We give intrinsic formula for the multi-dimensional Schwarzian derivative on a manifold M endowed with a projective connection. This Schwarzian derivative is naturally related to the space of second-order linear differential operators acting on tensor densities on M. 1

Let Σ be an open Riemann surface and Hol(Σ) be the Lie algebra of holomorphic vector fields on Σ. We fix a projective structure (i.e. a local SL2(C)−structure) on Σ. We calculate the first group of cohomology of Hol(Σ) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra sl2(C). The result is independent on the choice of the projective structure. We give explicit formulæ of 1-cocycles generating this cohomology group. 1

The main topic of this paper is two folds. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(RP 1), with coefficients in the space of bilinear differential operators that act on tensor densities, Dλ,ν;µ, vanishing on the Lie algebra sl(2, R). Second, we compute the first cohomology group of the Lie algebra sl(2, R) with coefficients in Dλ,ν;µ. 1

Let (M, g) be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of M related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric g. In particular, if the manifold (M, g) is conformally flat, these derivatives vanish on the conformal group O(p + 1, q + 1), where dim(M) = p + q. This work is a continuation of [2, 4] where the Schwarzian derivative was defined on a manifold endowed with a projective connection. 1

We consider the universal central extension of the Lie algebra Vect(S 1 )nC 1 (S 1 ). The coadjoint representation of this Lie algebra has a natural geometric interpretation by matrix analogues of the Sturm-Liouville operators. This approach leads to new Lie superalgebras generalizing the well-known Neveu-Schwartz algebra. 1 Introduction 1.1 Sturm-Liouville operators and the action of Vect(S 1 ). Let us recall some well-known definitions (cf. e.g. [9],[8]). Consider the Sturm-Liouville operator: L = \Gamma2c d 2 dx 2 + u(x) (1) where c 2 R and u is a periodic potential: u(x + 2ß) = u(x) 2 C 1 (R). Let Vect(S 1 ) be the Lie algebra of smooth vector field on S 1 : f = f(x) d dx ; where f(x + 2ß) = f(x), with the commutator [f(x) d dx ; g(x) d dx ] = (f(x)g 0 (x) \Gamma f 0 (x)g(x)) d dx : We define a Vect(S 1 )-action on the space of Sturm-Liouville operators. Consider a 1-parameter family of Vect(S 1 )-actions on the space of smooth functions C 1 (S 1...

Abstract. Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form α is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2). These two operators lead to a decomposition of the space of symbols (except for some critical density weights), which generalizes a splitting proposed by V. Ovsienko in [18]. 1.

In this paper we find an explicit formula for the most general vector evolution of curves on RP n\Gamma1 invariant under the projective action of SL(n; R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that the formula we found gives another alternative definition of the second KdV Hamiltonian evolution under appropriate conditions; that is, both evolutions are identical. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n; R), namely, the SL(n; R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n.

We formulate the uniformisation problem underlying the geometry of W n - gravity using the differential equation approach to W -algebras. We construct W n -space (analogous to superspace in supersymmetry) as a (n \Gamma 1) dimensional complex manifold using isomonodromic deformations of linear differential equations. The W n -manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n; R) which acts properly discontinuously on a simply connected domain in C P n\Gamma1 . The requirement that a deformation be isomonodromic furnishes relations which enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W n -manifold. We discuss how the Teichmuller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W n - manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisation of the Schwarzian. This construc...

The module of differential operators over the Lie algebra of smooth vector fields is a deformation of the corresponding module of symbols in the sense of Richardson-Neijenhuis theory of deformation, as recently pointed out by Duval-Lecomte-Ovsienko. In this paper, we compute the second (differentiable) cohomology group H 2 (Vect(RP 1), Homdiff(Fλ, Fµ)), where Fλ is the space of λ-densities, arising in this context. This cohomology group measures the obstruction to extend any infinitesimal deformation to a formal one. 1

This paper classifies locally projectively homogeneous surfaces in P