Reducibility of homogeneous Riemannian structures of the class \({\mathcal T}_ 3\) in low dimension
In Duke Math. J. 25, 647-669 (1958; Zbl 0134.178) \textit{W. Ambrose} and \textit{I. M. Singer} proved that a connected complete and simply connected Riemannian manifold (M,g) is homogeneous if and only if there exists a tensor field T of type (1,2) such that for X, Y, Z being vector fields on M the following three equations hold: \[ 1.\quad g(T_ XY,Z)+g(Y,T_ XZ)=0,\quad 2.\quad (\nabla_ XR)_{YZ}=[T_ X,R_{YZ}]-R_{T_ XYZ}-R_{YT_ XZ},\quad 3.\quad (\nabla_ XT)_ Y=[T_ X,T_ Y]- T_{T_ XY}. \] Here \(\nabla\) denotes the Levi-Cività connection and R the Riemann curvature tensor. The homogeneous structure T belongs to the class \({\mathcal T}_ 3\) of the classification given by Tricerri and Vanhecke (equivalently, (M,g) is naturally reductive) if it also satisfies \(T_ XY+T_ YX=0.\) Let \({\tilde \nabla}=\nabla -T\) be the Ambrose-Singer connection and \(\tilde R\) its curvature tensor. When R verifies all the symmetries of a Riemannian curvature tensor, including the first Bianchi identity, \(\tilde R\) is an algebraic tensor field. The author shows that if (M,g) is a connected, complete and simply connected Riemannian manifold of dimension \(\geq 4\) equipped with a homogeneous structure \(T\neq 0\), belonging to the class \({\mathcal T}_ 3\) then (i) \(\tilde R=0\) implies M is a compact symmetric space with \(b_ 3\geq 1\) (Theorem 1), (ii) \(\tilde R\) is algebraic implies that M is a Riemannian product (Theorem 2). Furthermore, in dimensions 4, 5 and 6, if \(\tilde R\) is algebraic and not zero then M is isometric to the direct product of two naturally reductive spaces, and at least one factor admits a non zero structure in the class \({\mathcal T}_ 3\) (Theorem 3).
scientific article; zbMATH DE number 4210986