De Rham decomposition of affinely connected manifolds (Q915146)

\textit{S. Kashiwabara} [Tôhoku Math. J., II. Ser. 8, 13--28 (1956; Zbl 0074.37904)] generalized the de Rham product decomposition theorem to the case of affinely connected manifolds without torsion. In the present paper the author extends the theorem of S. Kashiwabara to affine connections which may have torsion. The main result is as follows. Let \(M\) be a connected manifold with affine connection \(\nabla\) and \(M\) be \(\nabla\)-complete. Let \(E\), \(F\) be two complementary, integrable and \(\nabla\)-parallel subbundles of \(TM\). Suppose that the conditions \[ (C_1)\quad T(u,v)=0\text{ for all } p\in M,\quad u\in E_p,\ v\in F_p \] and \[ (C_2)\quad R(u,v)=0\text{ for all } p\in M,\quad u\in E_p,\ v\in F_p \] are satisfied. Then all the leaves of \(E\) and \(F\) are simply connected submanifolds of \(M\) and for every point \(p_0\in M\) there exists an affine diffeomorphism \[ \phi: E(p_0)\times F(p_0)\to M, \] such that for all \(a\in E(p_0)\) and \(b\in F(p_0)\), \(\phi(E(p_0)\times \{b\})\) and \(\phi(\{a\}\times F(p_0))\) are leaves of \(E\) and \(F\), respectively.