The Manifold of Conformally Equivalent Metrics

Fischer, A. E. and J. E. Marsden

Can. J. Math., XXIX, No. 1, 1977, 193-209

Abstract:

Ebin gives a thorough study of the space $\mathcal {M}$ of riemannian metrics on a compact manifold M and of the action of the diffeomorphism group $\mathcal {D}$ of M on $\mathcal {M}$. The purpose of this paper is to study the action of the larger group $\mathcal {C}$ of conformomorphisms, or conformal transformations, on $\mathcal {M}$ and on T^*$\mathcal {M}$. On $\mathcal {M}$, the L₂-orthogonal decomposition induced by the action of $\mathcal {C}$ gives a splitting of symmetric tensors into three summands introduced by York. We find submanifolds of $\mathcal {M}$ tangent to the pieces of this decomposition.

The action of $\mathcal {C}$ on T^*$\mathcal {M}$ is symplectic and may be reduced following Marsden-Weinstein. This process parametrizes the space $\mathcal {E}$ of true gravitational degrees of freedom. The space is shown to be an (infinite dimensional, weak) symplectic manifold near those points (g,$\pi$) with no simultaneous conformal Killing fields. It is argued that near other points, $\mathcal {E}$ has singularities.