Institute for Mathematical Physics on a Global Conformal Invariant of Initial Data Sets 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 in a Lorentzian spacetime naturally inherits from the spacetime metric a diierentiation D e , the so-called real Sen connection, which turns out to be determined completely by the initial data h ab and ab induced on , and coincides, in the case of vanishing second fundamental form ab , with the Levi-Civita covariant derivation D e of the induced metric h ab. Y is built from the real Sen connection D e in the similar way as the standard Chern{Simons invariant is built from D e. The number Y is invariant with respect to changes of h ab and ab 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.