Naturally reductive Riemannian \(S\)-manifolds
Let \((M,g)\) be a naturally reductive Riemannian space and \(S\) a tensor field of type (1,1) such that \(g\) and \(\nabla S\) are \(S\)-invariant and \(I-S\) is non-singular. \(S\) naturally defines an \(S\)-invariant almost complex structure \(J\) on the orthogonal complement of the distribution \({\mathcal D}_ 0 = \{X \in TM\mid SX = -X\}\). Now, the main theorem says that if the tensor \(\nabla R\) vanishes on the eigenspace distribution \({\mathcal D_ 0}\) and if the function \((\nabla_ XR)(X,JX,X,JX)\) vanishes on each other eigenspace distribution of \(S\), then \((M,g,S)\) gives rise to a locally regular \(s\)-structure, i.e., \(S\) is the tangent field of a system \(\{s_ X\}_{x\in M}\) of local isometries satisfying \(s_ x(x) = x\) and \(s_ x \circ s_ y = s_{s_ x(y)} \circ s_ x\) for every \(x,y \in M\). A trivial case occurs when \(S = -\text{Id}\) and \(\nabla R = 0\), then the result means that \((M,g)\) is locally symmetric. For the case \(S^ 3 = \text{Id}\) one obtains an earlier result by A. Gray on nearly- Kähler 3-symmetric spaces. Despite of the technical character, the main theorem seems to have a deep geometrical meaning.
scientific article; zbMATH DE number 119919