Prolongations, invariants, and fundamental identities of geometric structures

Abstract

Working in the framework of nilpotent geometry, we give a unified scheme for the equivalence problem of geometric structures which extends and integrates the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto.By giving a new formulation of the higher order geometric structures and the universal frame bundles, we reconstruct the step prolongation of Singer-Sternberg and Tanaka. We then investigate the structure function γ of the complete step prolongation of a normal geometric structure by expanding it into components γ = κ + τ + σ and establish the fundamental identities for κ, τ , σ. This then enables us to study the equivalence problem of geometric structures in full generality and to extend applications largely to the geometric structures which have not necessarily Cartan connections.Among all we give an algorithm to construct a complete system of invariants for any higher order normal geometric structure of constant symbol by making use of generalized Spencer cohomology group associated to the symbol of the geometric structure. We then discuss thoroughly the equivalence problem for geometric structure in both cases of infinite and finite type.We also give a characterization of the Cartan connections by means of the structure function τ and make clear where the Cartan connections are placed in the perspective of the step prolongations. 1 5.3. Transitive models of geometric structures 37 6. Equivalence problems 39 6.1. Formal equivalence 39 6.2. Involutive geometric structures 41 6.3. Analytic geometric structures of finite type 43 7. Cartan connections from the view-point of step prolongation 44 7.1. L'espace généralisé 44 7.2. Principal geometric structures and principal prolongations 45 7.3. The structure function τ and Cartan connections 48 7.4. Condition (C) 53 7.5. Inductive vanishing of the structure function κ 54 8. Applications to subriemannian geometry and complex geometry 56 8.1. Subriemannian geometry 56 8.2. Complex geometry 58 Appendix 59 References 62

Characterizations of smooth projective horospherical varieties of Picard number one

“…Prolongation methods. We review the theory of prolongation methods in [4]. Let m = p<0 g p be a fundamental graded Lie algebra and G 0 be a connected subgroup of G 0 (m) with Lie algebra g 0 .…”

“…For the definition of the universal frame bundle S (i+1) P (i) of P (i) of order i + 1, see Definition 2.1 of [4]. The property we use is that a map between two geometric structures P (i) , Q (i) of order i induces a map between their universal frame bundles S (i+1) P (i) , S (i+1) Q (i) of order i + 1.…”

“…We write a sequence P (ℓ) −→ P (ℓ−1) −→ • • • −→ P (0) −→ M of principal bundles simply as P (ℓ) when no confusion can arise, and define the equivalence of two geometric structures P (ℓ) and Q (ℓ) inductively as follows. Two geometric structures P (ℓ) and Q (ℓ) are equivalent if their truncations P (ℓ−1) and Q (ℓ−1) are equivalent and the lifting S (ℓ) P (ℓ−1) → S (ℓ) Q (ℓ−1) of the equivalence [4]). Let P be a G 0 -structure on a filtered manifold (M, D) of type m. Then for each ℓ ≥ 1, there is a geometric structure…”

“…Given a G 0 -structure P on a filtered manifold (M, D) of type m, for k ≥ 1, define a vector bundle H 2 k on M by H 2 k := P × G 0 H 2 (m, g) k . Proposition 3.8 (Theorem 7.4 of [4]). Let a P be a G 0 -structure on a filtered manifold (M, D) of type m. If H 0 (M, H 2 k ) is zero for all k ≥ 1, then the W -normal complete step prolongation S W P of P is a Cartan connection of type G/G 0 which is flat, and P is locally isomorphic to the standard G 0 -structure on G/G 0 .…”

“…For the geometric structures associated to them, the second author suggested a way to construct a Cartan connection which solves the local equivalence problem by showing that the condition (C) in [17] is satisfied (Proposition 53 of [14]). Due to a gap in the proof of Theorem 17 of [14], instead, we use the prolongation methods developed in [4].…”