative to 〈,〉 and «,»

\(dJ\left( {x_0 } \right)^\dag :\mathfrak{g}^ * \to T_{x_0 } P\)

adjoint ofdJ(x0) relative to (,) and «,»

\(T_{x_0 } P = Range\left[ {\mathbb{J} \circ dJ\left( {x_0 } \right)^ * } \right] \oplus Range\left[ {dJ\left( {x_0 } \right)^ * } \right] \oplus \left[ {\ker \left( {dJ\left( {x_0 } \right) \circ \mathbb{J}} \right) \cap \ker dJ\left( {x_0 } \right)} \right]\)

Moncrief's decomposition

ℙ:g*→RangeddJ(x0)

orthogonal projection

\(\mathcal{C} = J^{ - 1} \left( 0 \right)\)

zero set ofJ (or constraint set)

\(\mathcal{C}_\mathbb{P} = \left( {\mathbb{P}J} \right)^{ - 1} \left( 0 \right)\)

zero set of ℙ∘J

\(N_{x_0 } \)

x's with the same orbit type asx0

\(\mathcal{N}_{x_0 } \)

\(N_{x_0 } \cap S_{x_0 } \) (Lemma 1)

\(\mathfrak{g}_{x_0 }^ * \)

elements in g* with same symmetry asx0 (Lemma 5)

f

\(f = \left( {Id - \mathbb{P}} \right) \circ J:\mathcal{C}_\mathbb{P} \cap S_{x_0 } \to \ker dJ\left( {x_0 } \right)^\dag \) (Lemma 8)

Δ=dJ(x0)∘dJ(x0)

“elliptic” operator associated withJ

G−1∘ℙ

Greens' function for Δ

F(x)=x+dJ(x0)GQ(h),h=xx0

Kuranishi map (Lemma 9)

\(C_{x_0 } \)

homogeneous cone associated withd2J(x0) (Theorem 3)

https://static-content.springer.com/image/art%3A10.1007%2FBF02046759/MediaObjects/220_2005_BF02046759_f1.jpg

orthogonal projection onto kerdJ(x0) (Theorem 3)

\(\mathfrak{h}\), ℋ

a Lie subalgebra of\(\mathfrak{s}_{x_0 } \), its Lie group

https://static-content.springer.com/image/art%3A10.1007%2FBF02046759/MediaObjects/220_2005_BF02046759_f2.jpg

points with symmetry (at least) ℋ (Theorem 4)

https://static-content.springer.com/image/art%3A10.1007%2FBF02046759/MediaObjects/220_2005_BF02046759_f3.jpg

https://static-content.springer.com/image/art%3A10.1007%2FBF02046759/MediaObjects/220_2005_BF02046759_f4.jpg

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© Springer-Verlag 1981