Mikhail A. Malakhaltsev
The Lie derivative and cohomology of $G$-structures
(Lobachevskii Journal of Mathematics,
Vol.3, pp.197-200)
J.F.~Pommaret constructed the so-called Spencer $P$-complex
for a differential operator. Applying this construction to the Lie derivative
associated with a general pseudogroup structure on a smooth manifold, he
defined the deformation cohomology of a pseudogroup structure. The aim
of this paper is to specify this complex for a particular case of pseudogroup
structure, namely, for a first-order $G$-structure, and to express this
complex in differential geometric form, i.\,e., in terms of tensor fields
and the covariant derivative. We show that the Pommaret construction provides
a powerful tool for associating a differential complex to a $G$-structure.
In a unified way one can obtain the Dolbeaut complex for the complex structure,
the Vaisman complex for the foliation structure, and the Vaisman--Molino
cohomology for the structure of manifold over an algebra.
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