A de Rham decomposition type theorem for contact sub-Riemannian manifolds

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 $$T(U/\xi )$$ T ( U / ξ ) induces the decomposition of $$U/\xi $$ U / ξ into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. We also give a version of the theorem for indefinite metrics.