A global variational calculus on Fr\'echet manifolds and the Lagrangian structure of the Einstein evolution equations

Abstract

We develop a variational calculus for curves on Fr\'echet manifolds and apply it to the Lagrangian formulation of the Einstein evolution equations on the manifold $\Mm$ of Riemannian metrics of a compact manifold. For mechanical Lagrangians associated with weak pseudo-Riemannian metrics, we establish the Euler--Lagrange equations, energy conservation, and an infinitesimal Noether theorem that does not require the symmetry vector field to generate a flow. Applied to the DeWitt Lagrangian, this framework identifies the momentum constraint with the vanishing of a Noether momentum map and gives a direct proof of its propagation. We also derive the pointwise transport identity \[ \partial_t\bigl(\Ham_\lambda\,\dv_g\bigr) = 2\alpha\,\delta_g(\delta_g^-k)\,\dv_g\,, \] which implies propagation of the Hamiltonian constraint. Finally, we prove a Maupertuis-- Jacobi theorem for weak pseudo-Riemannian metrics on Fr\'echet manifolds and show that, away from the zero set of the total scalar-curvature potential, constrained Einstein evolutions are reparametrized unit-speed geodesics of the conformal DeWitt metric $-4\alpha S_\lambda G^-$.