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A de Rham decomposition type theorem for contact sub-Riemannian manifolds by Marek Grochowski is a Mathematics article available to read on EtoBox.

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## Abstract In this paper we prove a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that (M, H, g) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field $$\xi $$ ξ is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point $$q\in M$$ q ∈ M such that the holonomy group $$\Psi (q)$$ Ψ ( q ) acts reducibly on H(q) yielding a decomposition $$H(q) = H\_1(q)\oplus \cdots \oplus H\_m(q)$$ H ( q ) = H 1 ( q ) ⊕ ⋯ ⊕ H m ( q ) into $$\Psi (q)$$ Ψ ( q ) -irreducible factors. Using parallel transport we obtain the decomposition $$H = H\_1\oplus \cdots \oplus H\_m$$ H = H 1 ⊕ ⋯ ⊕ H m of H into sub-distributions $$H\_i$$ H i . Unlike the Riemannian case, the distributions $$H\_i$$ H i are not integrable, however they induce integrable distributions $$\Delta \_i$$ Δ i on $$M/\xi $$ M / ξ , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that $$T(U/\xi )=\Delta \_1\oplus \cdots \oplus \Delta \_m$$ T ( U / ξ ) = Δ 1 ⊕ ⋯ ⊕ Δ m , and the latter decomposition of $$T(U/\xi )$

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Author

Marek Grochowski

Publisher

Springer Science and Business Media LLC

Published

2021

Language

EN

Field

Mathematics (Physical Sciences)