Riemannian submersion with isometric reflections with respect to the fibers (Q1321776)

A Riemannian submersion \(\pi:M \to N\) generates a family of local reflections w.r.t. the fibers. These are local geodesic symmetries for horizontal geodesics. A Sasakian structure \((\varphi,V,\eta)\) on a Riemannian manifold \(M\) is \(\varphi\)-symmetric when \(\varphi^ 2 ((\nabla_ XR) (Y,Z)H)=0\) for all \(X,Y,Z\) and all \(H\) orthogonal to \(V\). The author proves that, under some additional assumptions on the integrability tensor \(A\) of \(\pi\), the reflections w.r.t. the fibers are isometries iff \(M\) admits a Sasakian \(\varphi\)-symmetric structure.