The analogs of the Riemann tensor for exceptional structures on supermanifolds

Abstract

H. Hertz called any manifold M with a given nonintegrable distribution {\it nonholonomic}. Vershik and Gershkovich stated and R. Montgomery proved that the space of germs of any nonholonomic distribution on M with an open and dense orbit of the diffeomorphism group is either (1) of codimension one or (2) an Engel distribution. No analog of this statement for supermanifolds is formulated yet, we only have some examples: our list (an analog of E.Cartan's classification) of simple Lie superalgebras of vector fields with polynomial coefficients and a particular (Weisfeiler) grading contains 16 series similar to contact ones and 11 exceptional algebras preserving nonholonomic structures. Here we compute the cohomology corresponding to the analog of the Riemann tensor for the SUPERmanifolds corresponding to the 15 exceptional simple vectorial Lie superalgebras, 11 of which are nonholonomic. The cohomology for analogs of the Riemann tensor for the manifolds with an exceptional Engel manifolds are computed in math.RT/0202213.