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Cartan connection associated with a subriemannian structure. (English) Zbl 1147.53027
Let $$M$$ 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)].

##### MSC:
 53C17 Sub-Riemannian geometry
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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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