Geometry of \(N=1\) supergravity. II. (English) Zbl 0644.53078

Rosly, A. A.; Schwarz, A. S.

Commun. Math. Phys. 96, 285-309 (1984).

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.

Reviewer: K. K. Lee (M.R. 86d:83059)

MSC:

53C80	Applications of global differential geometry to the sciences
83E50	Supergravity

Keywords:

torsion; curvature; G-structure; integrability conditions; constraints on the internal geometry

Citations:

Zbl 0644.53077; Zbl 0644.53078

References:

[1]	Rosly, A. A., Schwarz, A. S.: Geometry ofN=1 supergravity. Commun. Math. Phys. · Zbl 0644.53077
[2]	Schwarz, A. S.: Supergravity, complex geometry andG-structures. Commun. Math. Phys.87, 37-63 (1982) · Zbl 0503.53048 · doi:10.1007/BF01211055
[3]	Rosly, A. A., Schwarz, A. S.: Geometry of non-minimal and alternative minimal supergravity. Yad. Fiz.37, 786-794 (1983)
[4]	Sternberg, S.: Lectures on differential geometry. Englewood Cliffs, N.Y.: Prentice Hall 1964 · Zbl 0129.13102
[5]	Kobayashi, S., Nomizu, K.: Foundations of differential geometry. New York, London, Sydney: Interscience Publishers 1969, Vol. 2 · Zbl 0175.48504
[6]	Goldschmidt, H.: Integrability criteria for systems of nonlinear partial differential equations. J. Diff. Geom.1, 269-307 (1967) · Zbl 0159.14101
[7]	Pommaret, J. F.: Systems of partial differential equations and Lie pseudogroups. New York: Gordon and Breach 1978 · Zbl 0418.35028
[8]	Kodaira, K., Spencer, D. C.: Multifoliate structures. Ann. Math.74, 52-100 (1961) · Zbl 0123.16401 · doi:10.2307/1970306