**Beiträge zur Algebra und Geometrie <BR> Contributions to Algebra and GeometryVol. 40, No. 2, pp. 459-474 (1999)
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Flow-symmetric Riemannian Manifolds

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Eric Boeckx, Peter Bueken, Lieven Vanhecke

Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium, e-mail: Eric.Boeckx@wis.kuleuven.ac.be, Peter.Bueken@wis.kuleuven.ac.be, Lieven.Vanhecke@wis.kuleuven.ac.be

**Abstract:** We introduce the notion of (locally) flow-symmetric Riemannian manifolds. These are Riemannian manifolds admitting a non-vanishing vector field $\xi$ such that all local reflections with respect to integral curves of this vector field are (local) isometries. Locally $\varphi$-symmetric Sasakian manifolds, strongly $\varphi$-symmetric contact metric manifolds and (locally) Killing-transversally symmetric spaces are examples of such spaces. We give necessary and sufficient conditions for an analytic Riemannian manifold to be locally flow-symmetric, and construct a number of additional examples. Finally, we present a complete local classification of two-dimensional locally flow-symmetric spaces.

**Keywords:** locally flow-symmetric spaces, KTS-spaces, $\varphi$-symmetric Sasakian spaces, strongly $\varphi$-symmetric contact metric manifolds

**Classification (MSC91):** 53C25

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