On manifolds admitting continuous foliations by geodesics (Q1818585)

The author presents some relations between a certain class of manifolds continuously foliated by geodesics and manifolds without conjugate points. \textit{H. Gluck} [Lect. Notes Math. 819, 190-215 (1980; Zbl 0448.58016)] and \textit{D. Sullivan} [J. Pure Appl. Algebra 13, 101-104 (1978; Zbl 0402.57015)] started studies of geodesible flows on a differentiable manifold. In particular, a differentiable flow \(\phi_t\) is geodesible if and only if there exists a distribution transversal to \(\phi_t\) which is preserved by some reparametrization of \(\phi_t\). The author proves the following theorem. Theorem 1. Let \((M,g)\) be a complete simply connected Riemannian manifold. Let \(X:M\to TM\) be a geodesic vector field (i.e., \(X\) is the unit tangent vector field of a continuous, oriented foliation of \(M\) by geodesics). Then 1) The distribution \(X^\bot\) is uniquely integrable, and the integral submanifolds of \(X^\bot\) form a \(C^0\) foliation of \(M\) by \(C^1\) leaves which are complete with respect to the metric induced by \(g\). 2) The orbits of \(X\) have no conjugate points. 3) The manifold \(M\) is homeomorphic to \(F\times\mathbb{R}\) where \(F\) is an integral submanifold of \(X^\bot\), and the integral leaves of \(X^\bot\) are simply connected and equidistant. From Theorem 1, the result about manifolds without conjugate points follows. Theorem 2. Let \((M,g)\) be a compact Riemannian manifold. Then \((M,g)\) has no conjugate points if and only if every geodesic in the universal covering \(\tilde M\) of \(M\) is an integral orbit of a continuous geodesic vector field \(X:\tilde M\to T\tilde M\) with integrable \(X^\bot\).