Tanaka structures (non holonomic G-structures) and Cartan connections
| Authors | |
|---|---|
| Year of publication | 2015 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Geometry and Physics |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.1016/j.geomphys.2015.01.018 |
| Field | General mathematics |
| Keywords | Tanaka structures; (normal) Cartan connections; Parabolic geometry; (prolongation of) G-structures |
| Description | Let h = h(-k) circle plus ... circle plus h(1) (k > 0, l >= 0) be a finite dimensional graded Lie algebra, with a Euclidean metric <., .> adapted to the gradation. The metric <., .> is called admissible if the codifferentials partial derivative*: Ck+1 (h(-), j) -> C-k(h(-), h) (k >= 0) are Q-invariant (Lie(Q) = h(0) circle plus h(+)). We find necessary and sufficient conditions for a Euclidean 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 from Cap and Slovak (2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in detail the case when h := t*(g) is the cotangent Lie algebra of a non-positively graded Lie algebra g. (C) 2015 Elsevier B.V. All rights reserved. |
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