Cohomology of Groups of Diffeomorphisms Related to the Modules of Differential Operators on a Smooth Manifold

Abstract

Let M be a manifold and T * M be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of M with values in the space of linear differential operators acting on C ∞ (T * M ). When M is the n-dimensional sphere, S n , we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of S n , with coefficients in the space of linear differential operators acting on contravariant tensor fields.

Projective and Conformal Schwarzian Derivatives and Cohomology of Lie Algebras Vector Fields Related to Differential Operators

“…(6.22) We state the following Theorem that generalizes the result of [3] for δ = 0. Theorem 6.11 For n = 2, 3, the first-cohomology group…”

“…The main result is to give a relation between the projective Schwarzian derivative (4.8) and the well-known Vey cocycle, answering a question raised in [3].…”

“…In the formula above, the subscript i runs from 1 to 2n, and ω stands for the standard symplectic structure on T * M, andX is the Hamiltonian lift of X. The following cocycle were introduced in [3], and interpreted as a group Vey cocycle: 26) wheref is the symplectic lift of f to T * M and L(f ) k ij are the components of the tensor (3.5) with respect to the lifted connection on T * M.…”