Geometry of \(N=1\) supergravity. II (Q1102570)

In a previous paper, the authors studied the geometry of various superspace formulations of \(N=1\) supergravity [Part I, ibid. 95, 161--184 (1984; Zbl 0644.53078)] (see the preceding review). In that paper, the supergravity torsion and curvature constraints were shown to be a particular case of constraints arising from a more general geometric situation. In this paper, the authors prove a theorem which gives the necessary and sufficient conditions for the given G-structure on a manifold to be realized on some surface as the one induced by the trivial G-structure in \(\mathbb R^n.\) In general, this problem is tantamount to the question of whether a certain system of nonlinear partial differential equations has a solution. The theorem describes the formal integrability conditions for that system in a convenient form of constraints on the internal geometry. There is a hierarchy of integrability conditions of increasing order and the number of nontrivial ones, which is always finite, is controlled by the Spencer cohomologies related to the problems. This theorem generalizes various well-known theorems (e.g., the Gauss-Codazzi theorem) and may be of interest in its own right.