# Domenico Giulini

# Idea and structures of geometrodynamics

** Abstract: **

The field equations of General Relativity (GR) can be cast into the form of (six underdetermined hyperbolic) evolution
equations with (four underdetermined elliptic) constraints.
The degrees of freedom for the gravitational field are
then fully encoded in terms of a three-dimensional
Riemannian geometry. The evolution equations for this
3-geometry are such that its time dependence may be
interpreted as coming about through an appropriate
motion (one-parameter family of embeddings) of a spacelike
hypersurface through a Lorentzian spacetime. Converseley,
the latter kinematical requirement may be used to motivate
Einstein's equations of GR. All this will be diuscussed
as well as certain interesting structural elements
concerning the geometry and topology of the space of
Riemannian 3-geometries which relates to the general
theory of 3-manifolds.