Flow-symmetric Riemannian manifolds (Q1567464)

The authors introduce the notion of locally flow-symmetric Riemannian manifolds as Riemannian manifolds admitting a non-vanishing vector field \(\xi\) such that the local reflections with respect to the integral curves of this vector field are local isometries. They give the necessary conditions (on the Riemann curvature tensor and its covariant derivatives) for a Riemannian manifold to be locally flow-symmetric and show that these conditions are also sufficient in the analytic case. The authors find that the manifold can only be locally flow-symmetric with respect to the eigenvector fields of the Ricci operator. They observe that all locally \(\phi\)-symmetric Sasakian manifolds, strongly \(\phi\)-symmetric contact metric manifolds and locally KTS-spaces are examples of locally flow-symmetric manifolds, and construct a number of new examples. The authors also prove that the warped product manifold \(\mathbb{R}\times_f M\) is locally flow-symmetric if and only if \(M\) is a locally symmetric manifold. Finally, using this fact, they present a complete local classification of two-dimensional locally flow-symmetric spaces.