Generalized curvature and the equations of D=11 supergravity

It is known that, for zero fermionic sector, psi(alpha)(mu) (x) = 0, the bosonic equations of Cremmer-Julia-Scherk eleven-dimensional supergravity can be collected in a compact expression, R-alpha b alpha(gamma) Gamma(b)(beta)(gamma) = 0, which is a condition on the curvature R-alpha(beta) of the generalized connection w. In this Letter we show that the equation R-bc alpha(gamma) Gamma(abc) (gamma beta) = 4i ((D psi)(bc) Gamma([abc))(beta)(psi d Gamma(d]))(alpha) where D is the covariant derivative for the generalized connection w, collects all the bosonic equations of D = 11 supergravity when the gravitino is nonvanishing, psi(alpha)(mu) (x) not equal 0. (c) 2005 Elsevier B.V. All rights reserved.