Cartan connection associated with a subriemannian structure.

*(English)*Zbl 1147.53027Let \(M\) be a differentiable manifold. A pair \((D,S)\) consisting of a subbundle \(D\) of the tangent bundle \(TM\) and a metric \(S\) on \(D\) is called a sub-Riemannian structure on \(M\). To investigate the equivalence problem for the sub-Riemannian manifolds \((M,D,S)\), the author treats sub-Riemannian manifolds with constant sub-Riemannian symbol and constructs canonically a Cartan connection in this case. First, the author introduces for every sub-Riemannian manifold \((M,D,S)\) satisfying some regularity conditions, a sub-Riemannian symbol, which assigns to every point \(x\in M\) a pair \((\text{gr\,}D_x,S_x)\) of a nilpotent graded Lie algebra \(\text{gr\,}D_x= \bigoplus_{p\leq -1} \text{gr}_p D_x\) and an inner product \(S_x\) on \(\text{gr}_{-1} D_x\).

Next, the author calls a pair \(({\mathfrak g}_-,\sigma)\) a sub-Riemannian graded Lie algebra if \({\mathfrak g}_-= \bigoplus_{p< 0}{\mathfrak g}_p\) is a nilpotent graded Lie algebra generated by \({\mathfrak g}_{-1}\) and if \(\sigma\) is an inner product on \({\mathfrak g}_{-1}\). Isomorphisms between sub-Riemannian graded algebra are naturally defined. Now, a sub-Riemannian manifold \((M,D,S)\) is said to have a constant sub-Riemannian symbol of type \(({\mathfrak g}_-,\sigma)\) if \((\text{gr\,}D_x, S_x)\) is isomorphic to the sub-Riemannian graded algebra \(({\mathfrak g}_-,\sigma)\) for all \(x\in M\).

The author proves the following theorem: To each sub-Riemannian manifold \((M,D,S)\) having a constant sub-Riemannian symbol of type \(({\mathfrak g},\sigma)\) there is canonically associated a Cartan connection \((P,M,\theta)\) of type \(({\mathfrak g}, G_0)\), where \(G_0\) is the automorphism group of \(({\mathfrak g}_-,\sigma)\) with Lie algebra \(g_0\) and where \({\mathfrak g}={\mathfrak g}_-\oplus{\mathfrak g}_0\), is a graded Lie algebra on which \(G_0\) acts naturally. The author recalls the precise definition of a Cartan connection and proves the theorem relying on the results of his previus paper [Hokkaido Math. J. 22, No. 3, 263–347 (1993; Zbl 0801.53019)].

Next, the author calls a pair \(({\mathfrak g}_-,\sigma)\) a sub-Riemannian graded Lie algebra if \({\mathfrak g}_-= \bigoplus_{p< 0}{\mathfrak g}_p\) is a nilpotent graded Lie algebra generated by \({\mathfrak g}_{-1}\) and if \(\sigma\) is an inner product on \({\mathfrak g}_{-1}\). Isomorphisms between sub-Riemannian graded algebra are naturally defined. Now, a sub-Riemannian manifold \((M,D,S)\) is said to have a constant sub-Riemannian symbol of type \(({\mathfrak g}_-,\sigma)\) if \((\text{gr\,}D_x, S_x)\) is isomorphic to the sub-Riemannian graded algebra \(({\mathfrak g}_-,\sigma)\) for all \(x\in M\).

The author proves the following theorem: To each sub-Riemannian manifold \((M,D,S)\) having a constant sub-Riemannian symbol of type \(({\mathfrak g},\sigma)\) there is canonically associated a Cartan connection \((P,M,\theta)\) of type \(({\mathfrak g}, G_0)\), where \(G_0\) is the automorphism group of \(({\mathfrak g}_-,\sigma)\) with Lie algebra \(g_0\) and where \({\mathfrak g}={\mathfrak g}_-\oplus{\mathfrak g}_0\), is a graded Lie algebra on which \(G_0\) acts naturally. The author recalls the precise definition of a Cartan connection and proves the theorem relying on the results of his previus paper [Hokkaido Math. J. 22, No. 3, 263–347 (1993; Zbl 0801.53019)].

Reviewer: Akihiko Morimoto (Nagoya)

##### MSC:

53C17 | Sub-Riemannian geometry |

##### Keywords:

Cartan connections; sub-Riemannian structures; sub-Riemannian symbol; sub-Riemannian graded Lie algebras
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\textit{T. Morimoto}, Differ. Geom. Appl. 26, No. 1, 75--78 (2008; Zbl 1147.53027)

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##### References:

[1] | Kobayashi, S., Transformation groups in differential geometry, (1972), Springer-Verlag · Zbl 0246.53031 |

[2] | Morimoto, T., Geometric structures on filtered manifolds, Hokkaido math. J., 22, 263-347, (1993) · Zbl 0801.53019 |

[3] | Morimoto, T., Lie algebras, geometric structures and differential equations on filtered manifolds, Adv. stud. pure math., 37, 205-252, (2002) · Zbl 1048.58015 |

[4] | Tanaka, N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido math. J., 8, 23-84, (1979) · Zbl 0409.17013 |

[5] | Yatsui, T., On pseudo-product graded Lie algebras, Hokkaido math. J., 17, 333-343, (1988) · Zbl 0658.17018 |

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