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 \(q\in M\) 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 we 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. We also give a version of the theorem for indefinite metrics.

在本文中，我们证明了一个结果，该结果可以看作是 de Rham 分解定理的亚黎曼版本。更准确地说，假设 ( M ,  H ,  g ) 是一个接触和定向的亚黎曼流形，使得 Reeb 向量场\(\xi \)是一个无穷小的等距。在这样的假设下，H上存在唯一的度量和无扭转连接。假设存在一个点\(q\in M\)使得完整群\(\Psi (q)\)可约化地作用于H ( q ) 产生分解\(H(q) = H_1(q)\ oplus \cdots \oplus H_m(q)\)变成\(\Psi (q)\)- 不可减少的因素。使用并行传输我们得到分解\（H = H_1 \ oplus \ cdots \ oplus H_m \）的ħ成子分布\（H_i \） 。与黎曼情况不同，分布\(H_i\)是不可积的，但是它们在\(M/\xi \)上引入了可积分布\(\Delta _i \)，这是局部光滑的流形。其结果是，在每一个点中号具有附近Ü使得\（T（U / \ⅹⅰ）= \德尔塔_1 \ oplus \ cdots \ oplus \德尔塔_m \） ，和后者分解\（T（U / \xi )\)引起\(U/\xi \)的分解转化为黎曼流形的乘积。可以将这一点重述如下：每个完整群作用可约的接触子黎曼流形，至少在局部，具有黎曼流形乘积上的纤维丛结构。我们还给出了不定度量的定理版本。