On a global conformal invariant of initial data sets

In the present paper a global conformal invariant Y of a closed initial data set is constructed. A spacelike hypersurface Σ in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation De, the so-called real Sen connection, which turns out to be determined completely by the initial data hab and χab induced on Σ, and coincides, in the case of vanishing second fundamental form χab, with the Levi-Civita covariant derivation De of the induced metric hab. Y is built from the real Sen connection De in the similar way as the standard Chern–Simons invariant is built from De. The number Y is invariant with respect to changes of hab and χab corresponding to conformal rescalings of the spacetime metric. In contrast the quantity Y built from the complex Ashtekar connection is not invariant in this sense. The critical points of our Y are precisely the initial data sets which are locally imbeddable into conformal Minkowski space.