The geometry of totally geodesic foliations admitting Killing field
This paper is concerned with results of preservation of codimension-one totally geodesic foliations admitting at least one compact leaf. By using De Rham's Decomposition Theorem for foliate manifolds and studying the expression of the Killing vector fields on the universal covering of the manifold, we give, first of all, a result of preservation for geodesic flows on surfaces. With similar techniques another proof of a recent preservation result due to \textit{G.-I. Oshikiri} [ibid. 38, 351-356 (1986; Zbl 0585.53025)] for closed manifolds is given, and finally we prove that on a 3-dimensional complete Riemannian manifold, with a codimension-one totally geodesic foliation, admitting at least one compact leaf, any Killing vector field preserves this foliation.
scientific article; zbMATH DE number 4073879