Let \(M\) be a complete Riemannian manifold with nonnegative sectional curvature. The authors prove: Theorem B. A singular Riemannian foliation in \(M\), without horizontal conjugate points, admits flat sections. The idea of the proof coincides with that used to prove the following well known result: if \(M\) is without conjugate points, then \(M\) is flat. The tools considered in this paper are contained in [B. Wilking, Geom. Funct. Anal. 17, No. 4, 1297–1320 (2007; Zbl 1139.53014), arXiv: math.DG/0606190].
A direct consequence of Theorem B is: Theorem A. A variationally complete action on \(M\) is hyperpolar. Variationally complete actions were introduced by R. Bott [Bull. Soc. Math. Fr. 84, 251–281 (1956; Zbl 0073.40001)] and studied by R. Bott and H. Samelson [Am. J. Math. 80, 964–1029 (1958); Correction. ibid. 83, 207–208 (1961; Zbl 0101.39702)]. The authors define the notion of “singular Riemannian foliation” in terms of P. Molino’s book “Riemannian foliations.” Progress in Mathematics, Vol. 73. Boston-Basel: Birkhäuser (1988; Zbl 0633.53001) and generalize in this context the property of an action to be variationally complete.