
Journal of Lie Theory 18 (2008), No. 4, 775809 Copyright Heldermann Verlag 2008 Algebraic Characterization of Differential Geometric Structures Serge Skryabin Chebotarev Research Institute, Universitetskaya St. 17, 420008 Kazan, Russia Serge.Skryabin@ksu.ru We consider purely algebraic data generalizing the notion of a smooth differentiable manifold. It is given by a triple X, R, W where X is a set, R a commutative associative algebra over the ground field, W a Lie subalgebra and an Rsubmodule in the derivation algebra of R. Geometric structures studied in differential geometry can be defined on such triples. The main result answers the question about the existence and the uniqueness of an Linvariant unimodular, hamiltonian, contact, or pseudoriemannian structure in terms of the isotropy subalgebras of points of X. The second major result generalizes a classical fact which says that the Lie algebra of infinitesimal automorphisms of a Riemann metric on a connected manifold is finite dimensional. Keywords: Unimodular structures, hamiltonian structures, contact structures, Riemann pseudometrics. MSC: 13N99 [ Fulltextpdf (336 KB)] for subscribers only. 