On a global conformal invariant of initial data sets

Abstract

A global conformal invariant Y of a closed initial data set is constructed. A spacelike hypersurface in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation , the so-called real Sen connection, which turns out to be determined completely by the initial data and induced on , and coincides, in the case of a vanishing second fundamental form , with the Levi-Civita covariant derivation of the induced metric is built from the real Sen connection in a similar way to how the standard Chern - Simons invariant is built from . The number Y is invariant with respect to changes of and 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.