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 is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on H. Suppose that there exists a point qin M such that the holonomy group Psi (q) acts reducibly on H(q) yielding a decomposition H(q) = H_1(q)oplus cdots oplus H_m(q) into Psi (q)-irreducible factors. Using parallel transport we obtain the decomposition H = H_1oplus cdots oplus H_m of H into sub-distributions H_i. Unlike the Riemannian case, the distributions H_i are not integrable, however they induce integrable distributions Delta _i on M/xi , which is locally a smooth manifold. As a result, every point in M has a neighborhood U such that T(U/xi )=Delta _1oplus cdots oplus Delta _m, and the latter decomposition of T(U/xi ) induces the decomposition of U/xi 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.
Highlights
Introduction and statement of resultsLet M be a smooth connected manifold
In this paper we deal with holonomy determined by a class of connections introduced in [13] for contact sub-Riemannian manifolds, and prove a theorem that can be considered as a version of de Rham decomposition theorem for Riemannian manifolds
Theorem 1.1 Suppose that (M, H, g) is a contact oriented sub-Riemannian manifold such that the Reeb vector field ξ is an infinitesimal isometry
Summary
Given a contact connected sub-Riemannian manifold (M, H , g) it is natural to consider the bundle of orthonormal horizontal frames OH,g(M) associated with it: OH,g(M) = {(q; v1, . Theorem 1.1 Suppose that (M, H , g) is a contact oriented sub-Riemannian manifold such that the Reeb vector field ξ is an infinitesimal isometry. Denote by the unique torsion-free connection on OH,g(M) and suppose that there exists a point q ∈ M such that the holonomy group (q) of acts reducibly on H (q) inducing the decomposition (1.1).
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