Projective and conformal Schwarzian derivatives and cohomology of Lie algebras vector fields related to differential operators

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Abstract

Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class of the metric [g], respectively. Furthermore, we compute the first cohomology group of Vect (M) with coefficients in the space of symmetric contravariant tensor fields valued in the space of δ-densities, and we compute the corresponding sl (n + 1, ℝ)-relative cohomology group.

Original languageEnglish (US)
Pages (from-to)667-696
Number of pages30
JournalInternational Journal of Geometric Methods in Modern Physics
Volume3
Issue number4
DOIs
StatePublished - Jun 1 2006

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differential operators
homology
algebra
tensors
operators
coefficients

Keywords

  • Gelfand-Fuchs cohomology
  • Invariant operators
  • Projective/conformal structures
  • The Schwarzian derivative

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)

Cite this

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abstract = "Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class of the metric [g], respectively. Furthermore, we compute the first cohomology group of Vect (M) with coefficients in the space of symmetric contravariant tensor fields valued in the space of δ-densities, and we compute the corresponding sl (n + 1, ℝ)-relative cohomology group.",
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AB - Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class of the metric [g], respectively. Furthermore, we compute the first cohomology group of Vect (M) with coefficients in the space of symmetric contravariant tensor fields valued in the space of δ-densities, and we compute the corresponding sl (n + 1, ℝ)-relative cohomology group.

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