Tanaka structures (non holonomic G-structures) and Cartan connections

Original · EN

Let = ₋ₖ⊕ ⊕ ₗ (k >0, l ≥ 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric ·, · adapted to the gradation. The metric ·, · is called admissible if the codifferentials ∂*: Cᵏ⁺¹(₋,) Cᵏ (₋,) (k≥ 0) are AdQ-invariant (Lie(Q) = ₀⊕ ₊). We find necessary and sufficient conditions for a Euclidian metric, adapted to the gradation, to be admissible, and we develop a theory of normal Cartan connections, when these conditions are satisfied. We show how the treatment by A. Cap and J. Slovak (Parabolic Geometry I, Mathematical Surveys and Monographs, vol. 154, 2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in some detail the case when = t* (≫) is the cotangent Lie algebra of a non-positively graded Lie algebra ≫.