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Subriemannian metrics and the metrizability of parabolic geometries. (English) Zbl 1486.53021

The authors present the linearized metrizability problem in the context of parabolic geometries and sub-Riemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville [J. de l’Éc. Polyt. cah. LIX. 7–76 (1889; JFM 21.0317.02)].
The paper under review is concerned with bracket-generating distributions arising in parabolic geometries, which are Cartan-Tanaka geometries modelled on homogeneous spaces \(G/P\), where \(G\) is a semisimple Lie group and \(P\) a parabolic subgroup of \(G\). On a manifold \(M\) equipped with such a parabolic geometry, each tangent space is modelled on the \(P\)-module \(g/p\).
First, motivating examples are provided:
1. Parabolic geometries and Weyl structures.
2. Projective parabolic geometries.
3. Parabolic geometries on filtered manifolds.
4. \(BGG\) operators, local metrizability of the homogeneous model, and normal solutions.
Next, a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies are given. These tools lead to natural sub-Riemannian metrics on generic distributions of interest in geometric control theory.
By using an algebraic linearization condition, the authors establish a linearization principle. The metrizability procedure is illustrated by showing how the well-known example of projective geometry fits into the general method. In addition, the metric tractor bundle is investigated.
Let \(p_0=p/{p^\perp}\) and \(h\) a \(p_0\)-module. The main result is a classification of metric parabolic geometries with irreducible \(h\).
Finally, examples of reducible cases where the linearized metrizability procedure works are given.

MSC:

53B15 Other connections
53C17 Sub-Riemannian geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E46 Semisimple Lie groups and their representations
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C30 Differential geometry of homogeneous manifolds
58A32 Natural bundles
58J70 Invariance and symmetry properties for PDEs on manifolds
93C10 Nonlinear systems in control theory

Citations:

JFM 21.0317.02
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References:

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