A de Rham decomposition type theorem for contact sub-Riemannian manifolds
The author proves 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 \(q\in M \) 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 the authors obtain the decomposition \(H = H_1 \oplus\cdots\oplus H_m\) 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 \(T (U/\xi) = \delta_1\oplus\cdots\oplus\delta_m\), 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. The authors also give a version of the theorem for indefinite metrics.
scientific article; zbMATH DE number 7438140
scientific article